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Theorem scmatscm 22469

Description: The multiplication of a matrix with a scalar matrix corresponds to a scalar multiplication. (Contributed by AV, 28-Dec-2019.)

**Hypotheses**
Ref	Expression
scmatscm.k	⊢ 𝐾 = (Base‘𝑅)
scmatscm.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
scmatscm.b	⊢ 𝐵 = (Base‘𝐴)
scmatscm.t	⊢ ∗ = ( ·_𝑠 ‘𝐴)
scmatscm.m	⊢ × = (._r‘𝐴)
scmatscm.c	⊢ 𝑆 = (𝑁 ScMat 𝑅)

**Assertion**
Ref	Expression
scmatscm	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 ∀𝑚 ∈ 𝐵 (𝐶 × 𝑚) = (𝑐 ∗ 𝑚))

Distinct variable groups: 𝐴,𝑚 𝐶,𝑐,𝑚 𝐾,𝑐,𝑚 𝑁,𝑐,𝑚 𝑅,𝑐,𝑚 𝑆,𝑐,𝑚 ∗ ,𝑚 Allowed substitution hints: 𝐴(𝑐) 𝐵(𝑚,𝑐) × (𝑚,𝑐) ∗ (𝑐)

Proof of Theorem scmatscm Dummy variables 𝑖 𝑗 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scmatscm.k	. . . 4 ⊢ 𝐾 = (Base‘𝑅)
2		scmatscm.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
3		scmatscm.b	. . . 4 ⊢ 𝐵 = (Base‘𝐴)
4		eqid 2737	. . . 4 ⊢ (1_r‘𝐴) = (1_r‘𝐴)
5		scmatscm.t	. . . 4 ⊢ ∗ = ( ·_𝑠 ‘𝐴)
6		scmatscm.c	. . . 4 ⊢ 𝑆 = (𝑁 ScMat 𝑅)
7	1, 2, 3, 4, 5, 6	scmatscmid 22462	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐶 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 𝐶 = (𝑐 ∗ (1_r‘𝐴)))
8	7	3expa 1119	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 𝐶 = (𝑐 ∗ (1_r‘𝐴)))
9		oveq1 7375	. . . . . 6 ⊢ (𝐶 = (𝑐 ∗ (1_r‘𝐴)) → (𝐶 × 𝑚) = ((𝑐 ∗ (1_r‘𝐴)) × 𝑚))
10		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
11	10	ad4antr 733	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring)
12		simpl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
13	12	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
14	2	matring 22399	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
15	3, 4	ringidcl 20212	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ Ring → (1_r‘𝐴) ∈ 𝐵)
16	14, 15	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐴) ∈ 𝐵)
17	16	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) → (1_r‘𝐴) ∈ 𝐵)
18	17	anim1ci 617	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → (𝑐 ∈ 𝐾 ∧ (1_r‘𝐴) ∈ 𝐵))
19	1, 2, 3, 5	matvscl 22387	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑐 ∈ 𝐾 ∧ (1_r‘𝐴) ∈ 𝐵)) → (𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵)
20	13, 18, 19	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → (𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵)
21	20	anim1i 616	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → ((𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵 ∧ 𝑚 ∈ 𝐵))
22	21	adantr 480	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵 ∧ 𝑚 ∈ 𝐵))
23		simpr 484	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
24		scmatscm.m	. . . . . . . . . . . 12 ⊢ × = (._r‘𝐴)
