Theorem mendlmod 43957

Description: The module endomorphism algebra is a left module. (Contributed by Mario Carneiro, 22-Sep-2015.)

Hypotheses

Ref	Expression
mendassa.a	⊢ 𝐴 = (MEndo‘𝑀)
mendassa.s	⊢ 𝑆 = (Scalar‘𝑀)

Assertion

Ref	Expression
mendlmod	⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ LMod)

Proof of Theorem mendlmod

Dummy variables 𝑥 𝑦 𝑧 𝑢 𝑘 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mendassa.a	. . . 4 ⊢ 𝐴 = (MEndo‘𝑀)
2	1	mendbas 43948	. . 3 ⊢ (𝑀 LMHom 𝑀) = (Base‘𝐴)
3	2	a1i 11	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (𝑀 LMHom 𝑀) = (Base‘𝐴))
4		eqidd 2767	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (+g‘𝐴) = (+g‘𝐴))
5		mendassa.s	. . . 4 ⊢ 𝑆 = (Scalar‘𝑀)
6	1, 5	mendsca 43953	. . 3 ⊢ 𝑆 = (Scalar‘𝐴)
7	6	a1i 11	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 = (Scalar‘𝐴))
8		eqidd 2767	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴))
9		eqidd 2767	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (Base‘𝑆) = (Base‘𝑆))
10		eqidd 2767	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (+g‘𝑆) = (+g‘𝑆))
11		eqidd 2767	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (.r‘𝑆) = (.r‘𝑆))
12		eqidd 2767	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → (1r‘𝑆) = (1r‘𝑆))
13		crngring 20358	. . 3 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring)
14	13	adantl 487	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝑆 ∈ Ring)
15	1	mendring 43956	. . . 4 ⊢ (𝑀 ∈ LMod → 𝐴 ∈ Ring)
16	15	adantr 486	. . 3 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ Ring)
17		ringgrp 20351	. . 3 ⊢ (𝐴 ∈ Ring → 𝐴 ∈ Grp)
18	16, 17	syl 18	. 2 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ Grp)
19		eqid 2766	. . . . 5 ⊢ ( ·𝑠 ‘𝑀) = ( ·𝑠 ‘𝑀)
20		eqid 2766	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
21		eqid 2766	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
22		eqid 2766	. . . . 5 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘𝐴)
23	1, 19, 2, 5, 20, 21, 22	mendvsca 43955	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))
24	23	3adant1 1148	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))
25	21, 19, 5, 20	lmhmvsca 21203	. . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
26	25	3adant1l 1195	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦) ∈ (𝑀 LMHom 𝑀))
27	24, 26	eqeltrd 2866	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
28		simpr2 1214	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (𝑀 LMHom 𝑀))
29		simpr3 1215	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
30		eqid 2766	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
31		eqid 2766	. . . . . 6 ⊢ (+g‘𝐴) = (+g‘𝐴)
32	1, 2, 30, 31	mendplusg 43950	. . . . 5 ⊢ ((𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g‘𝐴)𝑧) = (𝑦 ∘f (+g‘𝑀)𝑧))
33	28, 29, 32	syl2anc 596	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(+g‘𝐴)𝑧) = (𝑦 ∘f (+g‘𝑀)𝑧))
34	33	oveq2d 7439	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)(𝑦(+g‘𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)(𝑦 ∘f (+g‘𝑀)𝑧)))
35		simpr1 1213	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (Base‘𝑆))
