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Theorem scmatsubcl 22439

Description: The difference of two scalar matrices is a scalar matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 19-Dec-2019.)

**Hypotheses**
Ref	Expression
scmatid.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
scmatid.b	⊢ 𝐵 = (Base‘𝐴)
scmatid.e	⊢ 𝐸 = (Base‘𝑅)
scmatid.0	⊢ 0 = (0_g‘𝑅)
scmatid.s	⊢ 𝑆 = (𝑁 ScMat 𝑅)

**Assertion**
Ref	Expression
scmatsubcl	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(-_g‘𝐴)𝑌) ∈ 𝑆)

Proof of Theorem scmatsubcl Dummy variables 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scmatid.e	. . . . 5 ⊢ 𝐸 = (Base‘𝑅)
2		scmatid.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
3		scmatid.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
4		eqid 2728	. . . . 5 ⊢ (1_r‘𝐴) = (1_r‘𝐴)
5		eqid 2728	. . . . 5 ⊢ ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘𝐴)
6		scmatid.s	. . . . 5 ⊢ 𝑆 = (𝑁 ScMat 𝑅)
7	1, 2, 3, 4, 5, 6	scmatscmid 22428	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑆) → ∃𝑐 ∈ 𝐸 𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)))
8	7	3expa 1115	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑋 ∈ 𝑆) → ∃𝑐 ∈ 𝐸 𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)))
9	8	adantrr 715	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → ∃𝑐 ∈ 𝐸 𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)))
10	1, 2, 3, 4, 5, 6	scmatscmid 22428	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑆) → ∃𝑑 ∈ 𝐸 𝑌 = (𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴)))
11	10	3expia 1118	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑌 ∈ 𝑆 → ∃𝑑 ∈ 𝐸 𝑌 = (𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴))))
12		oveq12 7435	. . . . . . . . . . 11 ⊢ ((𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)) ∧ 𝑌 = (𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴))) → (𝑋(-_g‘𝐴)𝑌) = ((𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴))(-_g‘𝐴)(𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴))))
13	12	adantl 480	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) ∧ (𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)) ∧ 𝑌 = (𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴)))) → (𝑋(-_g‘𝐴)𝑌) = ((𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴))(-_g‘𝐴)(𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴))))
14		eqid 2728	. . . . . . . . . . . . . 14 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
15		eqid 2728	. . . . . . . . . . . . . 14 ⊢ (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
16		eqid 2728	. . . . . . . . . . . . . 14 ⊢ (-_g‘𝐴) = (-_g‘𝐴)
17		eqid 2728	. . . . . . . . . . . . . 14 ⊢ (-_g‘(Scalar‘𝐴)) = (-_g‘(Scalar‘𝐴))
18	2	matlmod 22351	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod)
19	18	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝐴 ∈ LMod)
20	2	matsca2 22342	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝐴))
21	20	fveq2d 6906	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑅) = (Base‘(Scalar‘𝐴)))
22	1, 21	eqtrid 2780	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐸 = (Base‘(Scalar‘𝐴)))
23	22	eleq2d 2815	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑐 ∈ 𝐸 ↔ 𝑐 ∈ (Base‘(Scalar‘𝐴))))
24	23	biimpd 228	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑐 ∈ 𝐸 → 𝑐 ∈ (Base‘(Scalar‘𝐴))))
25	24	adantr 479	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) → (𝑐 ∈ 𝐸 → 𝑐 ∈ (Base‘(Scalar‘𝐴))))
26	25	imp 405	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑐 ∈ (Base‘(Scalar‘𝐴)))
27	22	eleq2d 2815	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑑 ∈ 𝐸 ↔ 𝑑 ∈ (Base‘(Scalar‘𝐴))))
28	27	biimpa 475	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) → 𝑑 ∈ (Base‘(Scalar‘𝐴)))
29	28	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑑 ∈ (Base‘(Scalar‘𝐴)))
30	2	matring 22365	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
31	3, 4	ringidcl 20209	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Ring → (1_r‘𝐴) ∈ 𝐵)
32	30, 31	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐴) ∈ 𝐵)
33	32	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (1_r‘𝐴) ∈ 𝐵)
34	3, 5, 14, 15, 16, 17, 19, 26, 29, 33	lmodsubdir 20810	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → ((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) = ((𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴))(-_g‘𝐴)(𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴))))
35	34	eqcomd 2734	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → ((𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴))(-_g‘𝐴)(𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴))) = ((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)))
36		simpll 765	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
37	20	eqcomd 2734	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝐴) = 𝑅)
38	37	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (Scalar‘𝐴) = 𝑅)
39	38	fveq2d 6906	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (-_g‘(Scalar‘𝐴)) = (-_g‘𝑅))
40	39	oveqd 7443	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (𝑐(-_g‘(Scalar‘𝐴))𝑑) = (𝑐(-_g‘𝑅)𝑑))
41		ringgrp 20185	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
42	41	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Grp)
43	42	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑅 ∈ Grp)
44		simpr 483	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑐 ∈ 𝐸)
45		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑑 ∈ 𝐸)
46		eqid 2728	. . . . . . . . . . . . . . . . 17 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
