psrcom

Description: Commutative law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015)

	Ref	Expression
Hypotheses	psrring.s	$⊢ S = I mPwSer R$
	psrring.i	$⊢ φ \to I \in V$
	psrring.r	$⊢ φ \to R \in Ring$
	psrass.d	$⊢ D = \{f \in ({ℕ_{0}}^{I}) \| f^{-1} [ℕ] \in Fin\}$
	psrass.t	$⊢ \underset{˙}{\times} = \cdot_{S}$
	psrass.b	$⊢ B = {Base}_{S}$
	psrass.x	$⊢ φ \to X \in B$
	psrass.y	$⊢ φ \to Y \in B$
	psrcom.c	$⊢ φ \to R \in CRing$
Assertion	psrcom	$⊢ φ \to X \underset{˙}{\times} Y = Y \underset{˙}{\times} X$

Proof

Step	Hyp	Ref	Expression
1		psrring.s	$⊢ S = I mPwSer R$
2		psrring.i	$⊢ φ \to I \in V$
3		psrring.r	$⊢ φ \to R \in Ring$
4		psrass.d	$⊢ D = \{f \in ({ℕ_{0}}^{I}) \| f^{-1} [ℕ] \in Fin\}$
5		psrass.t	$⊢ \underset{˙}{\times} = \cdot_{S}$
6		psrass.b	$⊢ B = {Base}_{S}$
7		psrass.x	$⊢ φ \to X \in B$
8		psrass.y	$⊢ φ \to Y \in B$
9		psrcom.c	$⊢ φ \to R \in CRing$
10		eqid	$⊢ {Base}_{R} = {Base}_{R}$
11		eqid	$⊢ 0_{R} = 0_{R}$
12		ringcmn	$⊢ R \in Ring \to R \in CMnd$
13	3 12	syl	$⊢ φ \to R \in CMnd$
14	13	adantr	$⊢ (φ \land x \in D) \to R \in CMnd$
15	4	psrbaglefi	$⊢ x \in D \to \{g \in D \| g \leq_{f} x\} \in Fin$
16	15	adantl	$⊢ (φ \land x \in D) \to \{g \in D \| g \leq_{f} x\} \in Fin$
17	3	ad2antrr	$⊢ ((φ \land x \in D) \land k \in \{g \in D \| g \leq_{f} x\}) \to R \in Ring$
18	1 10 4 6 7	psrelbas	$⊢ φ \to X : D ⟶ {Base}_{R}$
19	18	ad2antrr	$⊢ ((φ \land x \in D) \land k \in \{g \in D \| g \leq_{f} x\}) \to X : D ⟶ {Base}_{R}$
20		simpr	$⊢ ((φ \land x \in D) \land k \in \{g \in D \| g \leq_{f} x\}) \to k \in \{g \in D \| g \leq_{f} x\}$
21		breq1	$⊢ g = k \to (g \leq_{f} x \leftrightarrow k \leq_{f} x)$
22	21	elrab	$⊢ k \in \{g \in D \| g \leq_{f} x\} \leftrightarrow (k \in D \land k \leq_{f} x)$
23	20 22	sylib	$⊢ ((φ \land x \in D) \land k \in \{g \in D \| g \leq_{f} x\}) \to (k \in D \land k \leq_{f} x)$
24	23	simpld	$⊢ ((φ \land x \in D) \land k \in \{g \in D \| g \leq_{f} x\}) \to k \in D$
25	19 24	ffvelrnd	$⊢ ((φ \land x \in D) \land k \in \{g \in D \| g \leq_{f} x\}) \to X (k) \in {Base}_{R}$
26	1 10 4 6 8	psrelbas	$⊢ φ \to Y : D ⟶ {Base}_{R}$
27	26	ad2antrr	$⊢ ((φ \land x \in D) \land k \in \{g \in D \| g \leq_{f} x\}) \to Y : D ⟶ {Base}_{R}$
28		simplr	$⊢ ((φ \land x \in D) \land k \in \{g \in D \| g \leq_{f} x\}) \to x \in D$
29	4	psrbagf	$⊢ k \in D \to k : I ⟶ ℕ_{0}$
30	24 29	syl	$⊢ ((φ \land x \in D) \land k \in \{g \in D \| g \leq_{f} x\}) \to k : I ⟶ ℕ_{0}$
31	23	simprd	$⊢ ((φ \land x \in D) \land k \in \{g \in D \| g \leq_{f} x\}) \to k \leq_{f} x$
32	4	psrbagcon	$⊢ (x \in D \land k : I ⟶ ℕ_{0} \land k \leq_{f} x) \to (x -_{f} k \in D \land (x -_{f} k) \leq_{f} x)$
33	28 30 31 32	syl3anc	$⊢ ((φ \land x \in D) \land k \in \{g \in D \| g \leq_{f} x\}) \to (x -_{f} k \in D \land (x -_{f} k) \leq_{f} x)$
34	33	simpld	$⊢ ((φ \land x \in D) \land k \in \{g \in D \| g \leq_{f} x\}) \to x -_{f} k \in D$
35	27 34	ffvelrnd	$⊢ ((φ \land x \in D) \land k \in \{g \in D \| g \leq_{f} x\}) \to Y (x -_{f} k) \in {Base}_{R}$
36		eqid	$⊢ \cdot_{R} = \cdot_{R}$
37	10 36	ringcl	$⊢ (R \in Ring \land X (k) \in {Base}_{R} \land Y (x -_{f} k) \in {Base}_{R}) \to X (k) \cdot_{R} Y (x -_{f} k) \in {Base}_{R}$
