Step |
Hyp |
Ref |
Expression |
1 |
|
lincvalsc0.b |
⊢ 𝐵 = ( Base ‘ 𝑀 ) |
2 |
|
lincvalsc0.s |
⊢ 𝑆 = ( Scalar ‘ 𝑀 ) |
3 |
|
lincvalsc0.0 |
⊢ 0 = ( 0g ‘ 𝑆 ) |
4 |
|
lincvalsc0.z |
⊢ 𝑍 = ( 0g ‘ 𝑀 ) |
5 |
|
lincvalsc0.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝑉 ↦ 0 ) |
6 |
|
simpl |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → 𝑀 ∈ LMod ) |
7 |
2
|
eqcomi |
⊢ ( Scalar ‘ 𝑀 ) = 𝑆 |
8 |
7
|
fveq2i |
⊢ ( Base ‘ ( Scalar ‘ 𝑀 ) ) = ( Base ‘ 𝑆 ) |
9 |
2 8 3
|
lmod0cl |
⊢ ( 𝑀 ∈ LMod → 0 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ) |
10 |
9
|
adantr |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → 0 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ) |
11 |
10
|
adantr |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) ∧ 𝑥 ∈ 𝑉 ) → 0 ∈ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ) |
12 |
11 5
|
fmptd |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → 𝐹 : 𝑉 ⟶ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ) |
13 |
|
fvexd |
⊢ ( 𝑀 ∈ LMod → ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∈ V ) |
14 |
|
elmapg |
⊢ ( ( ( Base ‘ ( Scalar ‘ 𝑀 ) ) ∈ V ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝐹 ∈ ( ( Base ‘ ( Scalar ‘ 𝑀 ) ) ↑m 𝑉 ) ↔ 𝐹 : 𝑉 ⟶ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ) ) |
15 |
13 14
|
sylan |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝐹 ∈ ( ( Base ‘ ( Scalar ‘ 𝑀 ) ) ↑m 𝑉 ) ↔ 𝐹 : 𝑉 ⟶ ( Base ‘ ( Scalar ‘ 𝑀 ) ) ) ) |
16 |
12 15
|
mpbird |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → 𝐹 ∈ ( ( Base ‘ ( Scalar ‘ 𝑀 ) ) ↑m 𝑉 ) ) |
17 |
1
|
pweqi |
⊢ 𝒫 𝐵 = 𝒫 ( Base ‘ 𝑀 ) |
18 |
17
|
eleq2i |
⊢ ( 𝑉 ∈ 𝒫 𝐵 ↔ 𝑉 ∈ 𝒫 ( Base ‘ 𝑀 ) ) |
19 |
18
|
biimpi |
⊢ ( 𝑉 ∈ 𝒫 𝐵 → 𝑉 ∈ 𝒫 ( Base ‘ 𝑀 ) ) |
20 |
19
|
adantl |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → 𝑉 ∈ 𝒫 ( Base ‘ 𝑀 ) ) |
21 |
|
lincval |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝐹 ∈ ( ( Base ‘ ( Scalar ‘ 𝑀 ) ) ↑m 𝑉 ) ∧ 𝑉 ∈ 𝒫 ( Base ‘ 𝑀 ) ) → ( 𝐹 ( linC ‘ 𝑀 ) 𝑉 ) = ( 𝑀 Σg ( 𝑣 ∈ 𝑉 ↦ ( ( 𝐹 ‘ 𝑣 ) ( ·𝑠 ‘ 𝑀 ) 𝑣 ) ) ) ) |
22 |
6 16 20 21
|