47	1, 46	grpsubcl 18983	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Grp ∧ 𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸) → (𝑐(-_g‘𝑅)𝑑) ∈ 𝐸)
48	43, 44, 45, 47	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (𝑐(-_g‘𝑅)𝑑) ∈ 𝐸)
49	40, 48	eqeltrd 2829	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (𝑐(-_g‘(Scalar‘𝐴))𝑑) ∈ 𝐸)
50	1, 2, 3, 5	matvscl 22353	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑐(-_g‘(Scalar‘𝐴))𝑑) ∈ 𝐸 ∧ (1_r‘𝐴) ∈ 𝐵)) → ((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) ∈ 𝐵)
51	36, 49, 33, 50	syl12anc 835	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → ((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) ∈ 𝐵)
52		oveq1 7433	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = (𝑐(-_g‘(Scalar‘𝐴))𝑑) → (𝑒( ·_𝑠 ‘𝐴)(1_r‘𝐴)) = ((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)))
53	52	eqeq2d 2739	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = (𝑐(-_g‘(Scalar‘𝐴))𝑑) → (((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) = (𝑒( ·_𝑠 ‘𝐴)(1_r‘𝐴)) ↔ ((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) = ((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴))))
54	53	adantl 480	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) ∧ 𝑒 = (𝑐(-_g‘(Scalar‘𝐴))𝑑)) → (((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) = (𝑒( ·_𝑠 ‘𝐴)(1_r‘𝐴)) ↔ ((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) = ((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴))))
55		eqidd 2729	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → ((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) = ((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)))
56	49, 54, 55	rspcedvd 3613	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → ∃𝑒 ∈ 𝐸 ((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) = (𝑒( ·_𝑠 ‘𝐴)(1_r‘𝐴)))
57	1, 2, 3, 4, 5, 6	scmatel 22427	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) ∈ 𝑆 ↔ (((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) ∈ 𝐵 ∧ ∃𝑒 ∈ 𝐸 ((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) = (𝑒( ·_𝑠 ‘𝐴)(1_r‘𝐴)))))
58	57	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) ∈ 𝑆 ↔ (((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) ∈ 𝐵 ∧ ∃𝑒 ∈ 𝐸 ((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) = (𝑒( ·_𝑠 ‘𝐴)(1_r‘𝐴)))))
59	51, 56, 58	mpbir2and 711	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → ((𝑐(-_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) ∈ 𝑆)
60	35, 59	eqeltrd 2829	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → ((𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴))(-_g‘𝐴)(𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴))) ∈ 𝑆)
61	60	adantr 479	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) ∧ (𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)) ∧ 𝑌 = (𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴)))) → ((𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴))(-_g‘𝐴)(𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴))) ∈ 𝑆)
62	13, 61	eqeltrd 2829	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) ∧ (𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)) ∧ 𝑌 = (𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴)))) → (𝑋(-_g‘𝐴)𝑌) ∈ 𝑆)
63	62	exp32 419	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)) → (𝑌 = (𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴)) → (𝑋(-_g‘𝐴)𝑌) ∈ 𝑆)))
64	63	rexlimdva 3152	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) → (∃𝑐 ∈ 𝐸 𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)) → (𝑌 = (𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴)) → (𝑋(-_g‘𝐴)𝑌) ∈ 𝑆)))
65	64	com23 86	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) → (𝑌 = (𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴)) → (∃𝑐 ∈ 𝐸 𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)) → (𝑋(-_g‘𝐴)𝑌) ∈ 𝑆)))
66	65	rexlimdva 3152	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (∃𝑑 ∈ 𝐸 𝑌 = (𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴)) → (∃𝑐 ∈ 𝐸 𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)) → (𝑋(-_g‘𝐴)𝑌) ∈ 𝑆)))
67	11, 66	syldc 48	. . . 4 ⊢ (𝑌 ∈ 𝑆 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (∃𝑐 ∈ 𝐸 𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)) → (𝑋(-_g‘𝐴)𝑌) ∈ 𝑆)))
68	67	adantl 480	. . 3 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (∃𝑐 ∈ 𝐸 𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)) → (𝑋(-_g‘𝐴)𝑌) ∈ 𝑆)))
69	68	impcom 406	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (∃𝑐 ∈ 𝐸 𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)) → (𝑋(-_g‘𝐴)𝑌) ∈ 𝑆))
70	9, 69	mpd 15	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(-_g‘𝐴)𝑌) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3067 ‘cfv 6553 (class class class)co 7426 Fincfn 8970 Basecbs 17187 Scalarcsca 17243 ·_𝑠 cvsca 17244 0_gc0g 17428 Grpcgrp 18897 -_gcsg 18899 1_rcur 20128 Ringcrg 20180 LModclmod 20750 Mat cmat 22327 ScMat cscmat 22411

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-ot 4641 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-hom 17264 df-cco 17265 df-0g 17430 df-gsum 17431 df-prds 17436 df-pws 17438 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-ghm 19175 df-cntz 19275 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-subrg 20515 df-lmod 20752 df-lss 20823 df-sra 21065 df-rgmod 21066 df-dsmm 21673 df-frlm 21688 df-mamu 22306 df-mat 22328 df-scmat 22413

This theorem is referenced by: scmatsgrp 22441 scmatsgrp1 22444

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