1arithidom

Description: Uniqueness of prime factorizations in an integral domain R . Given two equal products F and G of prime elements, F and G are equal up to a renumbering w and a multiplication by units u . See also 1arith . Chapter VII, Paragraph 3, Section 3, Proposition 2 of BourbakiCAlg2, p. 228. (Contributed by Thierry Arnoux, 27-May-2025)

	Ref	Expression
Hypotheses	1arithidom.u	$⊢ U = Unit (R)$
	1arithidom.i	$⊢ P = RPrime (R)$
	1arithidom.m	$⊢ M = {mulGrp}_{R}$
	1arithidom.t	$⊢ \underset{˙}{\cdot} = \cdot_{R}$
	1arithidom.j	$⊢ J = (0 ..^\|F\|)$
	1arithidom.r	$⊢ φ \to R \in IDomn$
	1arithidom.f	$⊢ φ \to F \in Word P$
	1arithidom.g	$⊢ φ \to G \in Word P$
	1arithidom.1	$⊢ φ \to \sum_{M} F = \sum_{M} G$
Assertion	1arithidom	$⊢ φ \to \exists w \exists u \in (U^{J}) (w : J \underset{1-1 onto}{⟶} J \land G = u {\underset{˙}{\cdot}}_{f} (F \circ w))$

Proof

Step	Hyp	Ref	Expression
1		1arithidom.u	$⊢ U = Unit (R)$
2		1arithidom.i	$⊢ P = RPrime (R)$
3		1arithidom.m	$⊢ M = {mulGrp}_{R}$
4		1arithidom.t	$⊢ \underset{˙}{\cdot} = \cdot_{R}$
5		1arithidom.j	$⊢ J = (0 ..^\|F\|)$
6		1arithidom.r	$⊢ φ \to R \in IDomn$
7		1arithidom.f	$⊢ φ \to F \in Word P$
8		1arithidom.g	$⊢ φ \to G \in Word P$
9		1arithidom.1	$⊢ φ \to \sum_{M} F = \sum_{M} G$
10	6	idomringd	$⊢ φ \to R \in Ring$
11		eqid	$⊢ 1_{R} = 1_{R}$
12	1 11	1unit	$⊢ R \in Ring \to 1_{R} \in U$
13	10 12	syl	$⊢ φ \to 1_{R} \in U$
14		oveq1	$⊢ k = 1_{R} \to k \underset{˙}{\cdot} (\sum_{M}, G) = 1_{R} \underset{˙}{\cdot} (\sum_{M}, G)$
15	14	adantl	$⊢ (φ \land k = 1_{R}) \to k \underset{˙}{\cdot} (\sum_{M}, G) = 1_{R} \underset{˙}{\cdot} (\sum_{M}, G)$
16		eqid	$⊢ {Base}_{R} = {Base}_{R}$
17	10	adantr	$⊢ (φ \land k = 1_{R}) \to R \in Ring$
18	3 16	mgpbas	$⊢ {Base}_{R} = {Base}_{M}$
19	3 11	ringidval	$⊢ 1_{R} = 0_{M}$
20		id	$⊢ R \in IDomn \to R \in IDomn$
21	20	idomcringd	$⊢ R \in IDomn \to R \in CRing$
22	3	crngmgp	$⊢ R \in CRing \to M \in CMnd$
23	21 22	syl	$⊢ R \in IDomn \to M \in CMnd$
24	6 23	syl	$⊢ φ \to M \in CMnd$
25		ovexd	$⊢ φ \to (0 ..^\|G\|) \in V$
26		eqidd	$⊢ φ \to \|G\| = \|G\|$
27	26 8	wrdfd	$⊢ φ \to G : (0 ..^\|G\|) ⟶ P$
28	6	adantr	$⊢ (φ \land p \in P) \to R \in IDomn$
29		simpr	$⊢ (φ \land p \in P) \to p \in P$
30	16 2 28 29	rprmcl	$⊢ (φ \land p \in P) \to p \in {Base}_{R}$
31	30	ex	$⊢ φ \to (p \in P \to p \in {Base}_{R})$
32	31	ssrdv	$⊢ φ \to P \subseteq {Base}_{R}$
33	27 32	fssd	$⊢ φ \to G : (0 ..^\|G\|) ⟶ {Base}_{R}$
34	13 8	wrdfsupp	$⊢ φ \to {finSupp}_{1_{R}} (G)$
35	18 19 24 25 33 34	gsumcl	$⊢ φ \to \sum_{M} G \in {Base}_{R}$
36	35	adantr	$⊢ (φ \land k = 1_{R}) \to \sum_{M} G \in {Base}_{R}$
37	16 4 11 17 36	ringlidmd	$⊢ (φ \land k = 1_{R}) \to 1_{R} \underset{˙}{\cdot} (\sum_{M}, G) = \sum_{M} G$
38	15 37	eqtrd	$⊢ (φ \land k = 1_{R}) \to k \underset{˙}{\cdot} (\sum_{M}, G) = \sum_{M} G$
39	38	eqeq2d	$⊢ (φ \land k = 1_{R}) \to (\sum_{M} F = k \underset{˙}{\cdot} (\sum_{M}, G) \leftrightarrow \sum_{M} F = \sum_{M} G)$
40	13 39 9	rspcedvd	$⊢ φ \to \exists k \in U \sum_{M} F = k \underset{˙}{\cdot} (\sum_{M}, G)$
41		oveq2	$⊢ g = G \to \sum_{M} g = \sum_{M} G$
42	41	oveq2d	$⊢ g = G \to k \underset{˙}{\cdot} (\sum_{M}, g) = k \underset{˙}{\cdot} (\sum_{M}, G)$
43	42	eqeq2d	$⊢ g = G \to (\sum_{M} F = k \underset{˙}{\cdot} (\sum_{M}, g) \leftrightarrow \sum_{M} F = k \underset{˙}{\cdot} (\sum_{M}, G))$
44	43	rexbidv	$⊢ g = G \to (\exists k \in U \sum_{M} F = k \underset{˙}{\cdot} (\sum_{M}, g) \leftrightarrow \exists k \in U \sum_{M} F = k \underset{˙}{\cdot} (\sum_{M}, G))$
45		eqeq1	$⊢ g = G \to (g = u {\underset{˙}{\cdot}}_{f} (F \circ w) \leftrightarrow G = u {\underset{˙}{\cdot}}_{f} (F \circ w))$
46	45	anbi2d	$⊢ g = G \to ((w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land g = u {\underset{˙}{\cdot}}_{f} (F \circ w)) \leftrightarrow (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land G = u {\underset{˙}{\cdot}}_{f} (F \circ w)))$
47	46	rexbidv	$⊢ g = G \to (\exists u \in (U^{(0 ..^\|F\|)}) (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land g = u {\underset{˙}{\cdot}}_{f} (F \circ w)) \leftrightarrow \exists u \in (U^{(0 ..^\|F\|)}) (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land G = u {\underset{˙}{\cdot}}_{f} (F \circ w)))$
48	47	exbidv	$⊢ g = G \to (\exists w \exists u \in (U^{(0 ..^\|F\|)}) (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land g = u {\underset{˙}{\cdot}}_{f} (F \circ w)) \leftrightarrow \exists w \exists u \in (U^{(0 ..^\|F\|)}) (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land G = u {\underset{˙}{\cdot}}_{f} (F \circ w)))$
49	44 48	imbi12d	$⊢ g = G \to ((\exists k \in U \sum_{M} F = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|F\|)}) (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land g = u {\underset{˙}{\cdot}}_{f} (F \circ w))) \leftrightarrow (\exists k \in U \sum_{M} F = k \underset{˙}{\cdot} (\sum_{M}, G) \to \exists w \exists u \in (U^{(0 ..^\|F\|)}) (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land G = u {\underset{˙}{\cdot}}_{f} (F \circ w))))$
50		oveq2	$⊢ h = \emptyset \to \sum_{M} h = \sum_{M} \emptyset$
51	50	eqeq1d	$⊢ h = \emptyset \to (\sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \leftrightarrow \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g))$
52	51	rexbidv	$⊢ h = \emptyset \to (\exists k \in U \sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \leftrightarrow \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g))$
53		fveq2	$⊢ h = \emptyset \to \|h\| = \|\emptyset\|$
54	53	oveq2d	$⊢ h = \emptyset \to (0 ..^\|h\|) = (0 ..^\|\emptyset\|)$
55	54	oveq2d	$⊢ h = \emptyset \to U^{(0 ..^\|h\|)} = U^{(0 ..^\|\emptyset\|)}$
56		eqidd	$⊢ h = \emptyset \to w = w$
57	56 54 54	f1oeq123d	$⊢ h = \emptyset \to (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \leftrightarrow w : (0 ..^\|\emptyset\|) \underset{1-1 onto}{⟶} (0 ..^\|\emptyset\|))$
58		coeq1	$⊢ h = \emptyset \to h \circ w = \emptyset \circ w$
59	58	oveq2d	$⊢ h = \emptyset \to u {\underset{˙}{\cdot}}_{f} (h \circ w) = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ w)$
60	59	eqeq2d	$⊢ h = \emptyset \to (g = u {\underset{˙}{\cdot}}_{f} (h \circ w) \leftrightarrow g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ w))$
61	57 60	anbi12d	$⊢ h = \emptyset \to ((w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w)) \leftrightarrow (w : (0 ..^\|\emptyset\|) \underset{1-1 onto}{⟶} (0 ..^\|\emptyset\|) \land g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ w)))$
62	55 61	rexeqbidv	$⊢ h = \emptyset \to (\exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w)) \leftrightarrow \exists u \in (U^{(0 ..^\|\emptyset\|)}) (w : (0 ..^\|\emptyset\|) \underset{1-1 onto}{⟶} (0 ..^\|\emptyset\|) \land g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ w)))$
63	62	exbidv	$⊢ h = \emptyset \to (\exists w \exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w)) \leftrightarrow \exists w \exists u \in (U^{(0 ..^\|\emptyset\|)}) (w : (0 ..^\|\emptyset\|) \underset{1-1 onto}{⟶} (0 ..^\|\emptyset\|) \land g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ w)))$
64	52 63	imbi12d	$⊢ h = \emptyset \to ((\exists k \in U \sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w))) \leftrightarrow (\exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|\emptyset\|)}) (w : (0 ..^\|\emptyset\|) \underset{1-1 onto}{⟶} (0 ..^\|\emptyset\|) \land g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ w))))$
65	64	ralbidv	$⊢ h = \emptyset \to (\forall g \in Word P (\exists k \in U \sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w))) \leftrightarrow \forall g \in Word P (\exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|\emptyset\|)}) (w : (0 ..^\|\emptyset\|) \underset{1-1 onto}{⟶} (0 ..^\|\emptyset\|) \land g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ w))))$
