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Theorem scmatscm 22551

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 2761	. . . 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 22544	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐶 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 𝐶 = (𝑐 ∗ (1_r‘𝐴)))
8	7	3expa 1130	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 𝐶 = (𝑐 ∗ (1_r‘𝐴)))
9		oveq1 7397	. . . . . 6 ⊢ (𝐶 = (𝑐 ∗ (1_r‘𝐴)) → (𝐶 × 𝑚) = ((𝑐 ∗ (1_r‘𝐴)) × 𝑚))
10		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
11	10	ad4antr 742	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Ring)
12		simpl 486	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
13	12	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
14	2	matring 22481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
15	3, 4	ringidcl 20292	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ Ring → (1_r‘𝐴) ∈ 𝐵)
16	14, 15	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐴) ∈ 𝐵)
17	16	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) → (1_r‘𝐴) ∈ 𝐵)
18	17	anim1ci 625	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → (𝑐 ∈ 𝐾 ∧ (1_r‘𝐴) ∈ 𝐵))
19	1, 2, 3, 5	matvscl 22469	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑐 ∈ 𝐾 ∧ (1_r‘𝐴) ∈ 𝐵)) → (𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵)
20	13, 18, 19	syl2anc 593	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → (𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵)
21	20	anim1i 624	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → ((𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵 ∧ 𝑚 ∈ 𝐵))
22	21	adantr 484	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → ((𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵 ∧ 𝑚 ∈ 𝐵))
23		simpr 488	. . . . . . . . . . 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 22483	. . . . . . . . . . 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 1389	. . . . . . . . . 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 624	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑐 ∈ 𝐾))
28		df-3an 1099	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑐 ∈ 𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑐 ∈ 𝐾))
29	27, 28	sylibr 236	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑐 ∈ 𝐾))
30	29	ad3antrrr 740	. . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
32	2, 1, 5, 31	matsc 22488	. . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑘) → (𝑥 = 𝑦 ↔ 𝑖 = 𝑘))
35	34	ifbid 4503	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑘) → if(𝑥 = 𝑦, 𝑐, (0_g‘𝑅)) = if(𝑖 = 𝑘, 𝑐, (0_g‘𝑅)))
36	35	adantl 485	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
38	37	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑖 ∈ 𝑁)
39	38	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑖 ∈ 𝑁)
40		simpr 488	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑘 ∈ 𝑁)
41		vex 3457	. . . . . . . . . . . . . . . . 17 ⊢ 𝑐 ∈ V
42		fvex 6874	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘𝑅) ∈ V
43	41, 42	ifex 4530	. . . . . . . . . . . . . . . 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 7542	. . . . . . . . . . . . . 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 7405	. . . . . . . . . . . . 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 5192	. . . . . . . . . . . 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 7406	. . . . . . . . . . 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 7488	. . . . . . . . . . . . . 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 797	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑅 ∈ Ring)
51		simplrr 787	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑗 ∈ 𝑁)
52		simpr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → 𝑚 ∈ 𝐵)
53	52	ad2antrr 736	. . . . . . . . . . . . . . . . 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 22464	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑚𝑗) ∈ 𝐾)
55		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (._r‘𝑅) = (._r‘𝑅)
56	1, 55, 31	ringlz 20320	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Ring ∧ (𝑘𝑚𝑗) ∈ 𝐾) → ((0_g‘𝑅)(._r‘𝑅)(𝑘𝑚𝑗)) = (0_g‘𝑅))
57	50, 54, 56	syl2anc 593	. . . . . . . . . . . . . . 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 4500	. . . . . . . . . . . . . 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 2808	. . . . . . . . . . . . 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 5192	. . . . . . . . . . . 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 7406	. . . . . . . . . . 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 20270	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
63	62	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Mnd)
64	63	ad4antr 742	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑅 ∈ Mnd)
65		simpl 486	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑁 ∈ Fin)
66	65	ad4antr 742	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → 𝑁 ∈ Fin)
67		equcom 2037	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑘 ↔ 𝑘 = 𝑖)
68		ifbi 4502	. . . . . . . . . . . . . 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 5195	. . . . . . . . . . . 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 2853	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ 𝐾 ↔ 𝑐 ∈ (Base‘𝑅))
72	71	bilani 508	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → 𝑐 ∈ (Base‘𝑅))
73	72	ad3antrrr 740	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → 𝑐 ∈ (Base‘𝑅))
74		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑅) = (Base‘𝑅)
75	2, 74, 3, 40, 51, 53	matecld 22464	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) ∧ 𝑘 ∈ 𝑁) → (𝑘𝑚𝑗) ∈ (Base‘𝑅))
76	74, 55	ringcl 20277	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝑐 ∈ (Base‘𝑅) ∧ (𝑘𝑚𝑗) ∈ (Base‘𝑅)) → (𝑐(._r‘𝑅)(𝑘𝑚𝑗)) ∈ (Base‘𝑅))
77	50, 73, 75, 76	syl3anc 1389	. . . . . . . . . . . . 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 3153	. . . . . . . . . . . 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 19986	. . . . . . . . . . 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 2800	. . . . . . . . . 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 7438	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌(𝑐(._r‘𝑅)(𝑘𝑚𝑗)) = (𝑐(._r‘𝑅)⦋𝑖 / 𝑘⦌(𝑘𝑚𝑗)))
82		csbov1g 7437	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌(𝑘𝑚𝑗) = (⦋𝑖 / 𝑘⦌𝑘𝑚𝑗))
83		csbvarg 4387	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌𝑘 = 𝑖)
84	83	oveq1d 7405	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ 𝑁 → (⦋𝑖 / 𝑘⦌𝑘𝑚𝑗) = (𝑖𝑚𝑗))
85	82, 84	eqtrd 2796	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌(𝑘𝑚𝑗) = (𝑖𝑚𝑗))
86	85	oveq2d 7406	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝑁 → (𝑐(._r‘𝑅)⦋𝑖 / 𝑘⦌(𝑘𝑚𝑗)) = (𝑐(._r‘𝑅)(𝑖𝑚𝑗)))
87	81, 86	eqtrd 2796	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝑁 → ⦋𝑖 / 𝑘⦌(𝑐(._r‘𝑅)(𝑘𝑚𝑗)) = (𝑐(._r‘𝑅)(𝑖𝑚𝑗)))
88	87	adantr 484	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ⦋𝑖 / 𝑘⦌(𝑐(._r‘𝑅)(𝑘𝑚𝑗)) = (𝑐(._r‘𝑅)(𝑖𝑚𝑗)))
89	88	adantl 485	. . . . . . . . . 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 2800	. . . . . . . . 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 488	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → 𝑐 ∈ 𝐾)
92	91	anim1i 624	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → (𝑐 ∈ 𝐾 ∧ 𝑚 ∈ 𝐵))
93	92	adantr 484	. . . . . . . . . 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 22474	. . . . . . . . . 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 1389	. . . . . . . . 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 2799	. . . . . . . 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 3204	. . . . . . 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 740	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → 𝐴 ∈ Ring)
99	20	adantr 484	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → (𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵)
100	3, 24	ringcl 20277	. . . . . . . . 9 ⊢ ((𝐴 ∈ Ring ∧ (𝑐 ∗ (1_r‘𝐴)) ∈ 𝐵 ∧ 𝑚 ∈ 𝐵) → ((𝑐 ∗ (1_r‘𝐴)) × 𝑚) ∈ 𝐵)
101	98, 99, 52, 100	syl3anc 1389	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → ((𝑐 ∗ (1_r‘𝐴)) × 𝑚) ∈ 𝐵)
102	12	ad2antrr 736	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
103	1, 2, 3, 5	matvscl 22469	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑐 ∈ 𝐾 ∧ 𝑚 ∈ 𝐵)) → (𝑐 ∗ 𝑚) ∈ 𝐵)
104	102, 92, 103	syl2anc 593	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → (𝑐 ∗ 𝑚) ∈ 𝐵)
105	2, 3	eqmat 22462	. . . . . . . 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 593	. . . . . . 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 259	. . . . . 6 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → ((𝑐 ∗ (1_r‘𝐴)) × 𝑚) = (𝑐 ∗ 𝑚))
108	9, 107	sylan9eqr 2818	. . . . 5 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) ∧ 𝐶 = (𝑐 ∗ (1_r‘𝐴))) → (𝐶 × 𝑚) = (𝑐 ∗ 𝑚))
109	108	ex 416	. . . 4 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) ∧ 𝑚 ∈ 𝐵) → (𝐶 = (𝑐 ∗ (1_r‘𝐴)) → (𝐶 × 𝑚) = (𝑐 ∗ 𝑚)))
110	109	ralrimdva 3161	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶 ∈ 𝑆) ∧ 𝑐 ∈ 𝐾) → (𝐶 = (𝑐 ∗ (1_r‘𝐴)) → ∀𝑚 ∈ 𝐵 (𝐶 × 𝑚) = (𝑐 ∗ 𝑚)))
111	110	reximdva 3174	. 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 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 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ∀wral 3075 ∃wrex 3085 Vcvv 3453 ⦋csb 3852 ifcif 4479 ↦ cmpt 5180 ‘cfv 6515 (class class class)co 7390 ∈ cmpo 7392 Fincfn 8921 Basecbs 17226 ._rcmulr 17268 ·_𝑠 cvsca 17271 0_gc0g 17449 Σ_g cgsu 17450 Mndcmnd 18749 1_rcur 20208 Ringcrg 20260 Mat cmat 22445 ScMat cscmat 22527