36	18	adantr 486	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝐴 ∈ Grp)
37	2, 31	grpcl 19039	. . . . 5 ⊢ ((𝐴 ∈ Grp ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦(+g‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
38	36, 28, 29, 37	syl3anc 1398	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦(+g‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
39	1, 19, 2, 5, 20, 21, 22	mendvsca 43955	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ (𝑦(+g‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)(𝑦(+g‘𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)(𝑦(+g‘𝐴)𝑧)))
40	35, 38, 39	syl2anc 596	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠 ‘𝐴)(𝑦(+g‘𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)(𝑦(+g‘𝐴)𝑧)))
41	35, 28, 23	syl2anc 596	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠 ‘𝐴)𝑦) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦))
42	1, 19, 2, 5, 20, 21, 22	mendvsca 43955	. . . . . 6 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑧))
43	35, 29, 42	syl2anc 596	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠 ‘𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑧))
44	41, 43	oveq12d 7441	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠 ‘𝐴)𝑦) ∘f (+g‘𝑀)(𝑥( ·𝑠 ‘𝐴)𝑧)) = ((((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦) ∘f (+g‘𝑀)(((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑧)))
45	27	3adant3r3 1203	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠 ‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀))
46		eleq1w 2849	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ (𝑀 LMHom 𝑀) ↔ 𝑧 ∈ (𝑀 LMHom 𝑀)))
47	46	3anbi3d 1470	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) ↔ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))))
48		oveq2 7431	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑥( ·𝑠 ‘𝐴)𝑦) = (𝑥( ·𝑠 ‘𝐴)𝑧))
49	48	eleq1d 2851	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝑥( ·𝑠 ‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀) ↔ (𝑥( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)))
50	47, 49	imbi12d 347	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀)) ↔ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))))
51	50, 27	chvarvv 2022	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
52	51	3adant3r2 1202	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
53	1, 2, 30, 31	mendplusg 43950	. . . . 5 ⊢ (((𝑥( ·𝑠 ‘𝐴)𝑦) ∈ (𝑀 LMHom 𝑀) ∧ (𝑥( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)) → ((𝑥( ·𝑠 ‘𝐴)𝑦)(+g‘𝐴)(𝑥( ·𝑠 ‘𝐴)𝑧)) = ((𝑥( ·𝑠 ‘𝐴)𝑦) ∘f (+g‘𝑀)(𝑥( ·𝑠 ‘𝐴)𝑧)))
54	45, 52, 53	syl2anc 596	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠 ‘𝐴)𝑦)(+g‘𝐴)(𝑥( ·𝑠 ‘𝐴)𝑧)) = ((𝑥( ·𝑠 ‘𝐴)𝑦) ∘f (+g‘𝑀)(𝑥( ·𝑠 ‘𝐴)𝑧)))
55		fvexd 6903	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
56		fconst6g 6774	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝑆) → ((Base‘𝑀) × {𝑥}):(Base‘𝑀)⟶(Base‘𝑆))
57	35, 56	syl 18	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑥}):(Base‘𝑀)⟶(Base‘𝑆))
58	21, 21	lmhmf 21192	. . . . . 6 ⊢ (𝑦 ∈ (𝑀 LMHom 𝑀) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
