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

Metamath Proof Explorer

Theorem psrcom

Proof