66	65	imbi2d	$⊢ h = \emptyset \to ((R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w)))) \leftrightarrow (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|\emptyset\|)}) (w : (0 ..^\|\emptyset\|) \underset{1-1 onto}{⟶} (0 ..^\|\emptyset\|) \land g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ w)))))$
67		oveq2	$⊢ h = f \to \sum_{M} h = \sum_{M} f$
68	67	eqeq1d	$⊢ h = f \to (\sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \leftrightarrow \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g))$
69	68	rexbidv	$⊢ h = f \to (\exists k \in U \sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \leftrightarrow \exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g))$
70		fveq2	$⊢ h = f \to \|h\| = \|f\|$
71	70	oveq2d	$⊢ h = f \to (0 ..^\|h\|) = (0 ..^\|f\|)$
72	71	oveq2d	$⊢ h = f \to U^{(0 ..^\|h\|)} = U^{(0 ..^\|f\|)}$
73		eqidd	$⊢ h = f \to w = w$
74	73 71 71	f1oeq123d	$⊢ h = f \to (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \leftrightarrow w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|))$
75		coeq1	$⊢ h = f \to h \circ w = f \circ w$
76	75	oveq2d	$⊢ h = f \to u {\underset{˙}{\cdot}}_{f} (h \circ w) = u {\underset{˙}{\cdot}}_{f} (f \circ w)$
77	76	eqeq2d	$⊢ h = f \to (g = u {\underset{˙}{\cdot}}_{f} (h \circ w) \leftrightarrow g = u {\underset{˙}{\cdot}}_{f} (f \circ w))$
78	74 77	anbi12d	$⊢ h = f \to ((w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w)) \leftrightarrow (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w)))$
79	72 78	rexeqbidv	$⊢ h = f \to (\exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w)) \leftrightarrow \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w)))$
80	79	exbidv	$⊢ h = f \to (\exists w \exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w)) \leftrightarrow \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w)))$
81	69 80	imbi12d	$⊢ h = f \to ((\exists k \in U \sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w))) \leftrightarrow (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))$
82	81	ralbidv	$⊢ h = f \to (\forall g \in Word P (\exists k \in U \sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w))) \leftrightarrow \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))$
83	82	imbi2d	$⊢ h = f \to ((R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w)))) \leftrightarrow (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w)))))$
84		oveq2	$⊢ h = f ++ ⟨“ p ”⟩ \to \sum_{M} h = \sum_{M} (f ++ ⟨“ p ”⟩)$
85	84	eqeq1d	$⊢ h = f ++ ⟨“ p ”⟩ \to (\sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \leftrightarrow \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, g))$
86	85	rexbidv	$⊢ h = f ++ ⟨“ p ”⟩ \to (\exists k \in U \sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \leftrightarrow \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, g))$
87		fveq2	$⊢ h = f ++ ⟨“ p ”⟩ \to \|h\| = \|f ++ ⟨“ p ”⟩\|$
88	87	oveq2d	$⊢ h = f ++ ⟨“ p ”⟩ \to (0 ..^\|h\|) = (0 ..^\|f ++ ⟨“ p ”⟩\|)$
89	88	oveq2d	$⊢ h = f ++ ⟨“ p ”⟩ \to U^{(0 ..^\|h\|)} = U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}$
90		eqidd	$⊢ h = f ++ ⟨“ p ”⟩ \to w = w$
91	90 88 88	f1oeq123d	$⊢ h = f ++ ⟨“ p ”⟩ \to (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \leftrightarrow w : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|))$
92		coeq1	$⊢ h = f ++ ⟨“ p ”⟩ \to h \circ w = (f ++ ⟨“ p ”⟩) \circ w$
93	92	oveq2d	$⊢ h = f ++ ⟨“ p ”⟩ \to u {\underset{˙}{\cdot}}_{f} (h \circ w) = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w)$
94	93	eqeq2d	$⊢ h = f ++ ⟨“ p ”⟩ \to (g = u {\underset{˙}{\cdot}}_{f} (h \circ w) \leftrightarrow g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w))$
95	91 94	anbi12d	$⊢ h = f ++ ⟨“ p ”⟩ \to ((w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w)) \leftrightarrow (w : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w)))$
96	89 95	rexeqbidv	$⊢ h = f ++ ⟨“ p ”⟩ \to (\exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w)) \leftrightarrow \exists u \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (w : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w)))$
97	96	exbidv	$⊢ h = f ++ ⟨“ p ”⟩ \to (\exists w \exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w)) \leftrightarrow \exists w \exists u \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (w : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w)))$
98	86 97	imbi12d	$⊢ h = f ++ ⟨“ p ”⟩ \to ((\exists k \in U \sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w))) \leftrightarrow (\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (w : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w))))$
99	98	ralbidv	$⊢ h = f ++ ⟨“ p ”⟩ \to (\forall g \in Word P (\exists k \in U \sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w))) \leftrightarrow \forall g \in Word P (\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (w : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w))))$
100	99	imbi2d	$⊢ h = f ++ ⟨“ p ”⟩ \to ((R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w)))) \leftrightarrow (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (w : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w)))))$
101		oveq2	$⊢ h = F \to \sum_{M} h = \sum_{M} F$
102	101	eqeq1d	$⊢ h = F \to (\sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \leftrightarrow \sum_{M} F = k \underset{˙}{\cdot} (\sum_{M}, g))$
103	102	rexbidv	$⊢ h = F \to (\exists k \in U \sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \leftrightarrow \exists k \in U \sum_{M} F = k \underset{˙}{\cdot} (\sum_{M}, g))$
104		fveq2	$⊢ h = F \to \|h\| = \|F\|$
105	104	oveq2d	$⊢ h = F \to (0 ..^\|h\|) = (0 ..^\|F\|)$
106	105	oveq2d	$⊢ h = F \to U^{(0 ..^\|h\|)} = U^{(0 ..^\|F\|)}$
107		eqidd	$⊢ h = F \to w = w$
108	107 105 105	f1oeq123d	$⊢ h = F \to (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \leftrightarrow w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|))$
109		coeq1	$⊢ h = F \to h \circ w = F \circ w$
110	109	oveq2d	$⊢ h = F \to u {\underset{˙}{\cdot}}_{f} (h \circ w) = u {\underset{˙}{\cdot}}_{f} (F \circ w)$
111	110	eqeq2d	$⊢ h = F \to (g = u {\underset{˙}{\cdot}}_{f} (h \circ w) \leftrightarrow g = u {\underset{˙}{\cdot}}_{f} (F \circ w))$
112	108 111	anbi12d	$⊢ h = F \to ((w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w)) \leftrightarrow (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land g = u {\underset{˙}{\cdot}}_{f} (F \circ w)))$
113	106 112	rexeqbidv	$⊢ h = F \to (\exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w)) \leftrightarrow \exists u \in (U^{(0 ..^\|F\|)}) (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land g = u {\underset{˙}{\cdot}}_{f} (F \circ w)))$
114	113	exbidv	$⊢ h = F \to (\exists w \exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w)) \leftrightarrow \exists w \exists u \in (U^{(0 ..^\|F\|)}) (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land g = u {\underset{˙}{\cdot}}_{f} (F \circ w)))$
115	103 114	imbi12d	$⊢ h = F \to ((\exists k \in U \sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w))) \leftrightarrow (\exists k \in U \sum_{M} F = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|F\|)}) (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land g = u {\underset{˙}{\cdot}}_{f} (F \circ w))))$
116	115	ralbidv	$⊢ h = F \to (\forall g \in Word P (\exists k \in U \sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w))) \leftrightarrow \forall g \in Word P (\exists k \in U \sum_{M} F = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|F\|)}) (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land g = u {\underset{˙}{\cdot}}_{f} (F \circ w))))$
