Metamath Proof Explorer

Theorem mendlmod

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

Ref Expression
Hypotheses mendassa.a ⊢ 𝐴 = ( MEndo ‘ 𝑀 )
mendassa.s ⊢ 𝑆 = ( Scalar ‘ 𝑀 )
Assertion mendlmod ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) → 𝐴 ∈ LMod )

Proof

Step Hyp Ref Expression
1 mendassa.a ⊢ 𝐴 = ( MEndo ‘ 𝑀 )
2 mendassa.s ⊢ 𝑆 = ( Scalar ‘ 𝑀 )
3 1 mendbas ⊢ ( 𝑀 LMHom 𝑀 ) = ( Base ‘ 𝐴 )
4 3 a1i ⊢ ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) → ( 𝑀 LMHom 𝑀 ) = ( Base ‘ 𝐴 ) )
5 eqidd ⊢ ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) → ( +g ‘ 𝐴 ) = ( +g ‘ 𝐴 ) )
6 1 2 mendsca ⊢ 𝑆 = ( Scalar ‘ 𝐴 )
7 6 a1i ⊢ ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) → 𝑆 = ( Scalar ‘ 𝐴 ) )
8 eqidd ⊢ ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) → ( ·𝑠 ‘ 𝐴 ) = ( ·𝑠 ‘ 𝐴 ) )
9 eqidd ⊢ ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) → ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) )
10 eqidd ⊢ ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) → ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) )
11 eqidd ⊢ ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) → ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 ) )
12 eqidd ⊢ ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) → ( 1r ‘ 𝑆 ) = ( 1r ‘ 𝑆 ) )
13 crngring ⊢ ( 𝑆 ∈ CRing → 𝑆 ∈ Ring )
14 13 adantl ⊢ ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) → 𝑆 ∈ Ring )
15 1 mendring ⊢ ( 𝑀 ∈ LMod → 𝐴 ∈ Ring )
16 15 adantr ⊢ ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) → 𝐴 ∈ Ring )
17 ringgrp ⊢ ( 𝐴 ∈ Ring → 𝐴 ∈ Grp )
18 16 17 syl ⊢ ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) → 𝐴 ∈ Grp )
19 eqid ⊢ ( ·𝑠 ‘ 𝑀 ) = ( ·𝑠 ‘ 𝑀 )
20 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
21 eqid ⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 )
22 eqid ⊢ ( ·𝑠 ‘ 𝐴 ) = ( ·𝑠 ‘ 𝐴 )
23 1 19 3 2 20 21 22 mendvsca ⊢ ( ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑦 ) = ( ( ( Base ‘ 𝑀 ) × { 𝑥 } ) ∘f ( ·𝑠 ‘ 𝑀 ) 𝑦 ) )
24 23 3adant1 ⊢ ( ( ( 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 2 20 lmhmvsca ⊢ ( ( 𝑆 ∈ CRing ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) ) → ( ( ( Base ‘ 𝑀 ) × { 𝑥 } ) ∘f ( ·𝑠 ‘ 𝑀 ) 𝑦 ) ∈ ( 𝑀 LMHom 𝑀 ) )
26 25 3adant1l ⊢ ( ( ( 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 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑦 ) ∈ ( 𝑀 LMHom 𝑀 ) )
28 simpr2 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) )
29 simpr3 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) )
30 eqid ⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 )
31 eqid ⊢ ( +g ‘ 𝐴 ) = ( +g ‘ 𝐴 )
32 1 3 30 31 mendplusg ⊢ ( ( 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) → ( 𝑦 ( +g ‘ 𝐴 ) 𝑧 ) = ( 𝑦 ∘f ( +g ‘ 𝑀 ) 𝑧 ) )
