Theorem matgsum 21931

Description: Finite commutative sums in a matrix algebra are taken componentwise. (Contributed by AV, 26-Sep-2019.)

Hypotheses

Ref	Expression
matgsum.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
matgsum.b	⊢ 𝐵 = (Base‘𝐴)
matgsum.z	⊢ 0 = (0g‘𝐴)
matgsum.i	⊢ (𝜑 → 𝑁 ∈ Fin)
matgsum.j	⊢ (𝜑 → 𝐽 ∈ 𝑊)
matgsum.r	⊢ (𝜑 → 𝑅 ∈ Ring)
matgsum.f	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈) ∈ 𝐵)
matgsum.w	⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) finSupp 0 )

Assertion

Ref	Expression
matgsum	⊢ (𝜑 → (𝐴 Σg (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈))))

Distinct variable groups:   𝑖,𝐽,𝑗,𝑦   𝑖,𝑁,𝑗,𝑦   𝑅,𝑖,𝑗,𝑦   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑖,𝑗)   𝐴(𝑦,𝑖,𝑗)   𝐵(𝑦,𝑖,𝑗)   𝑈(𝑦,𝑖,𝑗)   𝑊(𝑦,𝑖,𝑗)   0 (𝑦,𝑖,𝑗)

Proof of Theorem matgsum

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		matgsum.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ 𝑊)
2	1	mptexd 7223	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) ∈ V)
3		matgsum.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
4	3	ovexi 7440	. . . 4 ⊢ 𝐴 ∈ V
5	4	a1i 11	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
6		ovexd 7441	. . 3 ⊢ (𝜑 → (𝑅 freeLMod (𝑁 × 𝑁)) ∈ V)
7		matgsum.i	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ Fin)
8		matgsum.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
9		eqid 2733	. . . . . 6 ⊢ (𝑅 freeLMod (𝑁 × 𝑁)) = (𝑅 freeLMod (𝑁 × 𝑁))
10	3, 9	matbas 21905	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘𝐴))
11	7, 8, 10	syl2anc 585	. . . 4 ⊢ (𝜑 → (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘𝐴))
12	11	eqcomd 2739	. . 3 ⊢ (𝜑 → (Base‘𝐴) = (Base‘(𝑅 freeLMod (𝑁 × 𝑁))))
13	3, 9	matplusg 21906	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (+g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (+g‘𝐴))
14	7, 8, 13	syl2anc 585	. . . 4 ⊢ (𝜑 → (+g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (+g‘𝐴))
15	14	eqcomd 2739	. . 3 ⊢ (𝜑 → (+g‘𝐴) = (+g‘(𝑅 freeLMod (𝑁 × 𝑁))))
16	2, 5, 6, 12, 15	gsumpropd 18594	. 2 ⊢ (𝜑 → (𝐴 Σg (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈))) = ((𝑅 freeLMod (𝑁 × 𝑁)) Σg (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈))))
17		mpompts 8048	. . . . . 6 ⊢ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈) = (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)
18	17	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈) = (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈))
19	18	mpteq2dv 5250	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) = (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)))
20	19	oveq2d 7422	. . 3 ⊢ (𝜑 → ((𝑅 freeLMod (𝑁 × 𝑁)) Σg (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈))) = ((𝑅 freeLMod (𝑁 × 𝑁)) Σg (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈))))
21		eqid 2733	. . . 4 ⊢ (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘(𝑅 freeLMod (𝑁 × 𝑁)))
22		eqid 2733	. . . 4 ⊢ (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (0g‘(𝑅 freeLMod (𝑁 × 𝑁)))
23		xpfi 9314	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
24	7, 7, 23	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝑁 × 𝑁) ∈ Fin)
25		matgsum.f	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈) ∈ 𝐵)
26		matgsum.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
27	25, 26	eleqtrdi 2844	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈) ∈ (Base‘𝐴))
28	17	eqcomi 2742	. . . . . 6 ⊢ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)
29	28	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈))
30	7, 8	jca 513	. . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
31	30	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
32	31, 10	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘𝐴))
33	27, 29, 32	3eltr4d 2849	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈) ∈ (Base‘(𝑅 freeLMod (𝑁 × 𝑁))))
34		matgsum.w	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) finSupp 0 )
35	28	mpteq2i 5253	. . . . . 6 ⊢ (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)) = (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈))
36		matgsum.z	. . . . . . 7 ⊢ 0 = (0g‘𝐴)
37	36	eqcomi 2742	. . . . . 6 ⊢ (0g‘𝐴) = 0
38	34, 35, 37	3brtr4g 5182	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)) finSupp (0g‘𝐴))
39	3, 9	mat0 21911	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (0g‘𝐴))
40	7, 8, 39	syl2anc 585	. . . . 5 ⊢ (𝜑 → (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (0g‘𝐴))
41	38, 40	breqtrrd 5176	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)) finSupp (0g‘(𝑅 freeLMod (𝑁 × 𝑁))))
42	9, 21, 22, 24, 1, 8, 33, 41	frlmgsum 21319	. . 3 ⊢ (𝜑 → ((𝑅 freeLMod (𝑁 × 𝑁)) Σg (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈))) = (𝑧 ∈ (𝑁 × 𝑁) ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈))))