38	17 25 35 37	syl3anc	$⊢ ((φ \land x \in D) \land k \in \{g \in D \| g \leq_{f} x\}) \to X (k) \cdot_{R} Y (x -_{f} k) \in {Base}_{R}$
39	38	fmpttd	$⊢ (φ \land x \in D) \to (k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k)) : \{g \in D \| g \leq_{f} x\} ⟶ {Base}_{R}$
40		ovex	$⊢ {ℕ_{0}}^{I} \in V$
41	4 40	rabex2	$⊢ D \in V$
42	41	a1i	$⊢ (φ \land x \in D) \to D \in V$
43		rabexg	$⊢ D \in V \to \{g \in D \| g \leq_{f} x\} \in V$
44	42 43	syl	$⊢ (φ \land x \in D) \to \{g \in D \| g \leq_{f} x\} \in V$
45	44	mptexd	$⊢ (φ \land x \in D) \to (k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k)) \in V$
46		funmpt	$⊢ Fun (k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k))$
47	46	a1i	$⊢ (φ \land x \in D) \to Fun (k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k))$
48		fvexd	$⊢ (φ \land x \in D) \to 0_{R} \in V$
49		suppssdm	$⊢ (k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k)) supp 0_{R} \subseteq dom (k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k))$
50		eqid	$⊢ (k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k)) = (k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k))$
51	50	dmmptss	$⊢ dom (k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k)) \subseteq \{g \in D \| g \leq_{f} x\}$
52	49 51	sstri	$⊢ (k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k)) supp 0_{R} \subseteq \{g \in D \| g \leq_{f} x\}$
53	52	a1i	$⊢ (φ \land x \in D) \to (k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k)) supp 0_{R} \subseteq \{g \in D \| g \leq_{f} x\}$
54		suppssfifsupp	$⊢ (((k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k)) \in V \land Fun (k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k)) \land 0_{R} \in V) \land (\{g \in D \| g \leq_{f} x\} \in Fin \land (k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k)) supp 0_{R} \subseteq \{g \in D \| g \leq_{f} x\})) \to {finSupp}_{0_{R}} ((k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k)))$
55	45 47 48 16 53 54	syl32anc	$⊢ (φ \land x \in D) \to {finSupp}_{0_{R}} ((k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k)))$
56		eqid	$⊢ \{g \in D \| g \leq_{f} x\} = \{g \in D \| g \leq_{f} x\}$
57	4 56	psrbagconf1o	$⊢ x \in D \to (j \in \{g \in D \| g \leq_{f} x\} ⟼ x -_{f} j) : \{g \in D \| g \leq_{f} x\} \underset{1-1 onto}{⟶} \{g \in D \| g \leq_{f} x\}$
58	57	adantl	$⊢ (φ \land x \in D) \to (j \in \{g \in D \| g \leq_{f} x\} ⟼ x -_{f} j) : \{g \in D \| g \leq_{f} x\} \underset{1-1 onto}{⟶} \{g \in D \| g \leq_{f} x\}$
59	10 11 14 16 39 55 58	gsumf1o	$⊢ (φ \land x \in D) \to \underset{k \in \{g \in D \| g \leq_{f} x\}}{\sum_{R}} (X (k) \cdot_{R} Y (x -_{f} k)) = \sum_{R} ((k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k)) \circ (j \in \{g \in D \| g \leq_{f} x\} ⟼ x -_{f} j))$
60		simplr	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to x \in D$
61		simpr	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to j \in \{g \in D \| g \leq_{f} x\}$
62	4 56	psrbagconcl	$⊢ (x \in D \land j \in \{g \in D \| g \leq_{f} x\}) \to x -_{f} j \in \{g \in D \| g \leq_{f} x\}$
63	60 61 62	syl2anc	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to x -_{f} j \in \{g \in D \| g \leq_{f} x\}$