33 28 29 32 syl2anc ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → 𝑥 ∈ ( Base ‘ 𝑆 ) )
36 18 adantr ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → 𝐴 ∈ Grp )
37 3 31 grpcl ⊢ ( ( 𝐴 ∈ Grp ∧ 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) → ( 𝑦 ( +g ‘ 𝐴 ) 𝑧 ) ∈ ( 𝑀 LMHom 𝑀 ) )
38 36 28 29 37 syl3anc ⊢ ( ( ( 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 3 2 20 21 22 mendvsca ⊢ ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 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 3 2 20 21 22 mendvsca ⊢ ( ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑧 ) = ( ( ( Base ‘ 𝑀 ) × { 𝑥 } ) ∘f ( ·𝑠 ‘ 𝑀 ) 𝑧 ) )
43 35 29 42 syl2anc ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑦 ) ∈ ( 𝑀 LMHom 𝑀 ) )
46 eleq1w ⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) ↔ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) )
47 46 3anbi3d ⊢ ( 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 ⊢ ( 𝑦 = 𝑧 → ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑦 ) = ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑧 ) )
49 48 eleq1d ⊢ ( 𝑦 = 𝑧 → ( ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑦 ) ∈ ( 𝑀 LMHom 𝑀 ) ↔ ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑧 ) ∈ ( 𝑀 LMHom 𝑀 ) ) )
50 47 49 imbi12d ⊢ ( 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 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑧 ) ∈ ( 𝑀 LMHom 𝑀 ) )
52 51 3adant3r2 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑧 ) ∈ ( 𝑀 LMHom 𝑀 ) )
53 1 3 30 31 mendplusg ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → ( Base ‘ 𝑀 ) ∈ V )
56 fconst6g ⊢ ( 𝑥 ∈ ( Base ‘ 𝑆 ) → ( ( Base ‘ 𝑀 ) × { 𝑥 } ) : ( Base ‘ 𝑀 ) ⟶ ( Base ‘ 𝑆 ) )
57 35 56 syl ⊢ ( ( ( 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 ⊢ ( 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) → 𝑦 : ( Base ‘ 𝑀 ) ⟶ ( Base ‘ 𝑀 ) )
59 28 58 syl ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → 𝑦 : ( Base ‘ 𝑀 ) ⟶ ( Base ‘ 𝑀 ) )
60 21 21 lmhmf ⊢ ( 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) → 𝑧 : ( Base ‘ 𝑀 ) ⟶ ( Base ‘ 𝑀 ) )
61 29 60 syl ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → 𝑧 : ( Base ‘ 𝑀 ) ⟶ ( Base ‘ 𝑀 ) )
62 simpll ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( 𝑀 LMHom 𝑀 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → 𝑀 ∈ LMod )
63 21 30 2 19 20 lmodvsdi ⊢ ( ( 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 ⊢ ( ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → ( Base ‘ 𝑀 ) ∈ V )
69 simpr3 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) )
70 69 60 syl ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → 𝑧 : ( Base ‘ 𝑀 ) ⟶ ( Base ‘ 𝑀 ) )
