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Theorem scmatscm 22858

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 2763	. . . 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 22851	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐶 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 𝐶 = (𝑐 ∗ (1_r‘𝐴)))
8	7	3expa 1136	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 𝐶 = (𝑐 ∗ (1_r‘𝐴)))
9		oveq1 7450	. . . . . 6 ⊢ (𝐶 = (𝑐 ∗ (1_r‘𝐴)) → (𝐶 × 𝑚) = ((𝑐 ∗ (1_r‘𝐴)) × 𝑚))
10		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
11	10	ad4antr 745	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring)
12		simpl 488	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
13	12	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
14	2	matring 22788	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
15	3, 4	ringidcl 20524	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ Ring → (1_r‘𝐴) ∈ 𝐵)
16	14, 15	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐴) ∈ 𝐵)
17	16	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) → (1_r‘𝐴) ∈ 𝐵)
18	17	anim1ci 628	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → (𝑐 ∈ 𝐾 ∧ (1_r‘𝐴) ∈ 𝐵))
19	1, 2, 3, 5	matvscl 22776	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑐 ∈ 𝐾 ∧ (1_r‘𝐴) ∈ 𝐵)) → (𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵)
20	13, 18, 19	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → (𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵)
21	20	anim1i 627	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → ((𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵 ∧ 𝑚 ∈ 𝐵))
22	21	adantr 486	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵 ∧ 𝑚 ∈ 𝐵))
23		simpr 490	. . . . . . . . . . 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 22790	. . . . . . . . . . 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 1399	. . . . . . . . . 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 627	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑐 ∈ 𝐾))
28		df-3an 1105	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑐 ∈ 𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑐 ∈ 𝐾))
29	27, 28	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑐 ∈ 𝐾))
30	29	ad3antrrr 743	. . . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
32	2, 1, 5, 31	matsc 22795	. . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑘) → (𝑥 = 𝑦 ↔ 𝑖 = 𝑘))
35	34	ifbid 4509	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑘) → if(𝑥 = 𝑦, 𝑐, (0_g‘𝑅)) = if(𝑖 = 𝑘, 𝑐, (0_g‘𝑅)))
36	35	adantl 487	. . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
38	37	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
39	38	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑖 ∈ 𝑁)
40		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁)
41		vex 3458	. . . . . . . . . . . . . . . . 17 ⊢ 𝑐 ∈ V
42		fvex 6921	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝑅) ∈ V
43	41, 42	ifex 4536	. . . . . . . . . . . . . . . 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 7596	. . . . . . . . . . . . . 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 7458	. . . . . . . . . . . . 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 5201	. . . . . . . . . . . 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 7459	. . . . . . . . . . 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 7541	. . . . . . . . . . . . . 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 800	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑅 ∈ Ring)
51		simplrr 790	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑗 ∈ 𝑁)
52		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → 𝑚 ∈ 𝐵)
