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Theorem scmatsubcl 22018

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 2732	. . . . 5 ⊢ (1_r‘𝐴) = (1_r‘𝐴)
5		eqid 2732	. . . . 5 ⊢ ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘𝐴)
6		scmatid.s	. . . . 5 ⊢ 𝑆 = (𝑁 ScMat 𝑅)
7	1, 2, 3, 4, 5, 6	scmatscmid 22007	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑆) → ∃𝑐 ∈ 𝐸 𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)))
8	7	3expa 1118	. . 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 22007	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑆) → ∃𝑑 ∈ 𝐸 𝑌 = (𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴)))
11	10	3expia 1121	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑌 ∈ 𝑆 → ∃𝑑 ∈ 𝐸 𝑌 = (𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴))))
12		oveq12 7417	. . . . . . . . . . 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 482	. . . . . . . . . 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 2732	. . . . . . . . . . . . . 14 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
15		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
16		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (-_g‘𝐴) = (-_g‘𝐴)
17		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (-_g‘(Scalar‘𝐴)) = (-_g‘(Scalar‘𝐴))
18	2	matlmod 21930	. . . . . . . . . . . . . . 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 21921	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝐴))
21	20	fveq2d 6895	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑅) = (Base‘(Scalar‘𝐴)))
22	1, 21	eqtrid 2784	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐸 = (Base‘(Scalar‘𝐴)))
23	22	eleq2d 2819	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) → (𝑐 ∈ 𝐸 → 𝑐 ∈ (Base‘(Scalar‘𝐴))))
26	25	imp 407	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑐 ∈ (Base‘(Scalar‘𝐴)))
27	22	eleq2d 2819	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑑 ∈ 𝐸 ↔ 𝑑 ∈ (Base‘(Scalar‘𝐴))))
28	27	biimpa 477	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) → 𝑑 ∈ (Base‘(Scalar‘𝐴)))
29	28	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑑 ∈ (Base‘(Scalar‘𝐴)))
30	2	matring 21944	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
31	3, 4	ringidcl 20082	. . . . . . . . . . . . . . . 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 20529	. . . . . . . . . . . . 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 2738	. . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . 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 6895	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (-_g‘(Scalar‘𝐴)) = (-_g‘𝑅))
40	39	oveqd 7425	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (𝑐(-_g‘(Scalar‘𝐴))𝑑) = (𝑐(-_g‘𝑅)𝑑))
41		ringgrp 20060	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
42	41	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Grp)
43	42	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑅 ∈ Grp)
44		simpr 485	. . . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . . 17 ⊢ (-_g‘𝑅) = (-_g‘𝑅)
47	1, 46	grpsubcl 18902	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Grp ∧ 𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸) → (𝑐(-_g‘𝑅)𝑑) ∈ 𝐸)
48	43, 44, 45, 47	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (𝑐(-_g‘𝑅)𝑑) ∈ 𝐸)
49	40, 48	eqeltrd 2833	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (𝑐(-_g‘(Scalar‘𝐴))𝑑) ∈ 𝐸)
50	1, 2, 3, 5	matvscl 21932	. . . . . . . . . . . . . 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 7415	. . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . 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 3614	. . . . . . . . . . . . 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 22006	. . . . . . . . . . . . . 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 2833	. . . . . . . . . . 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 481	. . . . . . . . . 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 2833	. . . . . . . . 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 421	. . . . . . . 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 3155	. . . . . . 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 3155	. . . . 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 482	. . 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 408	. 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 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 ‘cfv 6543 (class class class)co 7408 Fincfn 8938 Basecbs 17143 Scalarcsca 17199 ·_𝑠 cvsca 17200 0_gc0g 17384 Grpcgrp 18818 -_gcsg 18820 1_rcur 20003 Ringcrg 20055 LModclmod 20470 Mat cmat 21906 ScMat cscmat 21990

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-fzo 13627 df-seq 13966 df-hash 14290 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-hom 17220 df-cco 17221 df-0g 17386 df-gsum 17387 df-prds 17392 df-pws 17394 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-submnd 18671 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mulg 18950 df-subg 19002 df-ghm 19089 df-cntz 19180 df-cmn 19649 df-abl 19650 df-mgp 19987 df-ur 20004 df-ring 20057 df-subrg 20316 df-lmod 20472 df-lss 20542 df-sra 20784 df-rgmod 20785 df-dsmm 21286 df-frlm 21301 df-mamu 21885 df-mat 21907 df-scmat 21992

This theorem is referenced by: scmatsgrp 22020 scmatsgrp1 22023

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