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
|
ffvelrnda |
⊢ ( ( ( ( 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 ) |