59	28, 58	syl 18	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦:(Base‘𝑀)⟶(Base‘𝑀))
60	21, 21	lmhmf 21192	. . . . . 6 ⊢ (𝑧 ∈ (𝑀 LMHom 𝑀) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
61	29, 60	syl 18	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
62		simpll 779	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑀 ∈ LMod)
63	21, 30, 5, 19, 20	lmodvsdi 21043	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ (𝑤 ∈ (Base‘𝑆) ∧ 𝑣 ∈ (Base‘𝑀) ∧ 𝑢 ∈ (Base‘𝑀))) → (𝑤( ·𝑠 ‘𝑀)(𝑣(+g‘𝑀)𝑢)) = ((𝑤( ·𝑠 ‘𝑀)𝑣)(+g‘𝑀)(𝑤( ·𝑠 ‘𝑀)𝑢)))
64	62, 63	sylan 592	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ (𝑤 ∈ (Base‘𝑆) ∧ 𝑣 ∈ (Base‘𝑀) ∧ 𝑢 ∈ (Base‘𝑀))) → (𝑤( ·𝑠 ‘𝑀)(𝑣(+g‘𝑀)𝑢)) = ((𝑤( ·𝑠 ‘𝑀)𝑣)(+g‘𝑀)(𝑤( ·𝑠 ‘𝑀)𝑢)))
65	55, 57, 59, 61, 64	caofdi 7729	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)(𝑦 ∘f (+g‘𝑀)𝑧)) = ((((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑦) ∘f (+g‘𝑀)(((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑧)))
66	44, 54, 65	3eqtr4d 2811	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠 ‘𝐴)𝑦)(+g‘𝐴)(𝑥( ·𝑠 ‘𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)(𝑦 ∘f (+g‘𝑀)𝑧)))
67	34, 40, 66	3eqtr4d 2811	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (𝑀 LMHom 𝑀) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠 ‘𝐴)(𝑦(+g‘𝐴)𝑧)) = ((𝑥( ·𝑠 ‘𝐴)𝑦)(+g‘𝐴)(𝑥( ·𝑠 ‘𝐴)𝑧)))
68		fvexd 6903	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (Base‘𝑀) ∈ V)
69		simpr3 1215	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 ∈ (𝑀 LMHom 𝑀))
70	69, 60	syl 18	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧:(Base‘𝑀)⟶(Base‘𝑀))
71		simpr1 1213	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑥 ∈ (Base‘𝑆))
72	71, 56	syl 18	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑥}):(Base‘𝑀)⟶(Base‘𝑆))
73		simpr2 1214	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑦 ∈ (Base‘𝑆))
74		fconst6g 6774	. . . . 5 ⊢ (𝑦 ∈ (Base‘𝑆) → ((Base‘𝑀) × {𝑦}):(Base‘𝑀)⟶(Base‘𝑆))
75	73, 74	syl 18	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑦}):(Base‘𝑀)⟶(Base‘𝑆))
76		simpll 779	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑀 ∈ LMod)
77		eqid 2766	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
78	21, 30, 5, 19, 20, 77	lmodvsdir 21044	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ (𝑤 ∈ (Base‘𝑆) ∧ 𝑣 ∈ (Base‘𝑆) ∧ 𝑢 ∈ (Base‘𝑀))) → ((𝑤(+g‘𝑆)𝑣)( ·𝑠 ‘𝑀)𝑢) = ((𝑤( ·𝑠 ‘𝑀)𝑢)(+g‘𝑀)(𝑣( ·𝑠 ‘𝑀)𝑢)))
79	76, 78	sylan 592	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ (𝑤 ∈ (Base‘𝑆) ∧ 𝑣 ∈ (Base‘𝑆) ∧ 𝑢 ∈ (Base‘𝑀))) → ((𝑤(+g‘𝑆)𝑣)( ·𝑠 ‘𝑀)𝑢) = ((𝑤( ·𝑠 ‘𝑀)𝑢)(+g‘𝑀)(𝑣( ·𝑠 ‘𝑀)𝑢)))
80	68, 70, 72, 75, 79	caofdir 7730	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((((Base‘𝑀) × {𝑥}) ∘f (+g‘𝑆)((Base‘𝑀) × {𝑦})) ∘f ( ·𝑠 ‘𝑀)𝑧) = ((((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑧) ∘f (+g‘𝑀)(((Base‘𝑀) × {𝑦}) ∘f ( ·𝑠 ‘𝑀)𝑧)))
81	14	adantr 486	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑆 ∈ Ring)
82	20, 77	ringacl 20393	. . . . . 6 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
83	81, 71, 73, 82	syl3anc 1398	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆))
