Theorem matgsum 21782

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 7171	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) ∈ V)
3		matgsum.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
4	3	ovexi 7388	. . . 4 ⊢ 𝐴 ∈ V
5	4	a1i 11	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
6		ovexd 7389	. . 3 ⊢ (𝜑 → (𝑅 freeLMod (𝑁 × 𝑁)) ∈ V)
7		matgsum.i	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ Fin)
8		matgsum.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
9		eqid 2736	. . . . . 6 ⊢ (𝑅 freeLMod (𝑁 × 𝑁)) = (𝑅 freeLMod (𝑁 × 𝑁))
10	3, 9	matbas 21756	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘𝐴))
11	7, 8, 10	syl2anc 584	. . . 4 ⊢ (𝜑 → (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘𝐴))
12	11	eqcomd 2742	. . 3 ⊢ (𝜑 → (Base‘𝐴) = (Base‘(𝑅 freeLMod (𝑁 × 𝑁))))
13	3, 9	matplusg 21757	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (+g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (+g‘𝐴))
14	7, 8, 13	syl2anc 584	. . . 4 ⊢ (𝜑 → (+g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (+g‘𝐴))
15	14	eqcomd 2742	. . 3 ⊢ (𝜑 → (+g‘𝐴) = (+g‘(𝑅 freeLMod (𝑁 × 𝑁))))
16	2, 5, 6, 12, 15	gsumpropd 18530	. 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 7994	. . . . . 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 5206	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) = (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)))
20	19	oveq2d 7370	. . 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 2736	. . . 4 ⊢ (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘(𝑅 freeLMod (𝑁 × 𝑁)))
22		eqid 2736	. . . 4 ⊢ (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (0g‘(𝑅 freeLMod (𝑁 × 𝑁)))
23		xpfi 9258	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
24	7, 7, 23	syl2anc 584	. . . 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 2745	. . . . . 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 512	. . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
31	30	adantr 481	. . . . . 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 5209	. . . . . 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 2745	. . . . . 6 ⊢ (0g‘𝐴) = 0
38	34, 35, 37	3brtr4g 5138	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)) finSupp (0g‘𝐴))
39	3, 9	mat0 21762	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (0g‘𝐴))
40	7, 8, 39	syl2anc 584	. . . . 5 ⊢ (𝜑 → (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (0g‘𝐴))
41	38, 40	breqtrrd 5132	. . . 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 21174	. . 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 2776	. 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 6853	. . . . . . . 8 ⊢ (2nd ‘𝑧) ∈ V
45		csbov2g 7400	. . . . . . . 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 3862	. . . . . 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 6853	. . . . . . 7 ⊢ (1st ‘𝑧) ∈ V
49		csbov2g 7400	. . . . . . 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 5514	. . . . . . . . . 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 3862	. . . . . . . 8 ⊢ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈) = ⦋(1st ‘𝑧) / 𝑖⦌(𝑦 ∈ 𝐽 ↦ ⦋(2nd ‘𝑧) / 𝑗⦌𝑈)
54		csbmpt2 5514	. . . . . . . . 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 2764	. . . . . . 7 ⊢ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈) = (𝑦 ∈ 𝐽 ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)
57	56	oveq2i 7365	. . . . . 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 2769	. . . . 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 5209	. . . 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 7994	. . . 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 2767	. . 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 2780	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 396   = wceq 1541   ∈ wcel 2106  Vcvv 3444  ⦋csb 3854   class class class wbr 5104   ↦ cmpt 5187   × cxp 5630  ‘cfv 6494  (class class class)co 7354   ∈ cmpo 7356  1^st c1st 7916  2^nd c2nd 7917  Fincfn 8880   finSupp cfsupp 9302  Basecbs 17080  +_gcplusg 17130  0_gc0g 17318   Σ_g cgsu 17319  Ringcrg 19960   freeLMod cfrlm 21148   Mat cmat 21750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7669  ax-cnex 11104  ax-resscn 11105  ax-1cn 11106  ax-icn 11107  ax-addcl 11108  ax-addrcl 11109  ax-mulcl 11110  ax-mulrcl 11111  ax-mulcom 11112  ax-addass 11113  ax-mulass 11114  ax-distr 11115  ax-i2m1 11116  ax-1ne0 11117  ax-1rid 11118  ax-rnegex 11119  ax-rrecex 11120  ax-cnre 11121  ax-pre-lttri 11122  ax-pre-lttrn 11123  ax-pre-ltadd 11124  ax-pre-mulgt0 11125

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4865  df-int 4907  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5530  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5587  df-se 5588  df-we 5589  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6252  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7310  df-ov 7357  df-oprab 7358  df-mpo 7359  df-om 7800  df-1st 7918  df-2nd 7919  df-supp 8090  df-frecs 8209  df-wrecs 8240  df-recs 8314  df-rdg 8353  df-1o 8409  df-er 8645  df-map 8764  df-ixp 8833  df-en 8881  df-dom 8882  df-sdom 8883  df-fin 8884  df-fsupp 9303  df-sup 9375  df-oi 9443  df-card 9872  df-pnf 11188  df-mnf 11189  df-xr 11190  df-ltxr 11191  df-le 11192  df-sub 11384  df-neg 11385  df-nn 12151  df-2 12213  df-3 12214  df-4 12215  df-5 12216  df-6 12217  df-7 12218  df-8 12219  df-9 12220  df-n0 12411  df-z 12497  df-dec 12616  df-uz 12761  df-fz 13422  df-fzo 13565  df-seq 13904  df-hash 14228  df-struct 17016  df-sets 17033  df-slot 17051  df-ndx 17063  df-base 17081  df-ress 17110  df-plusg 17143  df-mulr 17144  df-sca 17146  df-vsca 17147  df-ip 17148  df-tset 17149  df-ple 17150  df-ds 17152  df-hom 17154  df-cco 17155  df-0g 17320  df-gsum 17321  df-prds 17326  df-pws 17328  df-mgm 18494  df-sgrp 18543  df-mnd 18554  df-mhm 18598  df-grp 18748  df-minusg 18749  df-sbg 18750  df-subg 18921  df-cntz 19093  df-cmn 19560  df-abl 19561  df-mgp 19893  df-ur 19910  df-ring 19962  df-subrg 20216  df-lmod 20320  df-lss 20389  df-sra 20629  df-rgmod 20630  df-dsmm 21134  df-frlm 21149  df-mat 21751

This theorem is referenced by:  decpmatmul  22117  pmatcollpw2  22123