117	116	imbi2d	$⊢ h = F \to ((R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} h = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|h\|)}) (w : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land g = u {\underset{˙}{\cdot}}_{f} (h \circ w)))) \leftrightarrow (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} F = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|F\|)}) (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land g = u {\underset{˙}{\cdot}}_{f} (F \circ w)))))$
118		0ex	$⊢ \emptyset \in V$
119	118	a1i	$⊢ ((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to \emptyset \in V$
120	118	snid	$⊢ \emptyset \in \{\emptyset\}$
121	1	fvexi	$⊢ U \in V$
122		mapdm0	$⊢ U \in V \to U^{\emptyset} = \{\emptyset\}$
123	121 122	ax-mp	$⊢ U^{\emptyset} = \{\emptyset\}$
124	120 123	eleqtrri	$⊢ \emptyset \in (U^{\emptyset})$
125	124	a1i	$⊢ ((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to \emptyset \in (U^{\emptyset})$
126		f1o0	$⊢ \emptyset : \emptyset \underset{1-1 onto}{⟶} \emptyset$
127	126	biantrur	$⊢ g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ \emptyset) \leftrightarrow (\emptyset : \emptyset \underset{1-1 onto}{⟶} \emptyset \land g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ \emptyset))$
128		co02	$⊢ \emptyset \circ \emptyset = \emptyset$
129	128	oveq2i	$⊢ u {\underset{˙}{\cdot}}_{f} (\emptyset \circ \emptyset) = u {\underset{˙}{\cdot}}_{f} \emptyset$
130		of0r	$⊢ u {\underset{˙}{\cdot}}_{f} \emptyset = \emptyset$
131	129 130	eqtri	$⊢ u {\underset{˙}{\cdot}}_{f} (\emptyset \circ \emptyset) = \emptyset$
132	131	eqeq2i	$⊢ g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ \emptyset) \leftrightarrow g = \emptyset$
133	127 132	bitr3i	$⊢ (\emptyset : \emptyset \underset{1-1 onto}{⟶} \emptyset \land g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ \emptyset)) \leftrightarrow g = \emptyset$
134	133	a1i	$⊢ (((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \land u = \emptyset) \to ((\emptyset : \emptyset \underset{1-1 onto}{⟶} \emptyset \land g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ \emptyset)) \leftrightarrow g = \emptyset)$
135		simpl	$⊢ (R \in IDomn \land g \in Word P) \to R \in IDomn$
136	135	idomcringd	$⊢ (R \in IDomn \land g \in Word P) \to R \in CRing$
137	136	ad2antrr	$⊢ (((R \in IDomn \land g \in Word P) \land k \in U) \land \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to R \in CRing$
138		simplr	$⊢ (((R \in IDomn \land g \in Word P) \land k \in U) \land \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to k \in U$
139	16 1	unitcl	$⊢ k \in U \to k \in {Base}_{R}$
140	138 139	syl	$⊢ (((R \in IDomn \land g \in Word P) \land k \in U) \land \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to k \in {Base}_{R}$
141	137 22	syl	$⊢ (((R \in IDomn \land g \in Word P) \land k \in U) \land \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to M \in CMnd$
142		ovexd	$⊢ (((R \in IDomn \land g \in Word P) \land k \in U) \land \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to (0 ..^\|g\|) \in V$
143		eqidd	$⊢ (((R \in IDomn \land g \in Word P) \land k \in U) \land \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to \|g\| = \|g\|$
144		simpl	$⊢ (R \in IDomn \land p \in P) \to R \in IDomn$
145		simpr	$⊢ (R \in IDomn \land p \in P) \to p \in P$
146	16 2 144 145	rprmcl	$⊢ (R \in IDomn \land p \in P) \to p \in {Base}_{R}$
147	146	ex	$⊢ R \in IDomn \to (p \in P \to p \in {Base}_{R})$
148	147	ssrdv	$⊢ R \in IDomn \to P \subseteq {Base}_{R}$
149		sswrd	$⊢ P \subseteq {Base}_{R} \to Word P \subseteq Word {Base}_{R}$
150	148 149	syl	$⊢ R \in IDomn \to Word P \subseteq Word {Base}_{R}$
151	150	sselda	$⊢ (R \in IDomn \land g \in Word P) \to g \in Word {Base}_{R}$
152	151	ad2antrr	$⊢ (((R \in IDomn \land g \in Word P) \land k \in U) \land \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to g \in Word {Base}_{R}$
153	143 152	wrdfd	$⊢ (((R \in IDomn \land g \in Word P) \land k \in U) \land \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to g : (0 ..^\|g\|) ⟶ {Base}_{R}$
154	135	idomringd	$⊢ (R \in IDomn \land g \in Word P) \to R \in Ring$
155	154 12	syl	$⊢ (R \in IDomn \land g \in Word P) \to 1_{R} \in U$
156	155	ad2antrr	$⊢ (((R \in IDomn \land g \in Word P) \land k \in U) \land \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to 1_{R} \in U$
157	156 152	wrdfsupp	$⊢ (((R \in IDomn \land g \in Word P) \land k \in U) \land \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to {finSupp}_{1_{R}} (g)$
158	18 19 141 142 153 157	gsumcl	$⊢ (((R \in IDomn \land g \in Word P) \land k \in U) \land \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to \sum_{M} g \in {Base}_{R}$
159		simpr	$⊢ (((R \in IDomn \land g \in Word P) \land k \in U) \land \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)$
160	19	gsum0	$⊢ \sum_{M} \emptyset = 1_{R}$
161	160 156	eqeltrid	$⊢ (((R \in IDomn \land g \in Word P) \land k \in U) \land \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to \sum_{M} \emptyset \in U$
162	159 161	eqeltrrd	$⊢ (((R \in IDomn \land g \in Word P) \land k \in U) \land \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to k \underset{˙}{\cdot} (\sum_{M}, g) \in U$
163	1 4 16	unitmulclb	$⊢ (R \in CRing \land k \in {Base}_{R} \land \sum_{M} g \in {Base}_{R}) \to (k \underset{˙}{\cdot} (\sum_{M}, g) \in U \leftrightarrow (k \in U \land \sum_{M} g \in U))$
164	163	biimpa	$⊢ ((R \in CRing \land k \in {Base}_{R} \land \sum_{M} g \in {Base}_{R}) \land k \underset{˙}{\cdot} (\sum_{M}, g) \in U) \to (k \in U \land \sum_{M} g \in U)$
165	137 140 158 162 164	syl31anc	$⊢ (((R \in IDomn \land g \in Word P) \land k \in U) \land \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to (k \in U \land \sum_{M} g \in U)$
166	165	simprd	$⊢ (((R \in IDomn \land g \in Word P) \land k \in U) \land \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to \sum_{M} g \in U$
167	166	r19.29an	$⊢ ((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to \sum_{M} g \in U$
168	16 1 3 136 151	unitprodclb	$⊢ (R \in IDomn \land g \in Word P) \to (\sum_{M} g \in U \leftrightarrow ran g \subseteq U)$
169	168	adantr	$⊢ ((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to (\sum_{M} g \in U \leftrightarrow ran g \subseteq U)$
170	167 169	mpbid	$⊢ ((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to ran g \subseteq U$
171	170	adantr	$⊢ (((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \land g \neq \emptyset) \to ran g \subseteq U$
172		eqidd	$⊢ (R \in IDomn \land g \in Word P) \to \|g\| = \|g\|$
173		simpr	$⊢ (R \in IDomn \land g \in Word P) \to g \in Word P$
174	172 173	wrdfd	$⊢ (R \in IDomn \land g \in Word P) \to g : (0 ..^\|g\|) ⟶ P$
175	174	freld	$⊢ (R \in IDomn \land g \in Word P) \to Rel g$
176	175	ad2antrr	$⊢ (((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \land g \neq \emptyset) \to Rel g$
177		simpr	$⊢ (((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \land g \neq \emptyset) \to g \neq \emptyset$
178		relrn0	$⊢ Rel g \to (g = \emptyset \leftrightarrow ran g = \emptyset)$
179	178	necon3bid	$⊢ Rel g \to (g \neq \emptyset \leftrightarrow ran g \neq \emptyset)$
180	179	biimpa	$⊢ (Rel g \land g \neq \emptyset) \to ran g \neq \emptyset$
181	176 177 180	syl2anc	$⊢ (((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \land g \neq \emptyset) \to ran g \neq \emptyset$
182		n0	$⊢ ran g \neq \emptyset \leftrightarrow \exists i i \in ran g$