84	1, 19, 2, 5, 20, 21, 22	mendvsca 43955	. . . . 5 ⊢ (((𝑥(+g‘𝑆)𝑦) ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥(+g‘𝑆)𝑦)( ·𝑠 ‘𝐴)𝑧) = (((Base‘𝑀) × {(𝑥(+g‘𝑆)𝑦)}) ∘f ( ·𝑠 ‘𝑀)𝑧))
85	83, 69, 84	syl2anc 596	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g‘𝑆)𝑦)( ·𝑠 ‘𝐴)𝑧) = (((Base‘𝑀) × {(𝑥(+g‘𝑆)𝑦)}) ∘f ( ·𝑠 ‘𝑀)𝑧))
86	68, 71, 73	ofc12 7717	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑥}) ∘f (+g‘𝑆)((Base‘𝑀) × {𝑦})) = ((Base‘𝑀) × {(𝑥(+g‘𝑆)𝑦)}))
87	86	oveq1d 7438	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((((Base‘𝑀) × {𝑥}) ∘f (+g‘𝑆)((Base‘𝑀) × {𝑦})) ∘f ( ·𝑠 ‘𝑀)𝑧) = (((Base‘𝑀) × {(𝑥(+g‘𝑆)𝑦)}) ∘f ( ·𝑠 ‘𝑀)𝑧))
88	85, 87	eqtr4d 2804	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g‘𝑆)𝑦)( ·𝑠 ‘𝐴)𝑧) = ((((Base‘𝑀) × {𝑥}) ∘f (+g‘𝑆)((Base‘𝑀) × {𝑦})) ∘f ( ·𝑠 ‘𝑀)𝑧))
89	51	3adant3r2 1202	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
90		eleq1w 2849	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (Base‘𝑆) ↔ 𝑦 ∈ (Base‘𝑆)))
91	90	3anbi2d 1469	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) ↔ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))))
92		oveq1 7430	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥( ·𝑠 ‘𝐴)𝑧) = (𝑦( ·𝑠 ‘𝐴)𝑧))
93	92	eleq1d 2851	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀) ↔ (𝑦( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)))
94	91, 93	imbi12d 347	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)) ↔ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))))
95	94, 51	chvarvv 2022	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
96	95	3adant3r1 1201	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀))
97	1, 2, 30, 31	mendplusg 43950	. . . . 5 ⊢ (((𝑥( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀) ∧ (𝑦( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)) → ((𝑥( ·𝑠 ‘𝐴)𝑧)(+g‘𝐴)(𝑦( ·𝑠 ‘𝐴)𝑧)) = ((𝑥( ·𝑠 ‘𝐴)𝑧) ∘f (+g‘𝑀)(𝑦( ·𝑠 ‘𝐴)𝑧)))
98	89, 96, 97	syl2anc 596	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠 ‘𝐴)𝑧)(+g‘𝐴)(𝑦( ·𝑠 ‘𝐴)𝑧)) = ((𝑥( ·𝑠 ‘𝐴)𝑧) ∘f (+g‘𝑀)(𝑦( ·𝑠 ‘𝐴)𝑧)))
99	71, 69, 42	syl2anc 596	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠 ‘𝐴)𝑧) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑧))
100	1, 19, 2, 5, 20, 21, 22	mendvsca 43955	. . . . . 6 ⊢ ((𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → (𝑦( ·𝑠 ‘𝐴)𝑧) = (((Base‘𝑀) × {𝑦}) ∘f ( ·𝑠 ‘𝑀)𝑧))
101	73, 69, 100	syl2anc 596	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦( ·𝑠 ‘𝐴)𝑧) = (((Base‘𝑀) × {𝑦}) ∘f ( ·𝑠 ‘𝑀)𝑧))
102	99, 101	oveq12d 7441	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠 ‘𝐴)𝑧) ∘f (+g‘𝑀)(𝑦( ·𝑠 ‘𝐴)𝑧)) = ((((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑧) ∘f (+g‘𝑀)(((Base‘𝑀) × {𝑦}) ∘f ( ·𝑠 ‘𝑀)𝑧)))
103	98, 102	eqtrd 2801	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥( ·𝑠 ‘𝐴)𝑧)(+g‘𝐴)(𝑦( ·𝑠 ‘𝐴)𝑧)) = ((((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)𝑧) ∘f (+g‘𝑀)(((Base‘𝑀) × {𝑦}) ∘f ( ·𝑠 ‘𝑀)𝑧)))
104	80, 88, 103	3eqtr4d 2811	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(+g‘𝑆)𝑦)( ·𝑠 ‘𝐴)𝑧) = ((𝑥( ·𝑠 ‘𝐴)𝑧)(+g‘𝐴)(𝑦( ·𝑠 ‘𝐴)𝑧)))
105		ovexd 7458	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → (𝑥(.r‘𝑆)𝑦) ∈ V)