64		eqidd	$⊢ (φ \land x \in D) \to (j \in \{g \in D \| g \leq_{f} x\} ⟼ x -_{f} j) = (j \in \{g \in D \| g \leq_{f} x\} ⟼ x -_{f} j)$
65		eqidd	$⊢ (φ \land x \in D) \to (k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k)) = (k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k))$
66		fveq2	$⊢ k = x -_{f} j \to X (k) = X (x -_{f} j)$
67		oveq2	$⊢ k = x -_{f} j \to x -_{f} k = x -_{f} (x -_{f} j)$
68	67	fveq2d	$⊢ k = x -_{f} j \to Y (x -_{f} k) = Y (x -_{f} (x -_{f} j))$
69	66 68	oveq12d	$⊢ k = x -_{f} j \to X (k) \cdot_{R} Y (x -_{f} k) = X (x -_{f} j) \cdot_{R} Y (x -_{f} (x -_{f} j))$
70	63 64 65 69	fmptco	$⊢ (φ \land x \in D) \to (k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k)) \circ (j \in \{g \in D \| g \leq_{f} x\} ⟼ x -_{f} j) = (j \in \{g \in D \| g \leq_{f} x\} ⟼ X (x -_{f} j) \cdot_{R} Y (x -_{f} (x -_{f} j)))$
71	4	psrbagf	$⊢ x \in D \to x : I ⟶ ℕ_{0}$
72	71	adantl	$⊢ (φ \land x \in D) \to x : I ⟶ ℕ_{0}$
73	72	adantr	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to x : I ⟶ ℕ_{0}$
74	73	ffvelrnda	$⊢ (((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \land z \in I) \to x (z) \in ℕ_{0}$
75		breq1	$⊢ g = j \to (g \leq_{f} x \leftrightarrow j \leq_{f} x)$
76	75	elrab	$⊢ j \in \{g \in D \| g \leq_{f} x\} \leftrightarrow (j \in D \land j \leq_{f} x)$
77	61 76	sylib	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to (j \in D \land j \leq_{f} x)$
78	77	simpld	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to j \in D$
79	4	psrbagf	$⊢ j \in D \to j : I ⟶ ℕ_{0}$
80	78 79	syl	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to j : I ⟶ ℕ_{0}$
81	80	ffvelrnda	$⊢ (((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \land z \in I) \to j (z) \in ℕ_{0}$
82		nn0cn	$⊢ x (z) \in ℕ_{0} \to x (z) \in ℂ$
83		nn0cn	$⊢ j (z) \in ℕ_{0} \to j (z) \in ℂ$
84		nncan	$⊢ (x (z) \in ℂ \land j (z) \in ℂ) \to x (z) - (x (z) - j (z)) = j (z)$
85	82 83 84	syl2an	$⊢ (x (z) \in ℕ_{0} \land j (z) \in ℕ_{0}) \to x (z) - (x (z) - j (z)) = j (z)$
86	74 81 85	syl2anc	$⊢ (((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \land z \in I) \to x (z) - (x (z) - j (z)) = j (z)$
87	86	mpteq2dva	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to (z \in I ⟼ x (z) - (x (z) - j (z))) = (z \in I ⟼ j (z))$
88	2	ad2antrr	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to I \in V$
89		ovex	$⊢ x (z) - j (z) \in V$
90	89	a1i	$⊢ (((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \land z \in I) \to x (z) - j (z) \in V$
91	73	feqmptd	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to x = (z \in I ⟼ x (z))$
92	80	feqmptd	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to j = (z \in I ⟼ j (z))$
93	88 74 81 91 92	offval2	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to x -_{f} j = (z \in I ⟼ x (z) - j (z))$
94	88 74 90 91 93	offval2	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to x -_{f} (x -_{f} j) = (z \in I ⟼ x (z) - (x (z) - j (z)))$
95	87 94 92	3eqtr4d	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to x -_{f} (x -_{f} j) = j$
96	95	fveq2d	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to Y (x -_{f} (x -_{f} j)) = Y (j)$