43	20, 42	eqtrd 2773	. 2 ⊢ (𝜑 → ((𝑅 freeLMod (𝑁 × 𝑁)) Σg (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈))) = (𝑧 ∈ (𝑁 × 𝑁) ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈))))
44		fvex 6902	. . . . . . . 8 ⊢ (2nd ‘𝑧) ∈ V
45		csbov2g 7452	. . . . . . . 8 ⊢ ((2nd ‘𝑧) ∈ V → ⦋(2nd ‘𝑧) / 𝑗⦌(𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)) = (𝑅 Σg ⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈)))
46	44, 45	ax-mp 5	. . . . . . 7 ⊢ ⦋(2nd ‘𝑧) / 𝑗⦌(𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)) = (𝑅 Σg ⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈))
47	46	csbeq2i 3901	. . . . . 6 ⊢ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌(𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)) = ⦋(1st ‘𝑧) / 𝑖⦌(𝑅 Σg ⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈))
48		fvex 6902	. . . . . . 7 ⊢ (1st ‘𝑧) ∈ V
49		csbov2g 7452	. . . . . . 7 ⊢ ((1st ‘𝑧) ∈ V → ⦋(1st ‘𝑧) / 𝑖⦌(𝑅 Σg ⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈)) = (𝑅 Σg ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈)))
50	48, 49	ax-mp 5	. . . . . 6 ⊢ ⦋(1st ‘𝑧) / 𝑖⦌(𝑅 Σg ⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈)) = (𝑅 Σg ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈))
51		csbmpt2 5558	. . . . . . . . . 10 ⊢ ((2nd ‘𝑧) ∈ V → ⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈) = (𝑦 ∈ 𝐽 ↦ ⦋(2nd ‘𝑧) / 𝑗⦌𝑈))
52	44, 51	ax-mp 5	. . . . . . . . 9 ⊢ ⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈) = (𝑦 ∈ 𝐽 ↦ ⦋(2nd ‘𝑧) / 𝑗⦌𝑈)
53	52	csbeq2i 3901	. . . . . . . 8 ⊢ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈) = ⦋(1st ‘𝑧) / 𝑖⦌(𝑦 ∈ 𝐽 ↦ ⦋(2nd ‘𝑧) / 𝑗⦌𝑈)
54		csbmpt2 5558	. . . . . . . . 9 ⊢ ((1st ‘𝑧) ∈ V → ⦋(1st ‘𝑧) / 𝑖⦌(𝑦 ∈ 𝐽 ↦ ⦋(2nd ‘𝑧) / 𝑗⦌𝑈) = (𝑦 ∈ 𝐽 ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈))
55	48, 54	ax-mp 5	. . . . . . . 8 ⊢ ⦋(1st ‘𝑧) / 𝑖⦌(𝑦 ∈ 𝐽 ↦ ⦋(2nd ‘𝑧) / 𝑗⦌𝑈) = (𝑦 ∈ 𝐽 ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)
56	53, 55	eqtri 2761	. . . . . . 7 ⊢ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈) = (𝑦 ∈ 𝐽 ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)
57	56	oveq2i 7417	. . . . . 6 ⊢ (𝑅 Σg ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈)) = (𝑅 Σg (𝑦 ∈ 𝐽 ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈))
58	47, 50, 57	3eqtrri 2766	. . . . 5 ⊢ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)) = ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌(𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈))
59	58	mpteq2i 5253	. . . 4 ⊢ (𝑧 ∈ (𝑁 × 𝑁) ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈))) = (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌(𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)))
60		mpompts 8048	. . . 4 ⊢ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈))) = (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌(𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)))
61	59, 60	eqtr4i 2764	. . 3 ⊢ (𝑧 ∈ (𝑁 × 𝑁) ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)))
62	61	a1i 11	. 2 ⊢ (𝜑 → (𝑧 ∈ (𝑁 × 𝑁) ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈))))
63	16, 43, 62	3eqtrd 2777	1 ⊢ (𝜑 → (𝐴 Σg (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈))))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475  ⦋csb 3893   class class class wbr 5148   ↦ cmpt 5231   × cxp 5674  ‘cfv 6541  (class class class)co 7406   ∈ cmpo 7408  1^st c1st 7970  2^nd c2nd 7971  Fincfn 8936   finSupp cfsupp 9358  Basecbs 17141  +_gcplusg 17194  0_gc0g 17382   Σ_g cgsu 17383  Ringcrg 20050   freeLMod cfrlm 21293   Mat cmat 21899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-ot 4637  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-supp 8144  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-ixp 8889  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-fsupp 9359  df-sup 9434  df-oi 9502  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-fz 13482  df-fzo 13625  df-seq 13964  df-hash 14288  df-struct 17077  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17142  df-ress 17171  df-plusg 17207  df-mulr 17208  df-sca 17210  df-vsca 17211  df-ip 17212  df-tset 17213  df-ple 17214  df-ds 17216  df-hom 17218  df-cco 17219  df-0g 17384  df-gsum 17385  df-prds 17390  df-pws 17392  df-mgm 18558  df-sgrp 18607  df-mnd 18623  df-mhm 18668  df-grp 18819  df-minusg 18820  df-sbg 18821  df-subg 18998  df-cntz 19176  df-cmn 19645  df-abl 19646  df-mgp 19983  df-ur 20000  df-ring 20052  df-subrg 20354  df-lmod 20466  df-lss 20536  df-sra 20778  df-rgmod 20779  df-dsmm 21279  df-frlm 21294  df-mat 21900

This theorem is referenced by:  decpmatmul  22266  pmatcollpw2  22272