106	70	ffvelcdmda 7086	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → (𝑧‘𝑘) ∈ (Base‘𝑀))
107		fconstmpt 5728	. . . . 5 ⊢ ((Base‘𝑀) × {(𝑥(.r‘𝑆)𝑦)}) = (𝑘 ∈ (Base‘𝑀) ↦ (𝑥(.r‘𝑆)𝑦))
108	107	a1i 11	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {(𝑥(.r‘𝑆)𝑦)}) = (𝑘 ∈ (Base‘𝑀) ↦ (𝑥(.r‘𝑆)𝑦)))
109	70	feqmptd 6956	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → 𝑧 = (𝑘 ∈ (Base‘𝑀) ↦ (𝑧‘𝑘)))
110	68, 105, 106, 108, 109	offval2 7707	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {(𝑥(.r‘𝑆)𝑦)}) ∘f ( ·𝑠 ‘𝑀)𝑧) = (𝑘 ∈ (Base‘𝑀) ↦ ((𝑥(.r‘𝑆)𝑦)( ·𝑠 ‘𝑀)(𝑧‘𝑘))))
111		eqid 2766	. . . . . 6 ⊢ (.r‘𝑆) = (.r‘𝑆)
112	20, 111	ringcl 20363	. . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(.r‘𝑆)𝑦) ∈ (Base‘𝑆))
113	81, 71, 73, 112	syl3anc 1398	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥(.r‘𝑆)𝑦) ∈ (Base‘𝑆))
114	1, 19, 2, 5, 20, 21, 22	mendvsca 43955	. . . 4 ⊢ (((𝑥(.r‘𝑆)𝑦) ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀)) → ((𝑥(.r‘𝑆)𝑦)( ·𝑠 ‘𝐴)𝑧) = (((Base‘𝑀) × {(𝑥(.r‘𝑆)𝑦)}) ∘f ( ·𝑠 ‘𝑀)𝑧))
115	113, 69, 114	syl2anc 596	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r‘𝑆)𝑦)( ·𝑠 ‘𝐴)𝑧) = (((Base‘𝑀) × {(𝑥(.r‘𝑆)𝑦)}) ∘f ( ·𝑠 ‘𝑀)𝑧))
116	71	adantr 486	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → 𝑥 ∈ (Base‘𝑆))
117		ovexd 7458	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → (𝑦( ·𝑠 ‘𝑀)(𝑧‘𝑘)) ∈ V)
118		fconstmpt 5728	. . . . . 6 ⊢ ((Base‘𝑀) × {𝑥}) = (𝑘 ∈ (Base‘𝑀) ↦ 𝑥)
119	118	a1i 11	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑥}) = (𝑘 ∈ (Base‘𝑀) ↦ 𝑥))
120		simplr2 1235	. . . . . . 7 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → 𝑦 ∈ (Base‘𝑆))
121		fconstmpt 5728	. . . . . . . 8 ⊢ ((Base‘𝑀) × {𝑦}) = (𝑘 ∈ (Base‘𝑀) ↦ 𝑦)
122	121	a1i 11	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((Base‘𝑀) × {𝑦}) = (𝑘 ∈ (Base‘𝑀) ↦ 𝑦))
123	68, 120, 106, 122, 109	offval2 7707	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑦}) ∘f ( ·𝑠 ‘𝑀)𝑧) = (𝑘 ∈ (Base‘𝑀) ↦ (𝑦( ·𝑠 ‘𝑀)(𝑧‘𝑘))))
124	101, 123	eqtrd 2801	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑦( ·𝑠 ‘𝐴)𝑧) = (𝑘 ∈ (Base‘𝑀) ↦ (𝑦( ·𝑠 ‘𝑀)(𝑧‘𝑘))))
125	68, 116, 117, 119, 124	offval2 7707	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)(𝑦( ·𝑠 ‘𝐴)𝑧)) = (𝑘 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠 ‘𝑀)(𝑦( ·𝑠 ‘𝑀)(𝑧‘𝑘)))))
126	1, 19, 2, 5, 20, 21, 22	mendvsca 43955	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ (𝑦( ·𝑠 ‘𝐴)𝑧) ∈ (𝑀 LMHom 𝑀)) → (𝑥( ·𝑠 ‘𝐴)(𝑦( ·𝑠 ‘𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)(𝑦( ·𝑠 ‘𝐴)𝑧)))
127	71, 96, 126	syl2anc 596	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠 ‘𝐴)(𝑦( ·𝑠 ‘𝐴)𝑧)) = (((Base‘𝑀) × {𝑥}) ∘f ( ·𝑠 ‘𝑀)(𝑦( ·𝑠 ‘𝐴)𝑧)))
128	76	adantr 486	. . . . . 6 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → 𝑀 ∈ LMod)
129	21, 5, 19, 20, 111	lmodvsass 21045	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ (𝑧‘𝑘) ∈ (Base‘𝑀))) → ((𝑥(.r‘𝑆)𝑦)( ·𝑠 ‘𝑀)(𝑧‘𝑘)) = (𝑥( ·𝑠 ‘𝑀)(𝑦( ·𝑠 ‘𝑀)(𝑧‘𝑘))))
130	128, 116, 120, 106, 129	syl13anc 1399	. . . . 5 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) ∧ 𝑘 ∈ (Base‘𝑀)) → ((𝑥(.r‘𝑆)𝑦)( ·𝑠 ‘𝑀)(𝑧‘𝑘)) = (𝑥( ·𝑠 ‘𝑀)(𝑦( ·𝑠 ‘𝑀)(𝑧‘𝑘))))