183	181 182	sylib	$⊢ (((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \land g \neq \emptyset) \to \exists i i \in ran g$
184		simpr	$⊢ ((((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \land g \neq \emptyset) \land i \in ran g) \to i \in ran g$
185	135	ad3antrrr	$⊢ ((((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \land g \neq \emptyset) \land i \in ran g) \to R \in IDomn$
186	174	frnd	$⊢ (R \in IDomn \land g \in Word P) \to ran g \subseteq P$
187	186	ad2antrr	$⊢ (((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \land g \neq \emptyset) \to ran g \subseteq P$
188	187	sselda	$⊢ ((((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \land g \neq \emptyset) \land i \in ran g) \to i \in P$
189	2 1 185 188	rprmnunit	$⊢ ((((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \land g \neq \emptyset) \land i \in ran g) \to \neg i \in U$
190		nelss	$⊢ (i \in ran g \land \neg i \in U) \to \neg ran g \subseteq U$
191	184 189 190	syl2anc	$⊢ ((((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \land g \neq \emptyset) \land i \in ran g) \to \neg ran g \subseteq U$
192	183 191	exlimddv	$⊢ (((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \land g \neq \emptyset) \to \neg ran g \subseteq U$
193	171 192	pm2.65da	$⊢ ((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to \neg g \neq \emptyset$
194		nne	$⊢ \neg g \neq \emptyset \leftrightarrow g = \emptyset$
195	193 194	sylib	$⊢ ((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to g = \emptyset$
196	125 134 195	rspcedvd	$⊢ ((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to \exists u \in (U^{\emptyset}) (\emptyset : \emptyset \underset{1-1 onto}{⟶} \emptyset \land g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ \emptyset))$
197		hash0	$⊢ \|\emptyset\| = 0$
198	197	oveq2i	$⊢ (0 ..^\|\emptyset\|) = (0 ..^0)$
199		fzo0	$⊢ (0 ..^0) = \emptyset$
200	198 199	eqtri	$⊢ (0 ..^\|\emptyset\|) = \emptyset$
201	200	oveq2i	$⊢ U^{(0 ..^\|\emptyset\|)} = U^{\emptyset}$
202	201	a1i	$⊢ w = \emptyset \to U^{(0 ..^\|\emptyset\|)} = U^{\emptyset}$
203		id	$⊢ w = \emptyset \to w = \emptyset$
204	200	a1i	$⊢ w = \emptyset \to (0 ..^\|\emptyset\|) = \emptyset$
205	203 204 204	f1oeq123d	$⊢ w = \emptyset \to (w : (0 ..^\|\emptyset\|) \underset{1-1 onto}{⟶} (0 ..^\|\emptyset\|) \leftrightarrow \emptyset : \emptyset \underset{1-1 onto}{⟶} \emptyset)$
206		coeq2	$⊢ w = \emptyset \to \emptyset \circ w = \emptyset \circ \emptyset$
207	206	oveq2d	$⊢ w = \emptyset \to u {\underset{˙}{\cdot}}_{f} (\emptyset \circ w) = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ \emptyset)$
208	207	eqeq2d	$⊢ w = \emptyset \to (g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ w) \leftrightarrow g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ \emptyset))$
209	205 208	anbi12d	$⊢ w = \emptyset \to ((w : (0 ..^\|\emptyset\|) \underset{1-1 onto}{⟶} (0 ..^\|\emptyset\|) \land g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ w)) \leftrightarrow (\emptyset : \emptyset \underset{1-1 onto}{⟶} \emptyset \land g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ \emptyset)))$
210	202 209	rexeqbidv	$⊢ w = \emptyset \to (\exists u \in (U^{(0 ..^\|\emptyset\|)}) (w : (0 ..^\|\emptyset\|) \underset{1-1 onto}{⟶} (0 ..^\|\emptyset\|) \land g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ w)) \leftrightarrow \exists u \in (U^{\emptyset}) (\emptyset : \emptyset \underset{1-1 onto}{⟶} \emptyset \land g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ \emptyset)))$
211	119 196 210	spcedv	$⊢ ((R \in IDomn \land g \in Word P) \land \exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g)) \to \exists w \exists u \in (U^{(0 ..^\|\emptyset\|)}) (w : (0 ..^\|\emptyset\|) \underset{1-1 onto}{⟶} (0 ..^\|\emptyset\|) \land g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ w))$
212	211	ex	$⊢ (R \in IDomn \land g \in Word P) \to (\exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|\emptyset\|)}) (w : (0 ..^\|\emptyset\|) \underset{1-1 onto}{⟶} (0 ..^\|\emptyset\|) \land g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ w)))$
213	212	ralrimiva	$⊢ R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} \emptyset = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|\emptyset\|)}) (w : (0 ..^\|\emptyset\|) \underset{1-1 onto}{⟶} (0 ..^\|\emptyset\|) \land g = u {\underset{˙}{\cdot}}_{f} (\emptyset \circ w)))$
214		eqid	$⊢ (0 ..^\|h\|) = (0 ..^\|h\|)$
215		simp-4r	$⊢ (((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \to j \in (0 ..^\|h\|)$
216		simp-4r	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \to h \in Word P$
217	216	ad2antrr	$⊢ (((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \to h \in Word P$
218		eqid	$⊢ (h \circ r) prefix (\|h\| - 1) = (h \circ r) prefix (\|h\| - 1)$
219	214 215 217 218	wrdpmtrlast	$⊢ (((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \to \exists r (r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩)$
220		eqid	$⊢ (0 ..^\|f\|) = (0 ..^\|f\|)$
221		simp-5r	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \to R \in IDomn$
222	221	ad6antr	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to R \in IDomn$
223		simp-5l	$⊢ (((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \to f \in Word P$
224	223	ad8antr	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to f \in Word P$
225		eqidd	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \sum_{M} f = \sum_{M} f$
226		simp-7r	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \to p \in P$
227	226	ad6antr	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to p \in P$
228		simplr	$⊢ (((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \to (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))$
229	228	ad10antr	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))$
230	222 229	mpd	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w)))$
231	217	ad4antr	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to h \in Word P$
232		simp-9r	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)$
233	215	ad4antr	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to j \in (0 ..^\|h\|)$
234		simpr	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \to p ∥_{r} (R) h (j)$
235	234	ad6antr	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to p ∥_{r} (R) h (j)$
236		simp-6r	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to t \in U$
237		simp-5r	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to t \underset{˙}{\cdot} p = h (j)$
238		simp-4r	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)$
239		simpllr	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩$
240		simplr	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to m \in U$
241		simpr	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)$
242	1 2 3 4 220 222 224 224 225 227 230 231 232 233 235 236 237 238 239 240 241	1arithidomlem1	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \exists c \exists d \in (U^{(0 ..^\|f\|)}) (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))$
243		ovexd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to c ++ ⟨“ \|f\| ”⟩ \in V$
244		vex	$⊢ r \in V$
245	244	cnvex	$⊢ r^{-1} \in V$
246	245	a1i	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to r^{-1} \in V$
247	243 246	coexd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to (c ++ ⟨“ \|f\| ”⟩) \circ r^{-1} \in V$
248		oveq1	$⊢ s = (d ++ ⟨“ t ”⟩) \circ r^{-1} \to s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ ((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1})) = ((d ++ ⟨“ t ”⟩) \circ r^{-1}) {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ ((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1}))$
