Theorem matgsum 21334

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 7040	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) ∈ V)
3		matgsum.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
4	3	ovexi 7247	. . . 4 ⊢ 𝐴 ∈ V
5	4	a1i 11	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
6		ovexd 7248	. . 3 ⊢ (𝜑 → (𝑅 freeLMod (𝑁 × 𝑁)) ∈ V)
7		matgsum.i	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ Fin)
8		matgsum.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
9		eqid 2737	. . . . . 6 ⊢ (𝑅 freeLMod (𝑁 × 𝑁)) = (𝑅 freeLMod (𝑁 × 𝑁))
10	3, 9	matbas 21310	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘𝐴))
11	7, 8, 10	syl2anc 587	. . . 4 ⊢ (𝜑 → (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘𝐴))
12	11	eqcomd 2743	. . 3 ⊢ (𝜑 → (Base‘𝐴) = (Base‘(𝑅 freeLMod (𝑁 × 𝑁))))
13	3, 9	matplusg 21311	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (+g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (+g‘𝐴))
14	7, 8, 13	syl2anc 587	. . . 4 ⊢ (𝜑 → (+g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (+g‘𝐴))
15	14	eqcomd 2743	. . 3 ⊢ (𝜑 → (+g‘𝐴) = (+g‘(𝑅 freeLMod (𝑁 × 𝑁))))
16	2, 5, 6, 12, 15	gsumpropd 18150	. 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 7835	. . . . . 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 5151	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) = (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)))
20	19	oveq2d 7229	. . 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 2737	. . . 4 ⊢ (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘(𝑅 freeLMod (𝑁 × 𝑁)))
22		eqid 2737	. . . 4 ⊢ (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (0g‘(𝑅 freeLMod (𝑁 × 𝑁)))
23		xpfi 8942	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
24	7, 7, 23	syl2anc 587	. . . 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 2848	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈) ∈ (Base‘𝐴))
28	17	eqcomi 2746	. . . . . 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 515	. . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
31	30	adantr 484	. . . . . 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 2853	. . . 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 5147	. . . . . 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 2746	. . . . . 6 ⊢ (0g‘𝐴) = 0
38	34, 35, 37	3brtr4g 5087	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)) finSupp (0g‘𝐴))
39	3, 9	mat0 21314	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (0g‘𝐴))
40	7, 8, 39	syl2anc 587	. . . . 5 ⊢ (𝜑 → (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (0g‘𝐴))
41	38, 40	breqtrrd 5081	. . . 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 20734	. . 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 2777	. 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 6730	. . . . . . . 8 ⊢ (2nd ‘𝑧) ∈ V
45		csbov2g 7259	. . . . . . . 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 3819	. . . . . 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 6730	. . . . . . 7 ⊢ (1st ‘𝑧) ∈ V
49		csbov2g 7259	. . . . . . 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 5439	. . . . . . . . . 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 3819	. . . . . . . 8 ⊢ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈) = ⦋(1st ‘𝑧) / 𝑖⦌(𝑦 ∈ 𝐽 ↦ ⦋(2nd ‘𝑧) / 𝑗⦌𝑈)
54		csbmpt2 5439	. . . . . . . . 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 2765	. . . . . . 7 ⊢ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈) = (𝑦 ∈ 𝐽 ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)
57	56	oveq2i 7224	. . . . . 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 2770	. . . . 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 5147	. . . 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 7835	. . . 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 2768	. . 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 2781	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 399   = wceq 1543   ∈ wcel 2110  Vcvv 3408  ⦋csb 3811   class class class wbr 5053   ↦ cmpt 5135   × cxp 5549  ‘cfv 6380  (class class class)co 7213   ∈ cmpo 7215  1^st c1st 7759  2^nd c2nd 7760  Fincfn 8626   finSupp cfsupp 8985  Basecbs 16760  +_gcplusg 16802  0_gc0g 16944   Σ_g cgsu 16945  Ringcrg 19562   freeLMod cfrlm 20708   Mat cmat 21304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-ot 4550  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-supp 7904  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-map 8510  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fsupp 8986  df-sup 9058  df-oi 9126  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-dec 12294  df-uz 12439  df-fz 13096  df-fzo 13239  df-seq 13575  df-hash 13897  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-sca 16818  df-vsca 16819  df-ip 16820  df-tset 16821  df-ple 16822  df-ds 16824  df-hom 16826  df-cco 16827  df-0g 16946  df-gsum 16947  df-prds 16952  df-pws 16954  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-mhm 18218  df-grp 18368  df-minusg 18369  df-sbg 18370  df-subg 18540  df-cntz 18711  df-cmn 19172  df-abl 19173  df-mgp 19505  df-ur 19517  df-ring 19564  df-subrg 19798  df-lmod 19901  df-lss 19969  df-sra 20209  df-rgmod 20210  df-dsmm 20694  df-frlm 20709  df-mat 21305

This theorem is referenced by:  decpmatmul  21669  pmatcollpw2  21675