25	2, 3, 24	matmulcell 22401	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ ((𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵 ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖((𝑐 ∗ (1_r‘𝐴)) × 𝑚)𝑗) = (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖(𝑐 ∗ (1_r‘𝐴))𝑘)(._r‘𝑅)(𝑘𝑚𝑗)))))
26	11, 22, 23, 25	syl3anc 1374	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖((𝑐 ∗ (1_r‘𝐴)) × 𝑚)𝑗) = (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖(𝑐 ∗ (1_r‘𝐴))𝑘)(._r‘𝑅)(𝑘𝑚𝑗)))))
27	12	anim1i 616	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑐 ∈ 𝐾))
28		df-3an 1089	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑐 ∈ 𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑐 ∈ 𝐾))
29	27, 28	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑐 ∈ 𝐾))
30	29	ad3antrrr 731	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑐 ∈ 𝐾))
31		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
32	2, 1, 5, 31	matsc 22406	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑐 ∈ 𝐾) → (𝑐 ∗ (1_r‘𝐴)) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑦, 𝑐, (0_g‘𝑅))))
33	30, 32	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑐 ∗ (1_r‘𝐴)) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ if(𝑥 = 𝑦, 𝑐, (0_g‘𝑅))))
34		eqeq12 2754	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑘) → (𝑥 = 𝑦 ↔ 𝑖 = 𝑘))
35	34	ifbid 4505	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑘) → if(𝑥 = 𝑦, 𝑐, (0_g‘𝑅)) = if(𝑖 = 𝑘, 𝑐, (0_g‘𝑅)))
36	35	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) ∧ (𝑥 = 𝑖 ∧ 𝑦 = 𝑘)) → if(𝑥 = 𝑦, 𝑐, (0_g‘𝑅)) = if(𝑖 = 𝑘, 𝑐, (0_g‘𝑅)))
37		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
38	37	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
39	38	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑖 ∈ 𝑁)
40		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁)
41		vex 3446	. . . . . . . . . . . . . . . . 17 ⊢ 𝑐 ∈ V
42		fvex 6855	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝑅) ∈ V
43	41, 42	ifex 4532	. . . . . . . . . . . . . . . 16 ⊢ if(𝑖 = 𝑘, 𝑐, (0_g‘𝑅)) ∈ V
44	43	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → if(𝑖 = 𝑘, 𝑐, (0_g‘𝑅)) ∈ V)
45	33, 36, 39, 40, 44	ovmpod 7520	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑖(𝑐 ∗ (1_r‘𝐴))𝑘) = if(𝑖 = 𝑘, 𝑐, (0_g‘𝑅)))
46	45	oveq1d 7383	. . . . . . . . . . . . 13 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → ((𝑖(𝑐 ∗ (1_r‘𝐴))𝑘)(._r‘𝑅)(𝑘𝑚𝑗)) = (if(𝑖 = 𝑘, 𝑐, (0_g‘𝑅))(._r‘𝑅)(𝑘𝑚𝑗)))
47	46	mpteq2dva 5193	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑘 ∈ 𝑁 ↦ ((𝑖(𝑐 ∗ (1_r‘𝐴))𝑘)(._r‘𝑅)(𝑘𝑚𝑗))) = (𝑘 ∈ 𝑁 ↦ (if(𝑖 = 𝑘, 𝑐, (0_g‘𝑅))(._r‘𝑅)(𝑘𝑚𝑗))))
48	47	oveq2d 7384	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖(𝑐 ∗ (1_r‘𝐴))𝑘)(._r‘𝑅)(𝑘𝑚𝑗)))) = (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ (if(𝑖 = 𝑘, 𝑐, (0_g‘𝑅))(._r‘𝑅)(𝑘𝑚𝑗)))))
