Theorem matgsum 22646

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 7229	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) ∈ V)
3		matgsum.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
4	3	ovexi 7453	. . . 4 ⊢ 𝐴 ∈ V
5	4	a1i 11	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
6		ovexd 7454	. . 3 ⊢ (𝜑 → (𝑅 freeLMod (𝑁 × 𝑁)) ∈ V)
7		matgsum.i	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ Fin)
8		matgsum.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
9		eqid 2765	. . . . . 6 ⊢ (𝑅 freeLMod (𝑁 × 𝑁)) = (𝑅 freeLMod (𝑁 × 𝑁))
10	3, 9	matbas 22622	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘𝐴))
11	7, 8, 10	syl2anc 596	. . . 4 ⊢ (𝜑 → (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘𝐴))
12	11	eqcomd 2771	. . 3 ⊢ (𝜑 → (Base‘𝐴) = (Base‘(𝑅 freeLMod (𝑁 × 𝑁))))
13	3, 9	matplusg 22623	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (+g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (+g‘𝐴))
14	7, 8, 13	syl2anc 596	. . . 4 ⊢ (𝜑 → (+g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (+g‘𝐴))
15	14	eqcomd 2771	. . 3 ⊢ (𝜑 → (+g‘𝐴) = (+g‘(𝑅 freeLMod (𝑁 × 𝑁))))
16	2, 5, 6, 12, 15	gsumpropd 18770	. 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 8068	. . . . . 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 5207	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) = (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)))
20	19	oveq2d 7435	. . 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 2765	. . . 4 ⊢ (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘(𝑅 freeLMod (𝑁 × 𝑁)))
22		eqid 2765	. . . 4 ⊢ (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (0g‘(𝑅 freeLMod (𝑁 × 𝑁)))
23		xpfi 9286	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
24	7, 7, 23	syl2anc 596	. . . 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 2875	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈) ∈ (Base‘𝐴))
28	17	eqcomi 2774	. . . . . 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 521	. . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
31	30	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
32	31, 10	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘𝐴))
33	27, 29, 32	3eltr4d 2880	. . . 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 2774	. . . . . 6 ⊢ (0g‘𝐴) = 0
38	34, 35, 37	3brtr4g 5147	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)) finSupp (0g‘𝐴))
39	3, 9	mat0 22626	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (0g‘𝐴))
40	7, 8, 39	syl2anc 596	. . . . 5 ⊢ (𝜑 → (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (0g‘𝐴))
41	38, 40	breqtrrd 5141	. . . 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 21974	. . 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 2800	. 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 6898	. . . . . . . 8 ⊢ (2nd ‘𝑧) ∈ V
45		csbov2g 7467	. . . . . . . 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 6898	. . . . . . 7 ⊢ (1st ‘𝑧) ∈ V
49		csbov2g 7467	. . . . . . 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 5545	. . . . . . . . . 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 5545	. . . . . . . . 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 2788	. . . . . . 7 ⊢ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈) = (𝑦 ∈ 𝐽 ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)
57	56	oveq2i 7430	. . . . . 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 2793	. . . . 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 8068	. . . 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 2791	. . 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 2804	1 ⊢ (𝜑 → (𝐴 Σg (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ⦋csb 3854   class class class wbr 5111   ↦ cmpt 5194   × cxp 5661  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  1^st c1st 7990  2^nd c2nd 7991  Fincfn 8949   finSupp cfsupp 9328  Basecbs 17293  +_gcplusg 17334  0_gc0g 17516   Σ_g cgsu 17517  Ringcrg 20361   freeLMod cfrlm 21948   Mat cmat 22616

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-supp 8163  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-fsupp 9329  df-sup 9409  df-oi 9479  df-card 9941  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-nn 12251  df-2 12320  df-3 12321  df-4 12322  df-5 12323  df-6 12324  df-7 12325  df-8 12326  df-9 12327  df-n0 12522  df-z 12609  df-dec 12730  df-uz 12881  df-fz 13554  df-fzo 13702  df-seq 14058  df-hash 14387  df-struct 17231  df-sets 17248  df-slot 17266  df-ndx 17278  df-base 17294  df-ress 17315  df-plusg 17347  df-mulr 17348  df-sca 17350  df-vsca 17351  df-ip 17352  df-tset 17353  df-ple 17354  df-ds 17356  df-hom 17358  df-cco 17359  df-0g 17518  df-gsum 17519  df-prds 17524  df-pws 17526  df-mgm 18722  df-sgrp 18811  df-mnd 18827  df-mhm 18880  df-grp 19049  df-minusg 19050  df-sbg 19051  df-subg 19235  df-cntz 19433  df-cmn 19898  df-abl 19899  df-mgp 20263  df-rng 20277  df-ur 20310  df-ring 20363  df-subrg 20721  df-lmod 21035  df-lss 21105  df-sra 21346  df-rgmod 21347  df-dsmm 21934  df-frlm 21949  df-mat 22617

This theorem is used by:  decpmatmul  22981  pmatcollpw2  22987