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7712 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-ot 4590 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7654 df-om 7841 df-1st 7964 df-2nd 7965 df-supp 8134 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-er 8671 df-map 8803 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9303 df-sup 9383 df-oi 9453 df-card 9892 df-pnf 11213 df-mnf 11214 df-xr 11215 df-ltxr 11216 df-le 11217 df-sub 11411 df-neg 11412 df-nn 12206 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12477 df-z 12564 df-dec 12684 df-uz 12835 df-fz 13508 df-fzo 13655 df-seq 14010 df-hash 14339 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17248 df-plusg 17280 df-mulr 17281 df-sca 17283 df-vsca 17284 df-ip 17285 df-tset 17286 df-ple 17287 df-ds 17289 df-hom 17291 df-cco 17292 df-0g 17451 df-gsum 17452 df-prds 17457 df-pws 17459 df-mre 17595 df-mrc 17596 df-acs 17598 df-mgm 18655 df-sgrp 18734 df-mnd 18750 df-mhm 18798 df-submnd 18799 df-grp 18959 df-minusg 18960 df-sbg 18961 df-mulg 19091 df-subg 19146 df-ghm 19235 df-cntz 19338 df-cmn 19803 df-abl 19804 df-mgp 20168 df-rng 20180 df-ur 20209 df-ring 20262 df-subrg 20597 df-lmod 20907 df-lss 20977 df-sra 21218 df-rgmod 21219 df-dsmm 21762 df-frlm 21777 df-mamu 22429 df-mat 22446 df-scmat 22529

This theorem is referenced by: (None)

W3C validator