53	52	ad2antrr 739	. . . . . . . . . . . . . . . . 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 22771	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑚𝑗) ∈ 𝐾)
55		eqid 2763	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑅) = (._r‘𝑅)
56	1, 55, 31	ringlz 20554	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ (𝑘𝑚𝑗) ∈ 𝐾) → ((0_g‘𝑅)(._r‘𝑅)(𝑘𝑚𝑗)) = (0_g‘𝑅))
57	50, 54, 56	syl2anc 596	. . . . . . . . . . . . . . 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 4506	. . . . . . . . . . . . . 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 2810	. . . . . . . . . . . . 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 5201	. . . . . . . . . . . 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 7459	. . . . . . . . . . 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 20500	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
63	62	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Mnd)
64	63	ad4antr 745	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Mnd)
65		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑁 ∈ Fin)
66	65	ad4antr 745	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑁 ∈ Fin)
67		equcom 2053	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑘 ↔ 𝑘 = 𝑖)
68		ifbi 4508	. . . . . . . . . . . . . 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 5204	. . . . . . . . . . . 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 2856	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ 𝐾 ↔ 𝑐 ∈ (Base‘𝑅))
72	71	bilani 510	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → 𝑐 ∈ (Base‘𝑅))
73	72	ad3antrrr 743	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑐 ∈ (Base‘𝑅))
74		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑅) = (Base‘𝑅)
75	2, 74, 3, 40, 51, 53	matecld 22771	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑚𝑗) ∈ (Base‘𝑅))
76	74, 55	ringcl 20507	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝑐 ∈ (Base‘𝑅) ∧ (𝑘𝑚𝑗) ∈ (Base‘𝑅)) → (𝑐(._r‘𝑅)(𝑘𝑚𝑗)) ∈ (Base‘𝑅))
77	50, 73, 75, 76	syl3anc 1399	. . . . . . . . . . . . 13 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑐(._r‘𝑅)(𝑘𝑚𝑗)) ∈ (Base‘𝑅))
78	77	ralrimiva 3158	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ∀𝑘 ∈ 𝑁 (𝑐(._r‘𝑅)(𝑘𝑚𝑗)) ∈ (Base‘𝑅))
79	31, 64, 66, 38, 70, 78	gsummpt1n0 20209	. . . . . . . . . . 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)))
80	48, 61, 79	3eqtrd 2802	. . . . . . . . . 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)))
81		csbov2g 7491	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌(𝑐(._r‘𝑅)(𝑘𝑚𝑗)) = (𝑐(._r‘𝑅)⦋𝑖 / 𝑘⦌(𝑘𝑚𝑗)))
82		csbov1g 7490	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌(𝑘𝑚𝑗) = (⦋𝑖 / 𝑘⦌𝑘𝑚𝑗))
83		csbvarg 4395	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌𝑘 = 𝑖)
84	83	oveq1d 7458	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑁 → (⦋𝑖 / 𝑘⦌𝑘𝑚𝑗) = (𝑖𝑚𝑗))
85	82, 84	eqtrd 2798	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌(𝑘𝑚𝑗) = (𝑖𝑚𝑗))
86	85	oveq2d 7459	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝑁 → (𝑐(._r‘𝑅)⦋𝑖 / 𝑘⦌(𝑘𝑚𝑗)) = (𝑐(._r‘𝑅)(𝑖𝑚𝑗)))
87	81, 86	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌(𝑐(._r‘𝑅)(𝑘𝑚𝑗)) = (𝑐(._r‘𝑅)(𝑖𝑚𝑗)))
88	87	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ⦋𝑖 / 𝑘⦌(𝑐(._r‘𝑅)(𝑘𝑚𝑗)) = (𝑐(._r‘𝑅)(𝑖𝑚𝑗)))
89	88	adantl 487	. . . . . . . . . 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)))
90	26, 80, 89	3eqtrd 2802	. . . . . . . . 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)))
91		simpr 490	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → 𝑐 ∈ 𝐾)
92	91	anim1i 627	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → (𝑐 ∈ 𝐾 ∧ 𝑚 ∈ 𝐵))
93	92	adantr 486	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑐 ∈ 𝐾 ∧ 𝑚 ∈ 𝐵))
