Step |
Hyp |
Ref |
Expression |
1 |
|
lmodcmn |
⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ CMnd ) |
2 |
|
eqid |
⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 ) |
3 |
2
|
lmodring |
⊢ ( 𝑊 ∈ LMod → ( Scalar ‘ 𝑊 ) ∈ Ring ) |
4 |
|
ringsrg |
⊢ ( ( Scalar ‘ 𝑊 ) ∈ Ring → ( Scalar ‘ 𝑊 ) ∈ SRing ) |
5 |
3 4
|
syl |
⊢ ( 𝑊 ∈ LMod → ( Scalar ‘ 𝑊 ) ∈ SRing ) |
6 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
7 |
|
eqid |
⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 ) |
8 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑊 ) = ( ·𝑠 ‘ 𝑊 ) |
9 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) ) |
10 |
|
eqid |
⊢ ( +g ‘ ( Scalar ‘ 𝑊 ) ) = ( +g ‘ ( Scalar ‘ 𝑊 ) ) |
11 |
|
eqid |
⊢ ( .r ‘ ( Scalar ‘ 𝑊 ) ) = ( .r ‘ ( Scalar ‘ 𝑊 ) ) |
12 |
|
eqid |
⊢ ( 1r ‘ ( Scalar ‘ 𝑊 ) ) = ( 1r ‘ ( Scalar ‘ 𝑊 ) ) |
13 |
6 7 8 2 9 10 11 12
|
islmod |
⊢ ( 𝑊 ∈ LMod ↔ ( 𝑊 ∈ Grp ∧ ( Scalar ‘ 𝑊 ) ∈ Ring ∧ ∀ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑤 ∈ ( Base ‘ 𝑊 ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑤 ( +g ‘ 𝑊 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = 𝑤 ) ) ) ) |
14 |
13
|
simp3bi |
⊢ ( 𝑊 ∈ LMod → ∀ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑤 ∈ ( Base ‘ 𝑊 ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑤 ( +g ‘ 𝑊 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = 𝑤 ) ) ) |
15 |
14
|
r19.21bi |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) → ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑤 ∈ ( Base ‘ 𝑊 ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑤 ( +g ‘ 𝑊 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = 𝑤 ) ) ) |
16 |
15
|
r19.21bi |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑤 ∈ ( Base ‘ 𝑊 ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑤 ( +g ‘ 𝑊 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = 𝑤 ) ) ) |
17 |
16
|
r19.21bi |
⊢ ( ( ( ( 𝑊 ∈ LMod ∧ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ∀ 𝑤 ∈ ( Base ‘ 𝑊 ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑤 ( +g ‘ 𝑊 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = 𝑤 ) ) ) |
18 |
17
|
r19.21bi |
⊢ ( ( ( ( ( 𝑊 ∈ LMod ∧ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝑊 ) ) → ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑤 ( +g ‘ 𝑊 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = 𝑤 ) ) ) |
19 |
18
|
simpld |
⊢ ( ( ( ( ( 𝑊 ∈ LMod ∧ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝑊 ) ) → ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑤 ( +g ‘ 𝑊 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ) ) |
20 |
18
|
simprd |
⊢ ( ( ( ( ( 𝑊 ∈ LMod ∧ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝑊 ) ) → ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = 𝑤 ) ) |
21 |
20
|
simpld |
⊢ ( ( ( ( ( 𝑊 ∈ LMod ∧ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝑊 ) ) → ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ) |
22 |
20
|
simprd |
⊢ ( ( ( ( ( 𝑊 ∈ LMod ∧ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝑊 ) ) → ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = 𝑤 ) |
23 |
|
simp-4l |
⊢ ( ( ( ( ( 𝑊 ∈ LMod ∧ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝑊 ) ) → 𝑊 ∈ LMod ) |
24 |
|
eqid |
⊢ ( 0g ‘ ( Scalar ‘ 𝑊 ) ) = ( 0g ‘ ( Scalar ‘ 𝑊 ) ) |