49		ovif 7466	. . . . . . . . . . . . . 14 ⊢ (if(𝑖 = 𝑘, 𝑐, (0_g‘𝑅))(._r‘𝑅)(𝑘𝑚𝑗)) = if(𝑖 = 𝑘, (𝑐(._r‘𝑅)(𝑘𝑚𝑗)), ((0_g‘𝑅)(._r‘𝑅)(𝑘𝑚𝑗)))
50		simp-6r 788	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑅 ∈ Ring)
51		simplrr 778	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑗 ∈ 𝑁)
52		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → 𝑚 ∈ 𝐵)
53	52	ad2antrr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑚 ∈ 𝐵)
54	2, 1, 3, 40, 51, 53	matecld 22382	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑚𝑗) ∈ 𝐾)
55		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑅) = (._r‘𝑅)
56	1, 55, 31	ringlz 20240	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ (𝑘𝑚𝑗) ∈ 𝐾) → ((0_g‘𝑅)(._r‘𝑅)(𝑘𝑚𝑗)) = (0_g‘𝑅))
57	50, 54, 56	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → ((0_g‘𝑅)(._r‘𝑅)(𝑘𝑚𝑗)) = (0_g‘𝑅))
58	57	ifeq2d 4502	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → if(𝑖 = 𝑘, (𝑐(._r‘𝑅)(𝑘𝑚𝑗)), ((0_g‘𝑅)(._r‘𝑅)(𝑘𝑚𝑗))) = if(𝑖 = 𝑘, (𝑐(._r‘𝑅)(𝑘𝑚𝑗)), (0_g‘𝑅)))
59	49, 58	eqtrid 2784	. . . . . . . . . . . . 13 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (if(𝑖 = 𝑘, 𝑐, (0_g‘𝑅))(._r‘𝑅)(𝑘𝑚𝑗)) = if(𝑖 = 𝑘, (𝑐(._r‘𝑅)(𝑘𝑚𝑗)), (0_g‘𝑅)))
60	59	mpteq2dva 5193	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑘 ∈ 𝑁 ↦ (if(𝑖 = 𝑘, 𝑐, (0_g‘𝑅))(._r‘𝑅)(𝑘𝑚𝑗))) = (𝑘 ∈ 𝑁 ↦ if(𝑖 = 𝑘, (𝑐(._r‘𝑅)(𝑘𝑚𝑗)), (0_g‘𝑅))))
61	60	oveq2d 7384	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ (if(𝑖 = 𝑘, 𝑐, (0_g‘𝑅))(._r‘𝑅)(𝑘𝑚𝑗)))) = (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ if(𝑖 = 𝑘, (𝑐(._r‘𝑅)(𝑘𝑚𝑗)), (0_g‘𝑅)))))
62		ringmnd 20190	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
63	62	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Mnd)
64	63	ad4antr 733	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Mnd)
65		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑁 ∈ Fin)
66	65	ad4antr 733	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑁 ∈ Fin)
67		equcom 2020	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑘 ↔ 𝑘 = 𝑖)
68		ifbi 4504	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = 𝑘 ↔ 𝑘 = 𝑖) → if(𝑖 = 𝑘, (𝑐(._r‘𝑅)(𝑘𝑚𝑗)), (0_g‘𝑅)) = if(𝑘 = 𝑖, (𝑐(._r‘𝑅)(𝑘𝑚𝑗)), (0_g‘𝑅)))
69	67, 68	ax-mp 5	. . . . . . . . . . . . 13 ⊢ if(𝑖 = 𝑘, (𝑐(._r‘𝑅)(𝑘𝑚𝑗)), (0_g‘𝑅)) = if(𝑘 = 𝑖, (𝑐(._r‘𝑅)(𝑘𝑚𝑗)), (0_g‘𝑅))
70	69	mpteq2i 5196	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝑁 ↦ if(𝑖 = 𝑘, (𝑐(._r‘𝑅)(𝑘𝑚𝑗)), (0_g‘𝑅))) = (𝑘 ∈ 𝑁 ↦ if(𝑘 = 𝑖, (𝑐(._r‘𝑅)(𝑘𝑚𝑗)), (0_g‘𝑅)))
71	1	eleq2i 2829	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ 𝐾 ↔ 𝑐 ∈ (Base‘𝑅))
72	71	biimpi 216	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ 𝐾 → 𝑐 ∈ (Base‘𝑅))
73	72	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → 𝑐 ∈ (Base‘𝑅))
74	73	ad3antrrr 731	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑐 ∈ (Base‘𝑅))
75		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑅) = (Base‘𝑅)