71 simpr1 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → 𝑥 ∈ ( Base ‘ 𝑆 ) )
72 71 56 syl ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → ( ( Base ‘ 𝑀 ) × { 𝑥 } ) : ( Base ‘ 𝑀 ) ⟶ ( Base ‘ 𝑆 ) )
73 simpr2 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → 𝑦 ∈ ( Base ‘ 𝑆 ) )
74 fconst6g ⊢ ( 𝑦 ∈ ( Base ‘ 𝑆 ) → ( ( Base ‘ 𝑀 ) × { 𝑦 } ) : ( Base ‘ 𝑀 ) ⟶ ( Base ‘ 𝑆 ) )
75 73 74 syl ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → ( ( Base ‘ 𝑀 ) × { 𝑦 } ) : ( Base ‘ 𝑀 ) ⟶ ( Base ‘ 𝑆 ) )
76 simpll ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → 𝑀 ∈ LMod )
77 eqid ⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 )
78 21 30 2 19 20 77 lmodvsdir ⊢ ( ( 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 ⊢ ( ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → 𝑆 ∈ Ring )
82 20 77 ringacl ⊢ ( ( 𝑆 ∈ Ring ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ∈ ( Base ‘ 𝑆 ) )
83 81 71 73 82 syl3anc ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) ∈ ( Base ‘ 𝑆 ) )
84 1 19 3 2 20 21 22 mendvsca ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑧 ) ∈ ( 𝑀 LMHom 𝑀 ) )
90 eleq1w ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ ( Base ‘ 𝑆 ) ↔ 𝑦 ∈ ( Base ‘ 𝑆 ) ) )
91 90 3anbi2d ⊢ ( 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 ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑧 ) = ( 𝑦 ( ·𝑠 ‘ 𝐴 ) 𝑧 ) )
93 92 eleq1d ⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ( ·𝑠 ‘ 𝐴 ) 𝑧 ) ∈ ( 𝑀 LMHom 𝑀 ) ↔ ( 𝑦 ( ·𝑠 ‘ 𝐴 ) 𝑧 ) ∈ ( 𝑀 LMHom 𝑀 ) ) )
94 91 93 imbi12d ⊢ ( 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 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) → ( 𝑦 ( ·𝑠 ‘ 𝐴 ) 𝑧 ) ∈ ( 𝑀 LMHom 𝑀 ) )
96 95 3adant3r1 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → ( 𝑦 ( ·𝑠 ‘ 𝐴 ) 𝑧 ) ∈ ( 𝑀 LMHom 𝑀 ) )
97 1 3 30 31 mendplusg ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 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 3 2 20 21 22 mendvsca ⊢ ( ( 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) → ( 𝑦 ( ·𝑠 ‘ 𝐴 ) 𝑧 ) = ( ( ( Base ‘ 𝑀 ) × { 𝑦 } ) ∘f ( ·𝑠 ‘ 𝑀 ) 𝑧 ) )
101 73 69 100 syl2anc ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( ( 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 ⊢ ( ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) ∧ 𝑘 ∈ ( Base ‘ 𝑀 ) ) → ( 𝑧 ‘ 𝑘 ) ∈ ( Base ‘ 𝑀 ) )
107 fconstmpt ⊢ ( ( Base ‘ 𝑀 ) × { ( 𝑥 ( .r ‘ 𝑆 ) 𝑦 ) } ) = ( 𝑘 ∈ ( Base ‘ 𝑀 ) ↦ ( 𝑥 ( .r ‘ 𝑆 ) 𝑦 ) )
108 107 a1i ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 )
112 20 111 ringcl ⊢ ( ( 𝑆 ∈ Ring ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝑥 ( .r ‘ 𝑆 ) 𝑦 ) ∈ ( Base ‘ 𝑆 ) )
113 81 71 73 112 syl3anc ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → ( 𝑥 ( .r ‘ 𝑆 ) 𝑦 ) ∈ ( Base ‘ 𝑆 ) )
