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Theorem scmataddcl 22373

Description: The sum of two scalar matrices is a scalar matrix. (Contributed by AV, 25-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
scmataddcl	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(+_g‘𝐴)𝑌) ∈ 𝑆)

Proof of Theorem scmataddcl 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 2726	. . . . 5 ⊢ (1_r‘𝐴) = (1_r‘𝐴)
5		eqid 2726	. . . . 5 ⊢ ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘𝐴)
6		scmatid.s	. . . . 5 ⊢ 𝑆 = (𝑁 ScMat 𝑅)
7	1, 2, 3, 4, 5, 6	scmatscmid 22363	. . . 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 714	. 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 22363	. . . . . 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 7414	. . . . . . . . . . 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 481	. . . . . . . . . 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	2	matlmod 22286	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod)
15	14	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝐴 ∈ LMod)
16	2	matsca2 22277	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝐴))
17	16	fveq2d 6889	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑅) = (Base‘(Scalar‘𝐴)))
18	1, 17	eqtrid 2778	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐸 = (Base‘(Scalar‘𝐴)))
19	18	eleq2d 2813	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑐 ∈ 𝐸 ↔ 𝑐 ∈ (Base‘(Scalar‘𝐴))))
20	19	biimpd 228	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑐 ∈ 𝐸 → 𝑐 ∈ (Base‘(Scalar‘𝐴))))
21	20	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) → (𝑐 ∈ 𝐸 → 𝑐 ∈ (Base‘(Scalar‘𝐴))))
22	21	imp 406	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑐 ∈ (Base‘(Scalar‘𝐴)))
23	18	eleq2d 2813	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑑 ∈ 𝐸 ↔ 𝑑 ∈ (Base‘(Scalar‘𝐴))))
24	23	biimpa 476	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) → 𝑑 ∈ (Base‘(Scalar‘𝐴)))
25	24	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑑 ∈ (Base‘(Scalar‘𝐴)))
26	2	matring 22300	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
27	3, 4	ringidcl 20165	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Ring → (1_r‘𝐴) ∈ 𝐵)
28	26, 27	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐴) ∈ 𝐵)
29	28	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (1_r‘𝐴) ∈ 𝐵)
30		eqid 2726	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝐴) = (+_g‘𝐴)
31		eqid 2726	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
32		eqid 2726	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
33		eqid 2726	. . . . . . . . . . . . . . 15 ⊢ (+_g‘(Scalar‘𝐴)) = (+_g‘(Scalar‘𝐴))
34	3, 30, 31, 5, 32, 33	lmodvsdir 20732	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ LMod ∧ (𝑐 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑑 ∈ (Base‘(Scalar‘𝐴)) ∧ (1_r‘𝐴) ∈ 𝐵)) → ((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	15, 22, 25, 29, 34	syl13anc 1369	. . . . . . . . . . . . 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))))
36	35	eqcomd 2732	. . . . . . . . . . . 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)))
37		simpll 764	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
38	16	eqcomd 2732	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝐴) = 𝑅)
39	38	ad2antrr 723	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (Scalar‘𝐴) = 𝑅)
40	39	fveq2d 6889	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (+_g‘(Scalar‘𝐴)) = (+_g‘𝑅))
41	40	oveqd 7422	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (𝑐(+_g‘(Scalar‘𝐴))𝑑) = (𝑐(+_g‘𝑅)𝑑))
42		ringgrp 20143	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
43	42	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Grp)
44	43	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑅 ∈ Grp)
45		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑐 ∈ 𝐸)
46		simplr 766	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑑 ∈ 𝐸)
47		eqid 2726	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
48	1, 47	grpcl 18871	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ Grp ∧ 𝑐 ∈ 𝐸 ∧ 𝑑 ∈ 𝐸) → (𝑐(+_g‘𝑅)𝑑) ∈ 𝐸)
49	44, 45, 46, 48	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (𝑐(+_g‘𝑅)𝑑) ∈ 𝐸)
50	41, 49	eqeltrd 2827	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (𝑐(+_g‘(Scalar‘𝐴))𝑑) ∈ 𝐸)
51	1, 2, 3, 5	matvscl 22288	. . . . . . . . . . . . . 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)
52	37, 50, 29, 51	syl12anc 834	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → ((𝑐(+_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) ∈ 𝐵)
53		oveq1 7412	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = (𝑐(+_g‘(Scalar‘𝐴))𝑑) → (𝑒( ·_𝑠 ‘𝐴)(1_r‘𝐴)) = ((𝑐(+_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)))
54	53	eqeq2d 2737	. . . . . . . . . . . . . . 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))))
55	54	adantl 481	. . . . . . . . . . . . . 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))))
56		eqidd 2727	. . . . . . . . . . . . . 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)))
57	50, 55, 56	rspcedvd 3608	. . . . . . . . . . . . 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)))
58	1, 2, 3, 4, 5, 6	scmatel 22362	. . . . . . . . . . . . . 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)))))
59	58	ad2antrr 723	. . . . . . . . . . . . 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)))))
60	52, 57, 59	mpbir2and 710	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → ((𝑐(+_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) ∈ 𝑆)
61	36, 60	eqeltrd 2827	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → ((𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴))(+_g‘𝐴)(𝑑( ·_𝑠 ‘𝐴)(1_r‘𝐴))) ∈ 𝑆)
62	61	adantr 480	. . . . . . . . . 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)
63	13, 62	eqeltrd 2827	. . . . . . . . 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)
64	63	exp32 420	. . . . . . . 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)))
65	64	rexlimdva 3149	. . . . . . 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)))
66	65	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)))
67	66	rexlimdva 3149	. . . . 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)))
68	11, 67	syldc 48	. . . 4 ⊢ (𝑌 ∈ 𝑆 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (∃𝑐 ∈ 𝐸 𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)) → (𝑋(+_g‘𝐴)𝑌) ∈ 𝑆)))
69	68	adantl 481	. . 3 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (∃𝑐 ∈ 𝐸 𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)) → (𝑋(+_g‘𝐴)𝑌) ∈ 𝑆)))
70	69	impcom 407	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (∃𝑐 ∈ 𝐸 𝑋 = (𝑐( ·_𝑠 ‘𝐴)(1_r‘𝐴)) → (𝑋(+_g‘𝐴)𝑌) ∈ 𝑆))
71	9, 70	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 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 ‘cfv 6537 (class class class)co 7405 Fincfn 8941 Basecbs 17153 +_gcplusg 17206 Scalarcsca 17209 ·_𝑠 cvsca 17210 0_gc0g 17394 Grpcgrp 18863 1_rcur 20086 Ringcrg 20138 LModclmod 20706 Mat cmat 22262 ScMat cscmat 22346

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-ot 4632 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-hom 17230 df-cco 17231 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-ghm 19139 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-subrg 20471 df-lmod 20708 df-lss 20779 df-sra 21021 df-rgmod 21022 df-dsmm 21627 df-frlm 21642 df-mamu 22241 df-mat 22263 df-scmat 22348

This theorem is referenced by: scmatlss 22382

W3C validator