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Theorem scmataddcl 22436

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 2725	. . . . 5 ⊢ (1_r‘𝐴) = (1_r‘𝐴)
5		eqid 2725	. . . . 5 ⊢ ( ·_𝑠 ‘𝐴) = ( ·_𝑠 ‘𝐴)
6		scmatid.s	. . . . 5 ⊢ 𝑆 = (𝑁 ScMat 𝑅)
7	1, 2, 3, 4, 5, 6	scmatscmid 22426	. . . 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 22426	. . . . . 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 7425	. . . . . . . . . . 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	2	matlmod 22349	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod)
15	14	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝐴 ∈ LMod)
16	2	matsca2 22340	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 = (Scalar‘𝐴))
17	16	fveq2d 6896	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑅) = (Base‘(Scalar‘𝐴)))
18	1, 17	eqtrid 2777	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐸 = (Base‘(Scalar‘𝐴)))
19	18	eleq2d 2811	. . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) → (𝑐 ∈ 𝐸 → 𝑐 ∈ (Base‘(Scalar‘𝐴))))
22	21	imp 405	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑐 ∈ (Base‘(Scalar‘𝐴)))
23	18	eleq2d 2811	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑑 ∈ 𝐸 ↔ 𝑑 ∈ (Base‘(Scalar‘𝐴))))
24	23	biimpa 475	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) → 𝑑 ∈ (Base‘(Scalar‘𝐴)))
25	24	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑑 ∈ (Base‘(Scalar‘𝐴)))
26	2	matring 22363	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
27	3, 4	ringidcl 20206	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Ring → (1_r‘𝐴) ∈ 𝐵)
28	26, 27	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐴) ∈ 𝐵)
29	28	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (1_r‘𝐴) ∈ 𝐵)
30		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (+_g‘𝐴) = (+_g‘𝐴)
31		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝐴) = (Scalar‘𝐴)
32		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
33		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (+_g‘(Scalar‘𝐴)) = (+_g‘(Scalar‘𝐴))
34	3, 30, 31, 5, 32, 33	lmodvsdir 20773	. . . . . . . . . . . . . 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 2731	. . . . . . . . . . . 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 765	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
38	16	eqcomd 2731	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Scalar‘𝐴) = 𝑅)
39	38	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (Scalar‘𝐴) = 𝑅)
40	39	fveq2d 6896	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (+_g‘(Scalar‘𝐴)) = (+_g‘𝑅))
41	40	oveqd 7433	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (𝑐(+_g‘(Scalar‘𝐴))𝑑) = (𝑐(+_g‘𝑅)𝑑))
42		ringgrp 20182	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
43	42	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Grp)
44	43	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑅 ∈ Grp)
45		simpr 483	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑐 ∈ 𝐸)
46		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → 𝑑 ∈ 𝐸)
47		eqid 2725	. . . . . . . . . . . . . . . . 17 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
48	1, 47	grpcl 18902	. . . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → (𝑐(+_g‘(Scalar‘𝐴))𝑑) ∈ 𝐸)
51	1, 2, 3, 5	matvscl 22351	. . . . . . . . . . . . . 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 835	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → ((𝑐(+_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) ∈ 𝐵)
53		oveq1 7423	. . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . 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 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))))
56		eqidd 2726	. . . . . . . . . . . . . 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 3603	. . . . . . . . . . . . 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 22425	. . . . . . . . . . . . . 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 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)))))
60	52, 57, 59	mpbir2and 711	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑑 ∈ 𝐸) ∧ 𝑐 ∈ 𝐸) → ((𝑐(+_g‘(Scalar‘𝐴))𝑑)( ·_𝑠 ‘𝐴)(1_r‘𝐴)) ∈ 𝑆)
61	36, 60	eqeltrd 2825	. . . . . . . . . . 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 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)
63	13, 62	eqeltrd 2825	. . . . . . . . 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 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)))
65	64	rexlimdva 3145	. . . . . . 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 3145	. . . . 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 480	. . 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 406	. 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 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3060 ‘cfv 6543 (class class class)co 7416 Fincfn 8962 Basecbs 17179 +_gcplusg 17232 Scalarcsca 17235 ·_𝑠 cvsca 17236 0_gc0g 17420 Grpcgrp 18894 1_rcur 20125 Ringcrg 20177 LModclmod 20747 Mat cmat 22325 ScMat cscmat 22409

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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-0g 17422 df-gsum 17423 df-prds 17428 df-pws 17430 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mulg 19028 df-subg 19082 df-ghm 19172 df-cntz 19272 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-subrg 20512 df-lmod 20749 df-lss 20820 df-sra 21062 df-rgmod 21063 df-dsmm 21670 df-frlm 21685 df-mamu 22309 df-mat 22326 df-scmat 22411

This theorem is referenced by: scmatlss 22445

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