114 1 19 3 2 20 21 22 mendvsca ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) ∧ 𝑘 ∈ ( Base ‘ 𝑀 ) ) → 𝑥 ∈ ( Base ‘ 𝑆 ) )
117 ovexd ⊢ ( ( ( ( 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 ⊢ ( ( Base ‘ 𝑀 ) × { 𝑥 } ) = ( 𝑘 ∈ ( Base ‘ 𝑀 ) ↦ 𝑥 )
119 118 a1i ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) → ( ( Base ‘ 𝑀 ) × { 𝑥 } ) = ( 𝑘 ∈ ( Base ‘ 𝑀 ) ↦ 𝑥 ) )
120 simplr2 ⊢ ( ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) ∧ 𝑘 ∈ ( Base ‘ 𝑀 ) ) → 𝑦 ∈ ( Base ‘ 𝑆 ) )
121 fconstmpt ⊢ ( ( Base ‘ 𝑀 ) × { 𝑦 } ) = ( 𝑘 ∈ ( Base ‘ 𝑀 ) ↦ 𝑦 )
122 121 a1i ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 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 3 2 20 21 22 mendvsca ⊢ ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ 𝑧 ∈ ( 𝑀 LMHom 𝑀 ) ) ) ∧ 𝑘 ∈ ( Base ‘ 𝑀 ) ) → 𝑀 ∈ LMod )
129 21 2 19 20 111 lmodvsass ⊢ ( ( 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 ⊢ ( ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 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 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ 𝑥 ∈ ( 𝑀 LMHom 𝑀 ) ) → 𝑆 ∈ Ring )
135 eqid ⊢ ( 1r ‘ 𝑆 ) = ( 1r ‘ 𝑆 )
136 20 135 ringidcl ⊢ ( 𝑆 ∈ Ring → ( 1r ‘ 𝑆 ) ∈ ( Base ‘ 𝑆 ) )
137 134 136 syl ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ 𝑥 ∈ ( 𝑀 LMHom 𝑀 ) ) → ( 1r ‘ 𝑆 ) ∈ ( Base ‘ 𝑆 ) )
138 1 19 3 2 20 21 22 mendvsca ⊢ ( ( ( 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 ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ 𝑥 ∈ ( 𝑀 LMHom 𝑀 ) ) → ( ( 1r ‘ 𝑆 ) ( ·𝑠 ‘ 𝐴 ) 𝑥 ) = ( ( ( Base ‘ 𝑀 ) × { ( 1r ‘ 𝑆 ) } ) ∘f ( ·𝑠 ‘ 𝑀 ) 𝑥 ) )
140 fvexd ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ 𝑥 ∈ ( 𝑀 LMHom 𝑀 ) ) → ( Base ‘ 𝑀 ) ∈ V )
141 21 21 lmhmf ⊢ ( 𝑥 ∈ ( 𝑀 LMHom 𝑀 ) → 𝑥 : ( Base ‘ 𝑀 ) ⟶ ( Base ‘ 𝑀 ) )
142 141 adantl ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ 𝑥 ∈ ( 𝑀 LMHom 𝑀 ) ) → 𝑥 : ( Base ‘ 𝑀 ) ⟶ ( Base ‘ 𝑀 ) )
143 simpll ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ 𝑥 ∈ ( 𝑀 LMHom 𝑀 ) ) → 𝑀 ∈ LMod )
144 21 2 19 135 lmodvs1 ⊢ ( ( 𝑀 ∈ LMod ∧ 𝑦 ∈ ( Base ‘ 𝑀 ) ) → ( ( 1r ‘ 𝑆 ) ( ·𝑠 ‘ 𝑀 ) 𝑦 ) = 𝑦 )
145 143 144 sylan ⊢ ( ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ 𝑥 ∈ ( 𝑀 LMHom 𝑀 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑀 ) ) → ( ( 1r ‘ 𝑆 ) ( ·𝑠 ‘ 𝑀 ) 𝑦 ) = 𝑦 )
146 140 142 137 145 caofid0l ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ 𝑥 ∈ ( 𝑀 LMHom 𝑀 ) ) → ( ( ( Base ‘ 𝑀 ) × { ( 1r ‘ 𝑆 ) } ) ∘f ( ·𝑠 ‘ 𝑀 ) 𝑥 ) = 𝑥 )
147 139 146 eqtrd ⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) ∧ 𝑥 ∈ ( 𝑀 LMHom 𝑀 ) ) → ( ( 1r ‘ 𝑆 ) ( ·𝑠 ‘ 𝐴 ) 𝑥 ) = 𝑥 )
148 4 5 7 8 9 10 11 12 14 18 27 67 104 133 147 islmodd ⊢ ( ( 𝑀 ∈ LMod ∧ 𝑆 ∈ CRing ) → 𝐴 ∈ LMod )