249	248	eqeq2d	$⊢ s = (d ++ ⟨“ t ”⟩) \circ r^{-1} \to (h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ ((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1})) \leftrightarrow h = ((d ++ ⟨“ t ”⟩) \circ r^{-1}) {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ ((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1})))$
250	249	anbi2d	$⊢ s = (d ++ ⟨“ t ”⟩) \circ r^{-1} \to ((((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1}) : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ ((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1}))) \leftrightarrow (((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1}) : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = ((d ++ ⟨“ t ”⟩) \circ r^{-1}) {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ ((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1}))))$
251	121	a1i	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to U \in V$
252		ovexd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to (0 ..^\|f ++ ⟨“ p ”⟩\|) \in V$
253		simplr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to d \in (U^{(0 ..^\|f\|)})$
254		elmapi	$⊢ d \in (U^{(0 ..^\|f\|)}) \to d : (0 ..^\|f\|) ⟶ U$
255	253 254	syl	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to d : (0 ..^\|f\|) ⟶ U$
256		iswrdi	$⊢ d : (0 ..^\|f\|) ⟶ U \to d \in Word U$
257	255 256	syl	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to d \in Word U$
258		ccatws1len	$⊢ d \in Word U \to \|d ++ ⟨“ t ”⟩\| = \|d\| + 1$
259	257 258	syl	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|d ++ ⟨“ t ”⟩\| = \|d\| + 1$
260		elmapfn	$⊢ d \in (U^{(0 ..^\|f\|)}) \to d Fn (0 ..^\|f\|)$
261		hashfn	$⊢ d Fn (0 ..^\|f\|) \to \|d\| = \|(0 ..^\|f\|)\|$
262	253 260 261	3syl	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|d\| = \|(0 ..^\|f\|)\|$
263	223	ad10antr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to f \in Word P$
264		lencl	$⊢ f \in Word P \to \|f\| \in ℕ_{0}$
265	263 264	syl	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|f\| \in ℕ_{0}$
266		hashfzo0	$⊢ \|f\| \in ℕ_{0} \to \|(0 ..^\|f\|)\| = \|f\|$
267	265 266	syl	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|(0 ..^\|f\|)\| = \|f\|$
268	262 267	eqtrd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|d\| = \|f\|$
269	268	oveq1d	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|d\| + 1 = \|f\| + 1$
270		simprr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c)$
271	270	dmeqd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to dom ((h \circ r) prefix (\|h\| - 1)) = dom (d {\underset{˙}{\cdot}}_{f} (f \circ c))$
272		f1of	$⊢ r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \to r : (0 ..^\|h\|) ⟶ (0 ..^\|h\|)$
273		iswrdi	$⊢ r : (0 ..^\|h\|) ⟶ (0 ..^\|h\|) \to r \in Word (0 ..^\|h\|)$
274	238 272 273	3syl	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to r \in Word (0 ..^\|h\|)$
275		eqidd	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \to \|h\| = \|h\|$
276	275 216	wrdfd	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \to h : (0 ..^\|h\|) ⟶ P$
277	276	ad6antr	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to h : (0 ..^\|h\|) ⟶ P$
278		wrdco	$⊢ (r \in Word (0 ..^\|h\|) \land h : (0 ..^\|h\|) ⟶ P) \to h \circ r \in Word P$
279	274 277 278	syl2anc	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to h \circ r \in Word P$
280	279	ad2antrr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to h \circ r \in Word P$
281		elfzo0	$⊢ j \in (0 ..^\|h\|) \leftrightarrow (j \in ℕ_{0} \land \|h\| \in ℕ \land j < \|h\|)$
282	281	simp2bi	$⊢ j \in (0 ..^\|h\|) \to \|h\| \in ℕ$
283		nnm1nn0	$⊢ \|h\| \in ℕ \to \|h\| - 1 \in ℕ_{0}$
284	233 282 283	3syl	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \|h\| - 1 \in ℕ_{0}$
285		lenco	$⊢ (r \in Word (0 ..^\|h\|) \land h : (0 ..^\|h\|) ⟶ P) \to \|h \circ r\| = \|r\|$
286	274 277 285	syl2anc	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \|h \circ r\| = \|r\|$
287		lencl	$⊢ r \in Word (0 ..^\|h\|) \to \|r\| \in ℕ_{0}$
288	274 287	syl	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \|r\| \in ℕ_{0}$
289	286 288	eqeltrd	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \|h \circ r\| \in ℕ_{0}$
290		lencl	$⊢ h \in Word P \to \|h\| \in ℕ_{0}$
291	231 290	syl	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \|h\| \in ℕ_{0}$
292	291	nn0red	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \|h\| \in ℝ$
293	292	lem1d	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \|h\| - 1 \leq \|h\|$
294	238 272	syl	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to r : (0 ..^\|h\|) ⟶ (0 ..^\|h\|)$
295		ffn	$⊢ r : (0 ..^\|h\|) ⟶ (0 ..^\|h\|) \to r Fn (0 ..^\|h\|)$
296		hashfn	$⊢ r Fn (0 ..^\|h\|) \to \|r\| = \|(0 ..^\|h\|)\|$
297	294 295 296	3syl	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \|r\| = \|(0 ..^\|h\|)\|$
298		hashfzo0	$⊢ \|h\| \in ℕ_{0} \to \|(0 ..^\|h\|)\| = \|h\|$
299	231 290 298	3syl	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \|(0 ..^\|h\|)\| = \|h\|$
300	286 297 299	3eqtrrd	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \|h\| = \|h \circ r\|$
301	293 300	breqtrd	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \|h\| - 1 \leq \|h \circ r\|$
302		elfz2nn0	$⊢ \|h\| - 1 \in (0 \dots \|h \circ r\|) \leftrightarrow (\|h\| - 1 \in ℕ_{0} \land \|h \circ r\| \in ℕ_{0} \land \|h\| - 1 \leq \|h \circ r\|)$
303	284 289 301 302	syl3anbrc	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \|h\| - 1 \in (0 \dots \|h \circ r\|)$
304	303	ad2antrr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|h\| - 1 \in (0 \dots \|h \circ r\|)$
305		pfxfn	$⊢ (h \circ r \in Word P \land \|h\| - 1 \in (0 \dots \|h \circ r\|)) \to ((h \circ r) prefix (\|h\| - 1)) Fn (0 ..^\|h\| - 1)$
306	280 304 305	syl2anc	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to ((h \circ r) prefix (\|h\| - 1)) Fn (0 ..^\|h\| - 1)$
307	306	fndmd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to dom ((h \circ r) prefix (\|h\| - 1)) = (0 ..^\|h\| - 1)$
308	222	idomringd	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to R \in Ring$
309	308	ad3antrrr	$⊢ ((((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \land (x \in U \land y \in P)) \to R \in Ring$
310		simprl	$⊢ ((((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \land (x \in U \land y \in P)) \to x \in U$
311	16 1	unitcl	$⊢ x \in U \to x \in {Base}_{R}$
312	310 311	syl	$⊢ ((((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \land (x \in U \land y \in P)) \to x \in {Base}_{R}$
313	222	ad3antrrr	$⊢ ((((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \land (x \in U \land y \in P)) \to R \in IDomn$
314		simprr	$⊢ ((((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \land (x \in U \land y \in P)) \to y \in P$
315	16 2 313 314	rprmcl	$⊢ ((((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \land (x \in U \land y \in P)) \to y \in {Base}_{R}$
316	16 4 309 312 315	ringcld	$⊢ ((((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \land (x \in U \land y \in P)) \to x \underset{˙}{\cdot} y \in {Base}_{R}$
317		eqidd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|f\| = \|f\|$
318	317 263	wrdfd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to f : (0 ..^\|f\|) ⟶ P$
319		simprl	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|)$
320		f1of	$⊢ c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \to c : (0 ..^\|f\|) ⟶ (0 ..^\|f\|)$
321	319 320	syl	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to c : (0 ..^\|f\|) ⟶ (0 ..^\|f\|)$
322	318 321	fcod	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to (f \circ c) : (0 ..^\|f\|) ⟶ P$
323		ovexd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to (0 ..^\|f\|) \in V$