131	130	mpteq2dva 5209	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑘 ∈ (Base‘𝑀) ↦ ((𝑥(.r‘𝑆)𝑦)( ·𝑠 ‘𝑀)(𝑧‘𝑘))) = (𝑘 ∈ (Base‘𝑀) ↦ (𝑥( ·𝑠 ‘𝑀)(𝑦( ·𝑠 ‘𝑀)(𝑧‘𝑘)))))
132	125, 127, 131	3eqtr4d 2811	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → (𝑥( ·𝑠 ‘𝐴)(𝑦( ·𝑠 ‘𝐴)𝑧)) = (𝑘 ∈ (Base‘𝑀) ↦ ((𝑥(.r‘𝑆)𝑦)( ·𝑠 ‘𝑀)(𝑧‘𝑘))))
133	110, 115, 132	3eqtr4d 2811	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆) ∧ 𝑧 ∈ (𝑀 LMHom 𝑀))) → ((𝑥(.r‘𝑆)𝑦)( ·𝑠 ‘𝐴)𝑧) = (𝑥( ·𝑠 ‘𝐴)(𝑦( ·𝑠 ‘𝐴)𝑧)))
134	14	adantr 486	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑆 ∈ Ring)
135		eqid 2766	. . . . . 6 ⊢ (1r‘𝑆) = (1r‘𝑆)
136	20, 135	ringidcl 20380	. . . . 5 ⊢ (𝑆 ∈ Ring → (1r‘𝑆) ∈ (Base‘𝑆))
137	134, 136	syl 18	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (1r‘𝑆) ∈ (Base‘𝑆))
138	1, 19, 2, 5, 20, 21, 22	mendvsca 43955	. . . 4 ⊢ (((1r‘𝑆) ∈ (Base‘𝑆) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((1r‘𝑆)( ·𝑠 ‘𝐴)𝑥) = (((Base‘𝑀) × {(1r‘𝑆)}) ∘f ( ·𝑠 ‘𝑀)𝑥))
139	137, 138	sylancom 600	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((1r‘𝑆)( ·𝑠 ‘𝐴)𝑥) = (((Base‘𝑀) × {(1r‘𝑆)}) ∘f ( ·𝑠 ‘𝑀)𝑥))
140		fvexd 6903	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (Base‘𝑀) ∈ V)
141	21, 21	lmhmf 21192	. . . . 5 ⊢ (𝑥 ∈ (𝑀 LMHom 𝑀) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
142	141	adantl 487	. . . 4 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑥:(Base‘𝑀)⟶(Base‘𝑀))
143		simpll 779	. . . . 5 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → 𝑀 ∈ LMod)
144	21, 5, 19, 135	lmodvs1 21048	. . . . 5 ⊢ ((𝑀 ∈ LMod ∧ 𝑦 ∈ (Base‘𝑀)) → ((1r‘𝑆)( ·𝑠 ‘𝑀)𝑦) = 𝑦)
145	143, 144	sylan 592	. . . 4 ⊢ ((((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) ∧ 𝑦 ∈ (Base‘𝑀)) → ((1r‘𝑆)( ·𝑠 ‘𝑀)𝑦) = 𝑦)
146	140, 142, 137, 145	caofid0l 7720	. . 3 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → (((Base‘𝑀) × {(1r‘𝑆)}) ∘f ( ·𝑠 ‘𝑀)𝑥) = 𝑥)
147	139, 146	eqtrd 2801	. 2 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) ∧ 𝑥 ∈ (𝑀 LMHom 𝑀)) → ((1r‘𝑆)( ·𝑠 ‘𝐴)𝑥) = 𝑥)
148	3, 4, 7, 8, 9, 10, 11, 12, 14, 18, 27, 67, 104, 133, 147	islmodd 21024	1 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ CRing) → 𝐴 ∈ LMod)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  Vcvv 3458  {csn 4594   ↦ cmpt 5197   × cxp 5664  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423   ∘_f cof 7685  Basecbs 17294  +_gcplusg 17335  ._rcmulr 17336  Scalarcsca 17338   ·_𝑠 cvsca 17339  Grpcgrp 19031  1_rcur 20294  Ringcrg 20346  CRingccrg 20347  LModclmod 21018   LMHom clmhm 21177  MEndocmend 43939

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-of 7687  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8703  df-map 8835  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-n0 12523  df-z 12610  df-uz 12881  df-fz 13554  df-struct 17232  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-plusg 17348  df-mulr 17349  df-sca 17351  df-vsca 17352  df-0g 17519  df-mgm 18723  df-sgrp 18806  df-mnd 18822  df-mhm 18872  df-grp 19034  df-minusg 19035  df-ghm 19315  df-cmn 19883  df-abl 19884  df-mgp 20248  df-rng 20262  df-ur 20295  df-ring 20348  df-cring 20349  df-lmod 21020  df-lmhm 21180  df-mend 43940

This theorem is used by:  mendassa  43958