Step |
Hyp |
Ref |
Expression |
1 |
|
tnglvec.t |
⊢ 𝑇 = ( 𝐺 toNrmGrp 𝑁 ) |
2 |
|
eqidd |
⊢ ( ( 𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉 ) → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) ) |
3 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
4 |
1 3
|
tngbas |
⊢ ( 𝑁 ∈ 𝑉 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝑇 ) ) |
5 |
4
|
adantl |
⊢ ( ( 𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉 ) → ( Base ‘ 𝐺 ) = ( Base ‘ 𝑇 ) ) |
6 |
|
ssidd |
⊢ ( ( 𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉 ) → ( Base ‘ 𝐺 ) ⊆ ( Base ‘ 𝐺 ) ) |
7 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
8 |
1 7
|
tngplusg |
⊢ ( 𝑁 ∈ 𝑉 → ( +g ‘ 𝐺 ) = ( +g ‘ 𝑇 ) ) |
9 |
8
|
adantl |
⊢ ( ( 𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉 ) → ( +g ‘ 𝐺 ) = ( +g ‘ 𝑇 ) ) |
10 |
9
|
oveqdr |
⊢ ( ( ( 𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝐺 ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑇 ) 𝑦 ) ) |
11 |
|
lveclmod |
⊢ ( 𝐺 ∈ LVec → 𝐺 ∈ LMod ) |
12 |
|
eqid |
⊢ ( Scalar ‘ 𝐺 ) = ( Scalar ‘ 𝐺 ) |
13 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝐺 ) = ( ·𝑠 ‘ 𝐺 ) |
14 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝐺 ) ) = ( Base ‘ ( Scalar ‘ 𝐺 ) ) |
15 |
3 12 13 14
|
lmodvscl |
⊢ ( ( 𝐺 ∈ LMod ∧ 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐺 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐺 ) 𝑦 ) ∈ ( Base ‘ 𝐺 ) ) |
16 |
15
|
3expb |
⊢ ( ( 𝐺 ∈ LMod ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐺 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐺 ) 𝑦 ) ∈ ( Base ‘ 𝐺 ) ) |
17 |
11 16
|
sylan |
⊢ ( ( 𝐺 ∈ LVec ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐺 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐺 ) 𝑦 ) ∈ ( Base ‘ 𝐺 ) ) |
18 |
17
|
adantlr |
⊢ ( ( ( 𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉 ) ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐺 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐺 ) 𝑦 ) ∈ ( Base ‘ 𝐺 ) ) |
19 |
1 13
|
tngvsca |
⊢ ( 𝑁 ∈ 𝑉 → ( ·𝑠 ‘ 𝐺 ) = ( ·𝑠 ‘ 𝑇 ) ) |
20 |
19
|
adantl |
⊢ ( ( 𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉 ) → ( ·𝑠 ‘ 𝐺 ) = ( ·𝑠 ‘ 𝑇 ) ) |
21 |
20
|
oveqdr |
⊢ ( ( ( 𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉 ) ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐺 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑥 ( ·𝑠 ‘ 𝐺 ) 𝑦 ) = ( 𝑥 ( ·𝑠 ‘ 𝑇 ) 𝑦 ) ) |
22 |
|
eqid |
⊢ ( Scalar ‘ 𝑇 ) = ( Scalar ‘ 𝑇 ) |
23 |
|
eqidd |
⊢ ( ( 𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉 ) → ( Base ‘ ( Scalar ‘ 𝐺 ) ) = ( Base ‘ ( Scalar ‘ 𝐺 ) ) ) |
24 |
1 12
|
tngsca |
⊢ ( 𝑁 ∈ 𝑉 → ( Scalar ‘ 𝐺 ) = ( Scalar ‘ 𝑇 ) ) |
25 |
24
|
adantl |
⊢ ( ( 𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉 ) → ( Scalar ‘ 𝐺 ) = ( Scalar ‘ 𝑇 ) ) |
26 |
25
|
fveq2d |
⊢ ( ( 𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉 ) → ( Base ‘ ( Scalar ‘ 𝐺 ) ) = ( Base ‘ ( Scalar ‘ 𝑇 ) ) ) |
27 |
25
|
fveq2d |
⊢ ( ( 𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉 ) → ( +g ‘ ( Scalar ‘ 𝐺 ) ) = ( +g ‘ ( Scalar ‘ 𝑇 ) ) ) |
28 |
27
|
oveqdr |
⊢ ( ( ( 𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉 ) ∧ ( 𝑥 ∈ ( Base ‘ ( Scalar ‘ 𝐺 ) ) ∧ 𝑦 ∈ ( Base ‘ ( Scalar ‘ 𝐺 ) ) ) ) → ( 𝑥 ( +g ‘ ( Scalar ‘ 𝐺 ) ) 𝑦 ) = ( 𝑥 ( +g ‘ ( Scalar ‘ 𝑇 ) ) 𝑦 ) ) |
29 |
|
simpl |
⊢ ( ( 𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉 ) → 𝐺 ∈ LVec ) |
30 |
1
|
tnglvec |
⊢ ( 𝑁 ∈ 𝑉 → ( 𝐺 ∈ LVec ↔ 𝑇 ∈ LVec ) ) |
31 |
30
|
biimpac |
⊢ ( ( 𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉 ) → 𝑇 ∈ LVec ) |
32 |
2 5 6 10 18 21 12 22 23 26 28 29 31
|
dimpropd |
⊢ ( ( 𝐺 ∈ LVec ∧ 𝑁 ∈ 𝑉 ) → ( dim ‘ 𝐺 ) = ( dim ‘ 𝑇 ) ) |