Step |
Hyp |
Ref |
Expression |
1 |
|
subgngp.h |
⊢ 𝐻 = ( 𝐺 ↾s 𝐴 ) |
2 |
|
subgnm.n |
⊢ 𝑁 = ( norm ‘ 𝐺 ) |
3 |
|
subgnm.m |
⊢ 𝑀 = ( norm ‘ 𝐻 ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
5 |
4
|
subgss |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → 𝐴 ⊆ ( Base ‘ 𝐺 ) ) |
6 |
5
|
resmptd |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( 𝑥 ( dist ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) ) ↾ 𝐴 ) = ( 𝑥 ∈ 𝐴 ↦ ( 𝑥 ( dist ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) ) ) |
7 |
1
|
subgbas |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → 𝐴 = ( Base ‘ 𝐻 ) ) |
8 |
|
eqid |
⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 ) |
9 |
1 8
|
ressds |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( dist ‘ 𝐺 ) = ( dist ‘ 𝐻 ) ) |
10 |
|
eqidd |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → 𝑥 = 𝑥 ) |
11 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
12 |
1 11
|
subg0 |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐻 ) ) |
13 |
9 10 12
|
oveq123d |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑥 ( dist ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) = ( 𝑥 ( dist ‘ 𝐻 ) ( 0g ‘ 𝐻 ) ) ) |
14 |
7 13
|
mpteq12dv |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑥 ∈ 𝐴 ↦ ( 𝑥 ( dist ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) ) = ( 𝑥 ∈ ( Base ‘ 𝐻 ) ↦ ( 𝑥 ( dist ‘ 𝐻 ) ( 0g ‘ 𝐻 ) ) ) ) |
15 |
6 14
|
eqtr2d |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑥 ∈ ( Base ‘ 𝐻 ) ↦ ( 𝑥 ( dist ‘ 𝐻 ) ( 0g ‘ 𝐻 ) ) ) = ( ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( 𝑥 ( dist ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) ) ↾ 𝐴 ) ) |
16 |
|
eqid |
⊢ ( Base ‘ 𝐻 ) = ( Base ‘ 𝐻 ) |
17 |
|
eqid |
⊢ ( 0g ‘ 𝐻 ) = ( 0g ‘ 𝐻 ) |
18 |
|
eqid |
⊢ ( dist ‘ 𝐻 ) = ( dist ‘ 𝐻 ) |
19 |
3 16 17 18
|
nmfval |
⊢ 𝑀 = ( 𝑥 ∈ ( Base ‘ 𝐻 ) ↦ ( 𝑥 ( dist ‘ 𝐻 ) ( 0g ‘ 𝐻 ) ) ) |
20 |
2 4 11 8
|
nmfval |
⊢ 𝑁 = ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( 𝑥 ( dist ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) ) |
21 |
20
|
reseq1i |
⊢ ( 𝑁 ↾ 𝐴 ) = ( ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( 𝑥 ( dist ‘ 𝐺 ) ( 0g ‘ 𝐺 ) ) ) ↾ 𝐴 ) |
22 |
15 19 21
|
3eqtr4g |
⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝐺 ) → 𝑀 = ( 𝑁 ↾ 𝐴 ) ) |