94	2, 3, 1, 5, 55	matvscacell 22781	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ (𝑐 ∈ 𝐾 ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑐 ∗ 𝑚)𝑗) = (𝑐(._r‘𝑅)(𝑖𝑚𝑗)))
95	11, 93, 23, 94	syl3anc 1399	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑐 ∗ 𝑚)𝑗) = (𝑐(._r‘𝑅)(𝑖𝑚𝑗)))
96	90, 95	eqtr4d 2801	. . . . . . . 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))
97	96	ralrimivva 3209	. . . . . . 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))
98	14	ad3antrrr 743	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → 𝐴 ∈ Ring)
99	20	adantr 486	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → (𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵)
100	3, 24	ringcl 20507	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ring ∧ (𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵 ∧ 𝑚 ∈ 𝐵) → ((𝑐 ∗ (1_r‘𝐴)) × 𝑚) ∈ 𝐵)
101	98, 99, 52, 100	syl3anc 1399	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → ((𝑐 ∗ (1_r‘𝐴)) × 𝑚) ∈ 𝐵)
102	12	ad2antrr 739	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
103	1, 2, 3, 5	matvscl 22776	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑐 ∈ 𝐾 ∧ 𝑚 ∈ 𝐵)) → (𝑐 ∗ 𝑚) ∈ 𝐵)
104	102, 92, 103	syl2anc 596	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → (𝑐 ∗ 𝑚) ∈ 𝐵)
105	2, 3	eqmat 22769	. . . . . . . 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)))
106	101, 104, 105	syl2anc 596	. . . . . . 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)))
107	97, 106	mpbird 260	. . . . . 6 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → ((𝑐 ∗ (1_r‘𝐴)) × 𝑚) = (𝑐 ∗ 𝑚))
108	9, 107	sylan9eqr 2820	. . . . 5 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ 𝐶 = (𝑐 ∗ (1_r‘𝐴))) → (𝐶 × 𝑚) = (𝑐 ∗ 𝑚))
109	108	ex 418	. . . 4 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → (𝐶 = (𝑐 ∗ (1_r‘𝐴)) → (𝐶 × 𝑚) = (𝑐 ∗ 𝑚)))
110	109	ralrimdva 3166	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → (𝐶 = (𝑐 ∗ (1_r‘𝐴)) → ∀𝑚 ∈ 𝐵 (𝐶 × 𝑚) = (𝑐 ∗ 𝑚)))
111	110	reximdva 3179	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) → (∃𝑐 ∈ 𝐾 𝐶 = (𝑐 ∗ (1_r‘𝐴)) → ∃𝑐 ∈ 𝐾 ∀𝑚 ∈ 𝐵 (𝐶 × 𝑚) = (𝑐 ∗ 𝑚)))
112	8, 111	mpd 16	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 ∀𝑚 ∈ 𝐵 (𝐶 × 𝑚) = (𝑐 ∗ 𝑚))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ∃wrex 3090 Vcvv 3454 ⦋csb 3851 ifcif 4485 ↦ cmpt 5189 ‘cfv 6558 (class class class)co 7443 ∈ cmpo 7445 Fincfn 9001 Basecbs 17416 ._rcmulr 17458 ·_𝑠 cvsca 17461 0_gc0g 17639 Σ_g cgsu 17640 Mndcmnd 18952 1_rcur 20437 Ringcrg 20489 Mat cmat 22752 ScMat cscmat 22834

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-ot 4862 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-of 7717 df-om 7902 df-1st 8025 df-2nd 8026 df-supp 8205 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-er 8745 df-map 8877 df-ixp 8954 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-fsupp 9382 df-sup 9461 df-oi 9531 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-z 12721 df-dec 12842 df-uz 12993 df-fz 13667 df-fzo 13816 df-seq 14172 df-hash 14502 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-sca 17473 df-vsca 17474 df-ip 17475 df-tset 17476 df-ple 17477 df-ds 17479 df-hom 17481 df-cco 17482 df-0g 17641 df-gsum 17642 df-prds 17647 df-pws 17649 df-mre 17785 df-mrc 17786 df-acs 17788 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-mhm 19007 df-submnd 19008 df-grp 19176 df-minusg 19177 df-sbg 19178 df-mulg 19307 df-subg 19362 df-ghm 19457 df-cntz 19561 df-cmn 20026 df-abl 20027 df-mgp 20391 df-rng 20405 df-ur 20438 df-ring 20491 df-subrg 20852 df-lmod 21167 df-lss 21237 df-sra 21478 df-rgmod 21479 df-dsmm 22068 df-frlm 22083 df-mamu 22736 df-mat 22753 df-scmat 22836

This theorem is used by: (None)

W3C validator