324		inidm	$⊢ (0 ..^\|f\|) \cap (0 ..^\|f\|) = (0 ..^\|f\|)$
325	316 255 322 323 323 324	off	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to (d {\underset{˙}{\cdot}}_{f} (f \circ c)) : (0 ..^\|f\|) ⟶ {Base}_{R}$
326	325	fdmd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to dom (d {\underset{˙}{\cdot}}_{f} (f \circ c)) = (0 ..^\|f\|)$
327	271 307 326	3eqtr3d	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to (0 ..^\|h\| - 1) = (0 ..^\|f\|)$
328	284	ad2antrr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|h\| - 1 \in ℕ_{0}$
329	328 265	fzo0opth	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to ((0 ..^\|h\| - 1) = (0 ..^\|f\|) \leftrightarrow \|h\| - 1 = \|f\|)$
330	327 329	mpbid	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|h\| - 1 = \|f\|$
331	330	oveq1d	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|h\| - 1 + 1 = \|f\| + 1$
332	282	ad10antlr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|h\| \in ℕ$
333	332	nncnd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|h\| \in ℂ$
334		npcan1	$⊢ \|h\| \in ℂ \to \|h\| - 1 + 1 = \|h\|$
335	333 334	syl	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|h\| - 1 + 1 = \|h\|$
336	331 335	eqtr3d	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|f\| + 1 = \|h\|$
337	259 269 336	3eqtrd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|d ++ ⟨“ t ”⟩\| = \|h\|$
338	337	oveq2d	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to (0 ..^\|d ++ ⟨“ t ”⟩\|) = (0 ..^\|h\|)$
339		eqidd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|d ++ ⟨“ t ”⟩\| = \|d ++ ⟨“ t ”⟩\|$
340	236	ad2antrr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to t \in U$
341		ccatws1cl	$⊢ (d \in Word U \land t \in U) \to d ++ ⟨“ t ”⟩ \in Word U$
342	257 340 341	syl2anc	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to d ++ ⟨“ t ”⟩ \in Word U$
343	339 342	wrdfd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to (d ++ ⟨“ t ”⟩) : (0 ..^\|d ++ ⟨“ t ”⟩\|) ⟶ U$
344	338 343	feq2dd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to (d ++ ⟨“ t ”⟩) : (0 ..^\|h\|) ⟶ U$
345		ccatws1len	$⊢ f \in Word P \to \|f ++ ⟨“ p ”⟩\| = \|f\| + 1$
346	263 345	syl	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|f ++ ⟨“ p ”⟩\| = \|f\| + 1$
347	346 336	eqtrd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \|f ++ ⟨“ p ”⟩\| = \|h\|$
348	347	oveq2d	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to (0 ..^\|f ++ ⟨“ p ”⟩\|) = (0 ..^\|h\|)$
349	348	eqcomd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to (0 ..^\|h\|) = (0 ..^\|f ++ ⟨“ p ”⟩\|)$
350	238	ad2antrr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)$
351		f1ocnv	$⊢ r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \to r^{-1} : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)$
352		f1of	$⊢ r^{-1} : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \to r^{-1} : (0 ..^\|h\|) ⟶ (0 ..^\|h\|)$
353	350 351 352	3syl	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to r^{-1} : (0 ..^\|h\|) ⟶ (0 ..^\|h\|)$
354	349 353	feq2dd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to r^{-1} : (0 ..^\|f ++ ⟨“ p ”⟩\|) ⟶ (0 ..^\|h\|)$
355	344 354	fcod	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to ((d ++ ⟨“ t ”⟩) \circ r^{-1}) : (0 ..^\|f ++ ⟨“ p ”⟩\|) ⟶ U$
356	251 252 355	elmapdd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to (d ++ ⟨“ t ”⟩) \circ r^{-1} \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)})$
357	222	ad2antrr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to R \in IDomn$
358		eqidd	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \sum_{M} f = \sum_{M} f$
359	227	ad2antrr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to p \in P$
360	230	ad2antrr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w)))$
361	231	ad2antrr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to h \in Word P$
362	232	ad2antrr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)$
363	233	ad2antrr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to j \in (0 ..^\|h\|)$
364	235	ad2antrr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to p ∥_{r} (R) h (j)$
365	237	ad2antrr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to t \underset{˙}{\cdot} p = h (j)$
366		simp-5r	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩$
367		simp-4r	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to m \in U$
368		simpllr	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)$
369	1 2 3 4 220 357 263 263 358 359 360 361 362 363 364 340 365 350 366 367 368 253 319 270	1arithidomlem2	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to (((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1}) : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = ((d ++ ⟨“ t ”⟩) \circ r^{-1}) {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ ((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1})))$
370	250 356 369	rspcedvdw	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1}) : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ ((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1})))$
371		f1oeq1	$⊢ v = (c ++ ⟨“ \|f\| ”⟩) \circ r^{-1} \to (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \leftrightarrow ((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1}) : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|))$
372		coeq2	$⊢ v = (c ++ ⟨“ \|f\| ”⟩) \circ r^{-1} \to (f ++ ⟨“ p ”⟩) \circ v = (f ++ ⟨“ p ”⟩) \circ ((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1})$
373	372	oveq2d	$⊢ v = (c ++ ⟨“ \|f\| ”⟩) \circ r^{-1} \to s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v) = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ ((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1}))$
374	373	eqeq2d	$⊢ v = (c ++ ⟨“ \|f\| ”⟩) \circ r^{-1} \to (h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v) \leftrightarrow h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ ((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1})))$
375	371 374	anbi12d	$⊢ v = (c ++ ⟨“ \|f\| ”⟩) \circ r^{-1} \to ((v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)) \leftrightarrow (((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1}) : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ ((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1}))))$
376	375	rexbidv	$⊢ v = (c ++ ⟨“ \|f\| ”⟩) \circ r^{-1} \to (\exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)) \leftrightarrow \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1}) : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ ((c ++ ⟨“ \|f\| ”⟩) \circ r^{-1}))))$
377	247 370 376	spcedv	$⊢ (((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land d \in (U^{(0 ..^\|f\|)})) \land (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))$
378	377	r19.29an	$⊢ ((((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \land \exists d \in (U^{(0 ..^\|f\|)}) (c : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land (h \circ r) prefix (\|h\| - 1) = d {\underset{˙}{\cdot}}_{f} (f \circ c))) \to \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))$
379	242 378	exlimddv	$⊢ (((((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))$
380		simp-7r	$⊢ (((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \to \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)$
381		oveq1	$⊢ k = m \to k \underset{˙}{\cdot} (\sum_{M}, h) = m \underset{˙}{\cdot} (\sum_{M}, h)$
382	381	eqeq2d	$⊢ k = m \to (\sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h) \leftrightarrow \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h))$
383	382	cbvrexvw	$⊢ \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h) \leftrightarrow \exists m \in U \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)$