97	96	oveq2d	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to X (x -_{f} j) \cdot_{R} Y (x -_{f} (x -_{f} j)) = X (x -_{f} j) \cdot_{R} Y (j)$
98	9	ad2antrr	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to R \in CRing$
99	18	ad2antrr	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to X : D ⟶ {Base}_{R}$
100	77	simprd	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to j \leq_{f} x$
101	4	psrbagcon	$⊢ (x \in D \land j : I ⟶ ℕ_{0} \land j \leq_{f} x) \to (x -_{f} j \in D \land (x -_{f} j) \leq_{f} x)$
102	60 80 100 101	syl3anc	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to (x -_{f} j \in D \land (x -_{f} j) \leq_{f} x)$
103	102	simpld	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to x -_{f} j \in D$
104	99 103	ffvelrnd	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to X (x -_{f} j) \in {Base}_{R}$
105	26	ad2antrr	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to Y : D ⟶ {Base}_{R}$
106	105 78	ffvelrnd	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to Y (j) \in {Base}_{R}$
107	10 36	crngcom	$⊢ (R \in CRing \land X (x -_{f} j) \in {Base}_{R} \land Y (j) \in {Base}_{R}) \to X (x -_{f} j) \cdot_{R} Y (j) = Y (j) \cdot_{R} X (x -_{f} j)$
108	98 104 106 107	syl3anc	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to X (x -_{f} j) \cdot_{R} Y (j) = Y (j) \cdot_{R} X (x -_{f} j)$
109	97 108	eqtrd	$⊢ ((φ \land x \in D) \land j \in \{g \in D \| g \leq_{f} x\}) \to X (x -_{f} j) \cdot_{R} Y (x -_{f} (x -_{f} j)) = Y (j) \cdot_{R} X (x -_{f} j)$
110	109	mpteq2dva	$⊢ (φ \land x \in D) \to (j \in \{g \in D \| g \leq_{f} x\} ⟼ X (x -_{f} j) \cdot_{R} Y (x -_{f} (x -_{f} j))) = (j \in \{g \in D \| g \leq_{f} x\} ⟼ Y (j) \cdot_{R} X (x -_{f} j))$
111	70 110	eqtrd	$⊢ (φ \land x \in D) \to (k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k)) \circ (j \in \{g \in D \| g \leq_{f} x\} ⟼ x -_{f} j) = (j \in \{g \in D \| g \leq_{f} x\} ⟼ Y (j) \cdot_{R} X (x -_{f} j))$
112	111	oveq2d	$⊢ (φ \land x \in D) \to \sum_{R} ((k \in \{g \in D \| g \leq_{f} x\} ⟼ X (k) \cdot_{R} Y (x -_{f} k)) \circ (j \in \{g \in D \| g \leq_{f} x\} ⟼ x -_{f} j)) = \underset{j \in \{g \in D \| g \leq_{f} x\}}{\sum_{R}} (Y (j) \cdot_{R} X (x -_{f} j))$
113	59 112	eqtrd	$⊢ (φ \land x \in D) \to \underset{k \in \{g \in D \| g \leq_{f} x\}}{\sum_{R}} (X (k) \cdot_{R} Y (x -_{f} k)) = \underset{j \in \{g \in D \| g \leq_{f} x\}}{\sum_{R}} (Y (j) \cdot_{R} X (x -_{f} j))$
114	113	mpteq2dva	$⊢ φ \to (x \in D ⟼ \underset{k \in \{g \in D \| g \leq_{f} x\}}{\sum_{R}} (X (k) \cdot_{R} Y (x -_{f} k))) = (x \in D ⟼ \underset{j \in \{g \in D \| g \leq_{f} x\}}{\sum_{R}} (Y (j) \cdot_{R} X (x -_{f} j)))$
115	1 6 36 5 4 7 8	psrmulfval	$⊢ φ \to X \underset{˙}{\times} Y = (x \in D ⟼ \underset{k \in \{g \in D \| g \leq_{f} x\}}{\sum_{R}} (X (k) \cdot_{R} Y (x -_{f} k)))$
116	1 6 36 5 4 8 7	psrmulfval	$⊢ φ \to Y \underset{˙}{\times} X = (x \in D ⟼ \underset{j \in \{g \in D \| g \leq_{f} x\}}{\sum_{R}} (Y (j) \cdot_{R} X (x -_{f} j)))$
117	114 115 116	3eqtr4d	$⊢ φ \to X \underset{˙}{\times} Y = Y \underset{˙}{\times} X$

Metamath Proof Explorer

Theorem psrcom

Proof