Theorem matgsum 22420

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 7168	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) ∈ V)
3		matgsum.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
4	3	ovexi 7390	. . . 4 ⊢ 𝐴 ∈ V
5	4	a1i 11	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
6		ovexd 7391	. . 3 ⊢ (𝜑 → (𝑅 freeLMod (𝑁 × 𝑁)) ∈ V)
7		matgsum.i	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ Fin)
8		matgsum.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
9		eqid 2739	. . . . . 6 ⊢ (𝑅 freeLMod (𝑁 × 𝑁)) = (𝑅 freeLMod (𝑁 × 𝑁))
10	3, 9	matbas 22396	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘𝐴))
11	7, 8, 10	syl2anc 590	. . . 4 ⊢ (𝜑 → (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘𝐴))
12	11	eqcomd 2745	. . 3 ⊢ (𝜑 → (Base‘𝐴) = (Base‘(𝑅 freeLMod (𝑁 × 𝑁))))
13	3, 9	matplusg 22397	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (+g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (+g‘𝐴))
14	7, 8, 13	syl2anc 590	. . . 4 ⊢ (𝜑 → (+g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (+g‘𝐴))
15	14	eqcomd 2745	. . 3 ⊢ (𝜑 → (+g‘𝐴) = (+g‘(𝑅 freeLMod (𝑁 × 𝑁))))
16	2, 5, 6, 12, 15	gsumpropd 18637	. 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 8007	. . . . . 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 5166	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈)) = (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)))
20	19	oveq2d 7372	. . 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 2739	. . . 4 ⊢ (Base‘(𝑅 freeLMod (𝑁 × 𝑁))) = (Base‘(𝑅 freeLMod (𝑁 × 𝑁)))
22		eqid 2739	. . . 4 ⊢ (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (0g‘(𝑅 freeLMod (𝑁 × 𝑁)))
23		xpfi 9220	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
24	7, 7, 23	syl2anc 590	. . . 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 2849	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 𝑈) ∈ (Base‘𝐴))
28	17	eqcomi 2748	. . . . . 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 516	. . . . . . 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 2854	. . . 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 5168	. . . . . 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 2748	. . . . . 6 ⊢ (0g‘𝐴) = 0
38	34, 35, 37	3brtr4g 5106	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑧 ∈ (𝑁 × 𝑁) ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)) finSupp (0g‘𝐴))
39	3, 9	mat0 22400	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (0g‘𝐴))
40	7, 8, 39	syl2anc 590	. . . . 5 ⊢ (𝜑 → (0g‘(𝑅 freeLMod (𝑁 × 𝑁))) = (0g‘𝐴))
41	38, 40	breqtrrd 5100	. . . 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 21747	. . 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 2774	. 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 6840	. . . . . . . 8 ⊢ (2nd ‘𝑧) ∈ V
45		csbov2g 7404	. . . . . . . 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 3839	. . . . . 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 6840	. . . . . . 7 ⊢ (1st ‘𝑧) ∈ V
49		csbov2g 7404	. . . . . . 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 5500	. . . . . . . . . 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 3839	. . . . . . . 8 ⊢ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈) = ⦋(1st ‘𝑧) / 𝑖⦌(𝑦 ∈ 𝐽 ↦ ⦋(2nd ‘𝑧) / 𝑗⦌𝑈)
54		csbmpt2 5500	. . . . . . . . 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 2762	. . . . . . 7 ⊢ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌(𝑦 ∈ 𝐽 ↦ 𝑈) = (𝑦 ∈ 𝐽 ↦ ⦋(1st ‘𝑧) / 𝑖⦌⦋(2nd ‘𝑧) / 𝑗⦌𝑈)
57	56	oveq2i 7367	. . . . . 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 2767	. . . . 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 5168	. . . 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 8007	. . . 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 2765	. . 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 2778	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 1547   ∈ wcel 2119  Vcvv 3431  ⦋csb 3831   class class class wbr 5072   ↦ cmpt 5153   × cxp 5616  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929  2^nd c2nd 7930  Fincfn 8883   finSupp cfsupp 9264  Basecbs 17170  +_gcplusg 17211  0_gc0g 17393   Σ_g cgsu 17394  Ringcrg 20205   freeLMod cfrlm 21721   Mat cmat 22390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-ot 4564  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-sup 9345  df-oi 9415  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-fzo 13600  df-seq 13955  df-hash 14284  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-hom 17235  df-cco 17236  df-0g 17395  df-gsum 17396  df-prds 17401  df-pws 17403  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-grp 18903  df-minusg 18904  df-sbg 18905  df-subg 19090  df-cntz 19283  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-subrg 20542  df-lmod 20852  df-lss 20922  df-sra 21163  df-rgmod 21164  df-dsmm 21707  df-frlm 21722  df-mat 22391

This theorem is referenced by:  decpmatmul  22755  pmatcollpw2  22761