25 |
|
eqid |
⊢ ( 0g ‘ 𝑊 ) = ( 0g ‘ 𝑊 ) |
26 |
6 2 8 24 25
|
lmod0vs |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑤 ∈ ( Base ‘ 𝑊 ) ) → ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 0g ‘ 𝑊 ) ) |
27 |
23 26
|
sylancom |
⊢ ( ( ( ( ( 𝑊 ∈ LMod ∧ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝑊 ) ) → ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 0g ‘ 𝑊 ) ) |
28 |
21 22 27
|
3jca |
⊢ ( ( ( ( ( 𝑊 ∈ LMod ∧ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝑊 ) ) → ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = 𝑤 ∧ ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 0g ‘ 𝑊 ) ) ) |
29 |
19 28
|
jca |
⊢ ( ( ( ( ( 𝑊 ∈ LMod ∧ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) ∧ 𝑤 ∈ ( Base ‘ 𝑊 ) ) → ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑤 ( +g ‘ 𝑊 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = 𝑤 ∧ ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 0g ‘ 𝑊 ) ) ) ) |
30 |
29
|
ralrimiva |
⊢ ( ( ( ( 𝑊 ∈ LMod ∧ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ ( Base ‘ 𝑊 ) ) → ∀ 𝑤 ∈ ( Base ‘ 𝑊 ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑤 ( +g ‘ 𝑊 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = 𝑤 ∧ ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 0g ‘ 𝑊 ) ) ) ) |
31 |
30
|
ralrimiva |
⊢ ( ( ( 𝑊 ∈ LMod ∧ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) ∧ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑤 ∈ ( Base ‘ 𝑊 ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑤 ( +g ‘ 𝑊 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = 𝑤 ∧ ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 0g ‘ 𝑊 ) ) ) ) |
32 |
31
|
ralrimiva |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ) → ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑤 ∈ ( Base ‘ 𝑊 ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑤 ( +g ‘ 𝑊 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = 𝑤 ∧ ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 0g ‘ 𝑊 ) ) ) ) |
33 |
32
|
ralrimiva |
⊢ ( 𝑊 ∈ LMod → ∀ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑤 ∈ ( Base ‘ 𝑊 ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑤 ( +g ‘ 𝑊 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = 𝑤 ∧ ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 0g ‘ 𝑊 ) ) ) ) |
34 |
6 7 8 25 2 9 10 11 12 24
|
isslmd |
⊢ ( 𝑊 ∈ SLMod ↔ ( 𝑊 ∈ CMnd ∧ ( Scalar ‘ 𝑊 ) ∈ SRing ∧ ∀ 𝑞 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑟 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∀ 𝑥 ∈ ( Base ‘ 𝑊 ) ∀ 𝑤 ∈ ( Base ‘ 𝑊 ) ( ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ∈ ( Base ‘ 𝑊 ) ∧ ( 𝑟 ( ·𝑠 ‘ 𝑊 ) ( 𝑤 ( +g ‘ 𝑊 ) 𝑥 ) ) = ( ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑥 ) ) ∧ ( ( 𝑞 ( +g ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( ( 𝑞 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ( +g ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ) ∧ ( ( ( 𝑞 ( .r ‘ ( Scalar ‘ 𝑊 ) ) 𝑟 ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 𝑞 ( ·𝑠 ‘ 𝑊 ) ( 𝑟 ( ·𝑠 ‘ 𝑊 ) 𝑤 ) ) ∧ ( ( 1r ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = 𝑤 ∧ ( ( 0g ‘ ( Scalar ‘ 𝑊 ) ) ( ·𝑠 ‘ 𝑊 ) 𝑤 ) = ( 0g ‘ 𝑊 ) ) ) ) ) |
35 |
1 5 33 34
|
syl3anbrc |
⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ SLMod ) |