76	2, 75, 3, 40, 51, 53	matecld 22382	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑚𝑗) ∈ (Base‘𝑅))
77	75, 55	ringcl 20197	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝑐 ∈ (Base‘𝑅) ∧ (𝑘𝑚𝑗) ∈ (Base‘𝑅)) → (𝑐(._r‘𝑅)(𝑘𝑚𝑗)) ∈ (Base‘𝑅))
78	50, 74, 76, 77	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑐(._r‘𝑅)(𝑘𝑚𝑗)) ∈ (Base‘𝑅))
79	78	ralrimiva 3130	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ∀𝑘 ∈ 𝑁 (𝑐(._r‘𝑅)(𝑘𝑚𝑗)) ∈ (Base‘𝑅))
80	31, 64, 66, 38, 70, 79	gsummpt1n0 19906	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ if(𝑖 = 𝑘, (𝑐(._r‘𝑅)(𝑘𝑚𝑗)), (0_g‘𝑅)))) = ⦋𝑖 / 𝑘⦌(𝑐(._r‘𝑅)(𝑘𝑚𝑗)))
81	48, 61, 80	3eqtrd 2776	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑅 Σ_g (𝑘 ∈ 𝑁 ↦ ((𝑖(𝑐 ∗ (1_r‘𝐴))𝑘)(._r‘𝑅)(𝑘𝑚𝑗)))) = ⦋𝑖 / 𝑘⦌(𝑐(._r‘𝑅)(𝑘𝑚𝑗)))
82		csbov2g 7416	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌(𝑐(._r‘𝑅)(𝑘𝑚𝑗)) = (𝑐(._r‘𝑅)⦋𝑖 / 𝑘⦌(𝑘𝑚𝑗)))
83		csbov1g 7415	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌(𝑘𝑚𝑗) = (⦋𝑖 / 𝑘⦌𝑘𝑚𝑗))
84		csbvarg 4388	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌𝑘 = 𝑖)
85	84	oveq1d 7383	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑁 → (⦋𝑖 / 𝑘⦌𝑘𝑚𝑗) = (𝑖𝑚𝑗))
86	83, 85	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌(𝑘𝑚𝑗) = (𝑖𝑚𝑗))
87	86	oveq2d 7384	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝑁 → (𝑐(._r‘𝑅)⦋𝑖 / 𝑘⦌(𝑘𝑚𝑗)) = (𝑐(._r‘𝑅)(𝑖𝑚𝑗)))
88	82, 87	eqtrd 2772	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌(𝑐(._r‘𝑅)(𝑘𝑚𝑗)) = (𝑐(._r‘𝑅)(𝑖𝑚𝑗)))
89	88	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ⦋𝑖 / 𝑘⦌(𝑐(._r‘𝑅)(𝑘𝑚𝑗)) = (𝑐(._r‘𝑅)(𝑖𝑚𝑗)))
90	89	adantl 481	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ⦋𝑖 / 𝑘⦌(𝑐(._r‘𝑅)(𝑘𝑚𝑗)) = (𝑐(._r‘𝑅)(𝑖𝑚𝑗)))
91	26, 81, 90	3eqtrd 2776	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖((𝑐 ∗ (1_r‘𝐴)) × 𝑚)𝑗) = (𝑐(._r‘𝑅)(𝑖𝑚𝑗)))
92		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → 𝑐 ∈ 𝐾)
93	92	anim1i 616	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → (𝑐 ∈ 𝐾 ∧ 𝑚 ∈ 𝐵))
94	93	adantr 480	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑐 ∈ 𝐾 ∧ 𝑚 ∈ 𝐵))
95	2, 3, 1, 5, 55	matvscacell 22392	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ (𝑐 ∈ 𝐾 ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑐 ∗ 𝑚)𝑗) = (𝑐(._r‘𝑅)(𝑖𝑚𝑗)))
96	11, 94, 23, 95	syl3anc 1374	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑐 ∗ 𝑚)𝑗) = (𝑐(._r‘𝑅)(𝑖𝑚𝑗)))
97	91, 96	eqtr4d 2775	. . . . . . . 8 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖((𝑐 ∗ (1_r‘𝐴)) × 𝑚)𝑗) = (𝑖(𝑐 ∗ 𝑚)𝑗))
98	97	ralrimivva 3181	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖((𝑐 ∗ (1_r‘𝐴)) × 𝑚)𝑗) = (𝑖(𝑐 ∗ 𝑚)𝑗))
99	14	ad3antrrr 731	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → 𝐴 ∈ Ring)
100	20	adantr 480	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → (𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵)