syl3anc |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝐹 ( linC ‘ 𝑀 ) 𝑉 ) = ( 𝑀 Σg ( 𝑣 ∈ 𝑉 ↦ ( ( 𝐹 ‘ 𝑣 ) ( ·𝑠 ‘ 𝑀 ) 𝑣 ) ) ) ) |
23 |
|
simpr |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) ∧ 𝑣 ∈ 𝑉 ) → 𝑣 ∈ 𝑉 ) |
24 |
3
|
fvexi |
⊢ 0 ∈ V |
25 |
|
eqidd |
⊢ ( 𝑥 = 𝑣 → 0 = 0 ) |
26 |
25 5
|
fvmptg |
⊢ ( ( 𝑣 ∈ 𝑉 ∧ 0 ∈ V ) → ( 𝐹 ‘ 𝑣 ) = 0 ) |
27 |
23 24 26
|
sylancl |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) ∧ 𝑣 ∈ 𝑉 ) → ( 𝐹 ‘ 𝑣 ) = 0 ) |
28 |
27
|
oveq1d |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) ∧ 𝑣 ∈ 𝑉 ) → ( ( 𝐹 ‘ 𝑣 ) ( ·𝑠 ‘ 𝑀 ) 𝑣 ) = ( 0 ( ·𝑠 ‘ 𝑀 ) 𝑣 ) ) |
29 |
6
|
adantr |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) ∧ 𝑣 ∈ 𝑉 ) → 𝑀 ∈ LMod ) |
30 |
|
elelpwi |
⊢ ( ( 𝑣 ∈ 𝑉 ∧ 𝑉 ∈ 𝒫 𝐵 ) → 𝑣 ∈ 𝐵 ) |
31 |
30
|
expcom |
⊢ ( 𝑉 ∈ 𝒫 𝐵 → ( 𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵 ) ) |
32 |
31
|
adantl |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝑣 ∈ 𝑉 → 𝑣 ∈ 𝐵 ) ) |
33 |
32
|
imp |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) ∧ 𝑣 ∈ 𝑉 ) → 𝑣 ∈ 𝐵 ) |
34 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑀 ) = ( ·𝑠 ‘ 𝑀 ) |
35 |
1 2 34 3 4
|
lmod0vs |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑣 ∈ 𝐵 ) → ( 0 ( ·𝑠 ‘ 𝑀 ) 𝑣 ) = 𝑍 ) |
36 |
29 33 35
|
syl2anc |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) ∧ 𝑣 ∈ 𝑉 ) → ( 0 ( ·𝑠 ‘ 𝑀 ) 𝑣 ) = 𝑍 ) |
37 |
28 36
|
eqtrd |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) ∧ 𝑣 ∈ 𝑉 ) → ( ( 𝐹 ‘ 𝑣 ) ( ·𝑠 ‘ 𝑀 ) 𝑣 ) = 𝑍 ) |
38 |
37
|
mpteq2dva |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝑣 ∈ 𝑉 ↦ ( ( 𝐹 ‘ 𝑣 ) ( ·𝑠 ‘ 𝑀 ) 𝑣 ) ) = ( 𝑣 ∈ 𝑉 ↦ 𝑍 ) ) |
39 |
38
|
oveq2d |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝑀 Σg ( 𝑣 ∈ 𝑉 ↦ ( ( 𝐹 ‘ 𝑣 ) ( ·𝑠 ‘ 𝑀 ) 𝑣 ) ) ) = ( 𝑀 Σg ( 𝑣 ∈ 𝑉 ↦ 𝑍 ) ) ) |
40 |
|
lmodgrp |
⊢ ( 𝑀 ∈ LMod → 𝑀 ∈ Grp ) |
41 |
40
|
grpmndd |
⊢ ( 𝑀 ∈ LMod → 𝑀 ∈ Mnd ) |
42 |
4
|
gsumz |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝑀 Σg ( 𝑣 ∈ 𝑉 ↦ 𝑍 ) ) = 𝑍 ) |
43 |
41 42
|
sylan |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝑀 Σg ( 𝑣 ∈ 𝑉 ↦ 𝑍 ) ) = 𝑍 ) |
44 |
22 39 43
|
3eqtrd |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ) → ( 𝐹 ( linC ‘ 𝑀 ) 𝑉 ) = 𝑍 ) |