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Theorem scmatscm 22735

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 2760	. . . 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 22728	. . 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 7420	. . . . . 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 22665	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
15	3, 4	ringidcl 20406	. . . . . . . . . . . . . . . . 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 22653	. . . . . . . . . . . . . 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 22667	. . . . . . . . . . 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 1398	. . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
32	2, 1, 5, 31	matsc 22672	. . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑘) → (𝑥 = 𝑦 ↔ 𝑖 = 𝑘))
35	34	ifbid 4506	. . . . . . . . . . . . . . . 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 3454	. . . . . . . . . . . . . . . . 17 ⊢ 𝑐 ∈ V
42		fvex 6891	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝑅) ∈ V
43	41, 42	ifex 4533	. . . . . . . . . . . . . . . 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 7565	. . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . 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 5198	. . . . . . . . . . . 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 7429	. . . . . . . . . . 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 7511	. . . . . . . . . . . . . 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 22648	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑚𝑗) ∈ 𝐾)
55		eqid 2760	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑅) = (._r‘𝑅)
56	1, 55, 31	ringlz 20435	. . . . . . . . . . . . . . . 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 4503	. . . . . . . . . . . . . 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 2807	. . . . . . . . . . . . 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 5198	. . . . . . . . . . . 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 7429	. . . . . . . . . . 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 20382	. . . . . . . . . . . . . 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 2051	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑘 ↔ 𝑘 = 𝑖)
68		ifbi 4505	. . . . . . . . . . . . . 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 5201	. . . . . . . . . . . 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 2852	. . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑅) = (Base‘𝑅)
75	2, 74, 3, 40, 51, 53	matecld 22648	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑚𝑗) ∈ (Base‘𝑅))
76	74, 55	ringcl 20389	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝑐 ∈ (Base‘𝑅) ∧ (𝑘𝑚𝑗) ∈ (Base‘𝑅)) → (𝑐(._r‘𝑅)(𝑘𝑚𝑗)) ∈ (Base‘𝑅))
77	50, 73, 75, 76	syl3anc 1398	. . . . . . . . . . . . 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 3154	. . . . . . . . . . . 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 20092	. . . . . . . . . . 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 2799	. . . . . . . . . 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 7461	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌(𝑐(._r‘𝑅)(𝑘𝑚𝑗)) = (𝑐(._r‘𝑅)⦋𝑖 / 𝑘⦌(𝑘𝑚𝑗)))
82		csbov1g 7460	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌(𝑘𝑚𝑗) = (⦋𝑖 / 𝑘⦌𝑘𝑚𝑗))
83		csbvarg 4392	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌𝑘 = 𝑖)
84	83	oveq1d 7428	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑁 → (⦋𝑖 / 𝑘⦌𝑘𝑚𝑗) = (𝑖𝑚𝑗))
85	82, 84	eqtrd 2795	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌(𝑘𝑚𝑗) = (𝑖𝑚𝑗))
86	85	oveq2d 7429	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝑁 → (𝑐(._r‘𝑅)⦋𝑖 / 𝑘⦌(𝑘𝑚𝑗)) = (𝑐(._r‘𝑅)(𝑖𝑚𝑗)))
87	81, 86	eqtrd 2795	. . . . . . . . . . . 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 2799	. . . . . . . . 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 22658	. . . . . . . . . 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 1398	. . . . . . . . 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 2798	. . . . . . . 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 3205	. . . . . . 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 20389	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ring ∧ (𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵 ∧ 𝑚 ∈ 𝐵) → ((𝑐 ∗ (1_r‘𝐴)) × 𝑚) ∈ 𝐵)
101	98, 99, 52, 100	syl3anc 1398	. . . . . . . 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 22653	. . . . . . . . 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 22646	. . . . . . . 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 2817	. . . . 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 3162	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → (𝐶 = (𝑐 ∗ (1_r‘𝐴)) → ∀𝑚 ∈ 𝐵 (𝐶 × 𝑚) = (𝑐 ∗ 𝑚)))
111	110	reximdva 3175	. 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 2145 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ⦋csb 3847 ifcif 4482 ↦ cmpt 5186 ‘cfv 6533 (class class class)co 7413 ∈ cmpo 7415 Fincfn 8952 Basecbs 17301 ._rcmulr 17343 ·_𝑠 cvsca 17346 0_gc0g 17524 Σ_g cgsu 17525 Mndcmnd 18836 1_rcur 20320 Ringcrg 20372 Mat cmat 22629 ScMat cscmat 22711

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 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 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-sup 9412 df-oi 9482 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-fz 13562 df-fzo 13710 df-seq 14066 df-hash 14395 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-hom 17366 df-cco 17367 df-0g 17526 df-gsum 17527 df-prds 17532 df-pws 17534 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-mhm 18891 df-submnd 18892 df-grp 19060 df-minusg 19061 df-sbg 19062 df-mulg 19191 df-subg 19246 df-ghm 19341 df-cntz 19444 df-cmn 19909 df-abl 19910 df-mgp 20274 df-rng 20288 df-ur 20321 df-ring 20374 df-subrg 20732 df-lmod 21046 df-lss 21116 df-sra 21357 df-rgmod 21358 df-dsmm 21945 df-frlm 21960 df-mamu 22613 df-mat 22630 df-scmat 22713

This theorem is used by: (None)

W3C validator