384	380 383	sylib	$⊢ (((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \to \exists m \in U \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)$
385	379 384	r19.29a	$⊢ (((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|)) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩) \to \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))$
386	385	anasss	$⊢ ((((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \land (r : (0 ..^\|h\|) \underset{1-1 onto}{⟶} (0 ..^\|h\|) \land h \circ r = ((h \circ r) prefix (\|h\| - 1)) ++ ⟨“ h (j) ”⟩)) \to \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))$
387	219 386	exlimddv	$⊢ (((((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \land t \in U) \land t \underset{˙}{\cdot} p = h (j)) \to \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))$
388		eqid	$⊢ ∥_{r} (R) = ∥_{r} (R)$
389		simplr	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \to j \in (0 ..^\|h\|)$
390	276 389	ffvelcdmd	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \to h (j) \in P$
391	16 2 388 221 226 234 390 4 1	rprmasso3	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \to \exists t \in U t \underset{˙}{\cdot} p = h (j)$
392	387 391	r19.29a	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land j \in (0 ..^\|h\|)) \land p ∥_{r} (R) h (j)) \to \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))$
393		suppssdm	$⊢ h supp 1_{R} \subseteq dom h$
394		eqidd	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \|h\| = \|h\|$
395		simpllr	$⊢ (((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \to R \in IDomn$
396	395 150	syl	$⊢ (((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \to Word P \subseteq Word {Base}_{R}$
397	396	ad2antrr	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to Word P \subseteq Word {Base}_{R}$
398		simp-4r	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to h \in Word P$
399	397 398	sseldd	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to h \in Word {Base}_{R}$
400	394 399	wrdfd	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to h : (0 ..^\|h\|) ⟶ {Base}_{R}$
401	393 400	fssdm	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to h supp 1_{R} \subseteq (0 ..^\|h\|)$
402	21	ad5antlr	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to R \in CRing$
403		simp-5r	$⊢ (((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \to p \in P$
404	403	ad2antrr	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to p \in P$
405		ovexd	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to (0 ..^\|h\|) \in V$
406		fvexd	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to 1_{R} \in V$
407	406 398	wrdfsupp	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to {finSupp}_{1_{R}} (h)$
408		simp-5r	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to R \in IDomn$
409		simplr	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to m \in U$
410	16 1	unitcl	$⊢ m \in U \to m \in {Base}_{R}$
411	409 410	syl	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to m \in {Base}_{R}$
412	23	ad5antlr	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to M \in CMnd$
413	18 19 412 405 400 407	gsumcl	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \sum_{M} h \in {Base}_{R}$
414	16 2 395 403	rprmcl	$⊢ (((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \to p \in {Base}_{R}$
415	414	ad2antrr	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to p \in {Base}_{R}$
416		ovexd	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to (0 ..^\|f\|) \in V$
417		eqidd	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \|f\| = \|f\|$
418	396 223	sseldd	$⊢ (((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \to f \in Word {Base}_{R}$
419	418	ad2antrr	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to f \in Word {Base}_{R}$
420	417 419	wrdfd	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to f : (0 ..^\|f\|) ⟶ {Base}_{R}$
421	406 419	wrdfsupp	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to {finSupp}_{1_{R}} (f)$
422	18 19 412 416 420 421	gsumcl	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \sum_{M} f \in {Base}_{R}$
423	16 388 4	dvdsrmul	$⊢ (p \in {Base}_{R} \land \sum_{M} f \in {Base}_{R}) \to p ∥_{r} (R) ((\sum_{M}, f) \underset{˙}{\cdot} p)$
424	415 422 423	syl2anc	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to p ∥_{r} (R) ((\sum_{M}, f) \underset{˙}{\cdot} p)$
425	20	idomringd	$⊢ R \in IDomn \to R \in Ring$
426	3	ringmgp	$⊢ R \in Ring \to M \in Mnd$
427	425 426	syl	$⊢ R \in IDomn \to M \in Mnd$
428	427	ad3antlr	$⊢ (((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \to M \in Mnd$
429	3 4	mgpplusg	$⊢ \underset{˙}{\cdot} = +_{M}$
430	18 429	gsumccatsn	$⊢ (M \in Mnd \land f \in Word {Base}_{R} \land p \in {Base}_{R}) \to \sum_{M} (f ++ ⟨“ p ”⟩) = (\sum_{M}, f) \underset{˙}{\cdot} p$
431	428 418 414 430	syl3anc	$⊢ (((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \to \sum_{M} (f ++ ⟨“ p ”⟩) = (\sum_{M}, f) \underset{˙}{\cdot} p$
432	431	ad2antrr	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \sum_{M} (f ++ ⟨“ p ”⟩) = (\sum_{M}, f) \underset{˙}{\cdot} p$
433		simpr	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)$
434	432 433	eqtr3d	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to (\sum_{M}, f) \underset{˙}{\cdot} p = m \underset{˙}{\cdot} (\sum_{M}, h)$
435	424 434	breqtrd	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to p ∥_{r} (R) (m \underset{˙}{\cdot} (\sum_{M}, h))$
436	16 2 388 4 408 404 411 413 435	rprmdvds	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to (p ∥_{r} (R) m \lor p ∥_{r} (R) (\sum_{M}, h))$
437	1 2 388 402 404 409	rprmndvdsru	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \neg p ∥_{r} (R) m$
438	436 437	orcnd	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to p ∥_{r} (R) (\sum_{M}, h)$
439	16 2 388 11 3 402 404 405 407 400 438	rprmdvdsprod	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \exists j \in {supp}_{1_{R}} (h) p ∥_{r} (R) h (j)$
440		ssrexv	$⊢ h supp 1_{R} \subseteq (0 ..^\|h\|) \to (\exists j \in {supp}_{1_{R}} (h) p ∥_{r} (R) h (j) \to \exists j \in (0 ..^\|h\|) p ∥_{r} (R) h (j))$
441	401 439 440	sylc	$⊢ (((((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \land m \in U) \land \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)) \to \exists j \in (0 ..^\|h\|) p ∥_{r} (R) h (j)$
442	383	bilani	$⊢ (((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \to \exists m \in U \sum_{M} (f ++ ⟨“ p ”⟩) = m \underset{˙}{\cdot} (\sum_{M}, h)$
443	441 442	r19.29a	$⊢ (((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \to \exists j \in (0 ..^\|h\|) p ∥_{r} (R) h (j)$
444	392 443	r19.29a	$⊢ (((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \land \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h)) \to \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))$
445	444	ex	$⊢ ((((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \land h \in Word P) \to (\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h) \to \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)))$
446	445	ralrimiva	$⊢ (((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \to \forall h \in Word P (\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h) \to \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)))$
447		f1oeq1	$⊢ w = v \to (w : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \leftrightarrow v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|))$
448		coeq2	$⊢ w = v \to (f ++ ⟨“ p ”⟩) \circ w = (f ++ ⟨“ p ”⟩) \circ v$
449	448	oveq2d	$⊢ w = v \to u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w) = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)$