101	3, 24	ringcl 20197	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ring ∧ (𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵 ∧ 𝑚 ∈ 𝐵) → ((𝑐 ∗ (1_r‘𝐴)) × 𝑚) ∈ 𝐵)
102	99, 100, 52, 101	syl3anc 1374	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → ((𝑐 ∗ (1_r‘𝐴)) × 𝑚) ∈ 𝐵)
103	12	ad2antrr 727	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
104	1, 2, 3, 5	matvscl 22387	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑐 ∈ 𝐾 ∧ 𝑚 ∈ 𝐵)) → (𝑐 ∗ 𝑚) ∈ 𝐵)
105	103, 93, 104	syl2anc 585	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → (𝑐 ∗ 𝑚) ∈ 𝐵)
106	2, 3	eqmat 22380	. . . . . . . 8 ⊢ ((((𝑐 ∗ (1_r‘𝐴)) × 𝑚) ∈ 𝐵 ∧ (𝑐 ∗ 𝑚) ∈ 𝐵) → (((𝑐 ∗ (1_r‘𝐴)) × 𝑚) = (𝑐 ∗ 𝑚) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖((𝑐 ∗ (1_r‘𝐴)) × 𝑚)𝑗) = (𝑖(𝑐 ∗ 𝑚)𝑗)))
107	102, 105, 106	syl2anc 585	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → (((𝑐 ∗ (1_r‘𝐴)) × 𝑚) = (𝑐 ∗ 𝑚) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖((𝑐 ∗ (1_r‘𝐴)) × 𝑚)𝑗) = (𝑖(𝑐 ∗ 𝑚)𝑗)))
108	98, 107	mpbird 257	. . . . . 6 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → ((𝑐 ∗ (1_r‘𝐴)) × 𝑚) = (𝑐 ∗ 𝑚))
109	9, 108	sylan9eqr 2794	. . . . 5 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ 𝐶 = (𝑐 ∗ (1_r‘𝐴))) → (𝐶 × 𝑚) = (𝑐 ∗ 𝑚))
110	109	ex 412	. . . 4 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → (𝐶 = (𝑐 ∗ (1_r‘𝐴)) → (𝐶 × 𝑚) = (𝑐 ∗ 𝑚)))
111	110	ralrimdva 3138	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → (𝐶 = (𝑐 ∗ (1_r‘𝐴)) → ∀𝑚 ∈ 𝐵 (𝐶 × 𝑚) = (𝑐 ∗ 𝑚)))
112	111	reximdva 3151	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) → (∃𝑐 ∈ 𝐾 𝐶 = (𝑐 ∗ (1_r‘𝐴)) → ∃𝑐 ∈ 𝐾 ∀𝑚 ∈ 𝐵 (𝐶 × 𝑚) = (𝑐 ∗ 𝑚)))
113	8, 112	mpd 15	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 ∀𝑚 ∈ 𝐵 (𝐶 × 𝑚) = (𝑐 ∗ 𝑚))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 Vcvv 3442 ⦋csb 3851 ifcif 4481 ↦ cmpt 5181 ‘cfv 6500 (class class class)co 7368 ∈ cmpo 7370 Fincfn 8895 Basecbs 17148 ._rcmulr 17190 ·_𝑠 cvsca 17193 0_gc0g 17371 Σ_g cgsu 17372 Mndcmnd 18671 1_rcur 20128 Ringcrg 20180 Mat cmat 22363 ScMat cscmat 22445

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-ot 4591 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 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 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-fzo 13583 df-seq 13937 df-hash 14266 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-hom 17213 df-cco 17214 df-0g 17373 df-gsum 17374 df-prds 17379 df-pws 17381 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19010 df-subg 19065 df-ghm 19154 df-cntz 19258 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-subrg 20515 df-lmod 20825 df-lss 20895 df-sra 21137 df-rgmod 21138 df-dsmm 21699 df-frlm 21714 df-mamu 22347 df-mat 22364 df-scmat 22447

This theorem is referenced by: (None)

W3C validator