450	449	eqeq2d	$⊢ w = v \to (g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w) \leftrightarrow g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))$
451	447 450	anbi12d	$⊢ w = v \to ((w : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w)) \leftrightarrow (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)))$
452	451	rexbidv	$⊢ w = v \to (\exists u \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (w : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w)) \leftrightarrow \exists u \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)))$
453	452	cbvexvw	$⊢ \exists w \exists u \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (w : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w)) \leftrightarrow \exists v \exists u \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))$
454		oveq1	$⊢ u = s \to u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v) = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)$
455	454	eqeq2d	$⊢ u = s \to (g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v) \leftrightarrow g = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))$
456	455	anbi2d	$⊢ u = s \to ((v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)) \leftrightarrow (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)))$
457	456	cbvrexvw	$⊢ \exists u \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)) \leftrightarrow \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))$
458	457	exbii	$⊢ \exists v \exists u \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)) \leftrightarrow \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))$
459	453 458	bitri	$⊢ \exists w \exists u \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (w : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w)) \leftrightarrow \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))$
460	459	imbi2i	$⊢ (\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (w : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w))) \leftrightarrow (\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)))$
461	460	ralbii	$⊢ \forall g \in Word P (\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (w : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w))) \leftrightarrow \forall g \in Word P (\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)))$
462		oveq2	$⊢ g = h \to \sum_{M} g = \sum_{M} h$
463	462	oveq2d	$⊢ g = h \to k \underset{˙}{\cdot} (\sum_{M}, g) = k \underset{˙}{\cdot} (\sum_{M}, h)$
464	463	eqeq2d	$⊢ g = h \to (\sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, g) \leftrightarrow \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h))$
465	464	rexbidv	$⊢ g = h \to (\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, g) \leftrightarrow \exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h))$
466		eqeq1	$⊢ g = h \to (g = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v) \leftrightarrow h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))$
467	466	anbi2d	$⊢ g = h \to ((v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)) \leftrightarrow (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)))$
468	467	rexbidv	$⊢ g = h \to (\exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)) \leftrightarrow \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)))$
469	468	exbidv	$⊢ g = h \to (\exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)) \leftrightarrow \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)))$
470	465 469	imbi12d	$⊢ g = h \to ((\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))) \leftrightarrow (\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h) \to \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))))$
471	470	cbvralvw	$⊢ \forall g \in Word P (\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v))) \leftrightarrow \forall h \in Word P (\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h) \to \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)))$
472	461 471	bitri	$⊢ \forall g \in Word P (\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (w : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w))) \leftrightarrow \forall h \in Word P (\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, h) \to \exists v \exists s \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (v : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land h = s {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ v)))$
473	446 472	sylibr	$⊢ (((f \in Word P \land p \in P) \land (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w))))) \land R \in IDomn) \to \forall g \in Word P (\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (w : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w)))$
474	473	exp31	$⊢ (f \in Word P \land p \in P) \to ((R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} f = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f\|)}) (w : (0 ..^\|f\|) \underset{1-1 onto}{⟶} (0 ..^\|f\|) \land g = u {\underset{˙}{\cdot}}_{f} (f \circ w)))) \to (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} (f ++ ⟨“ p ”⟩) = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|f ++ ⟨“ p ”⟩\|)}) (w : (0 ..^\|f ++ ⟨“ p ”⟩\|) \underset{1-1 onto}{⟶} (0 ..^\|f ++ ⟨“ p ”⟩\|) \land g = u {\underset{˙}{\cdot}}_{f} ((f ++ ⟨“ p ”⟩) \circ w)))))$
475	66 83 100 117 213 474	wrdind	$⊢ F \in Word P \to (R \in IDomn \to \forall g \in Word P (\exists k \in U \sum_{M} F = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|F\|)}) (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land g = u {\underset{˙}{\cdot}}_{f} (F \circ w))))$
476	7 6 475	sylc	$⊢ φ \to \forall g \in Word P (\exists k \in U \sum_{M} F = k \underset{˙}{\cdot} (\sum_{M}, g) \to \exists w \exists u \in (U^{(0 ..^\|F\|)}) (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land g = u {\underset{˙}{\cdot}}_{f} (F \circ w)))$
477	49 476 8	rspcdva	$⊢ φ \to (\exists k \in U \sum_{M} F = k \underset{˙}{\cdot} (\sum_{M}, G) \to \exists w \exists u \in (U^{(0 ..^\|F\|)}) (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land G = u {\underset{˙}{\cdot}}_{f} (F \circ w)))$
478	40 477	mpd	$⊢ φ \to \exists w \exists u \in (U^{(0 ..^\|F\|)}) (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land G = u {\underset{˙}{\cdot}}_{f} (F \circ w))$
479	5	oveq2i	$⊢ U^{J} = U^{(0 ..^\|F\|)}$
480		f1oeq23	$⊢ (J = (0 ..^\|F\|) \land J = (0 ..^\|F\|)) \to (w : J \underset{1-1 onto}{⟶} J \leftrightarrow w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|))$
481	5 5 480	mp2an	$⊢ w : J \underset{1-1 onto}{⟶} J \leftrightarrow w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|)$
482	481	anbi1i	$⊢ (w : J \underset{1-1 onto}{⟶} J \land G = u {\underset{˙}{\cdot}}_{f} (F \circ w)) \leftrightarrow (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land G = u {\underset{˙}{\cdot}}_{f} (F \circ w))$
483	479 482	rexeqbii	$⊢ \exists u \in (U^{J}) (w : J \underset{1-1 onto}{⟶} J \land G = u {\underset{˙}{\cdot}}_{f} (F \circ w)) \leftrightarrow \exists u \in (U^{(0 ..^\|F\|)}) (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land G = u {\underset{˙}{\cdot}}_{f} (F \circ w))$
484	483	exbii	$⊢ \exists w \exists u \in (U^{J}) (w : J \underset{1-1 onto}{⟶} J \land G = u {\underset{˙}{\cdot}}_{f} (F \circ w)) \leftrightarrow \exists w \exists u \in (U^{(0 ..^\|F\|)}) (w : (0 ..^\|F\|) \underset{1-1 onto}{⟶} (0 ..^\|F\|) \land G = u {\underset{˙}{\cdot}}_{f} (F \circ w))$
485	478 484	sylibr	$⊢ φ \to \exists w \exists u \in (U^{J}) (w : J \underset{1-1 onto}{⟶} J \land G = u {\underset{˙}{\cdot}}_{f} (F \circ w))$

Metamath Proof Explorer

Theorem 1arithidom

Proof