Step |
Hyp |
Ref |
Expression |
1 |
|
nmpropd2.1 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) ) |
2 |
|
nmpropd2.2 |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) ) |
3 |
|
nmpropd2.3 |
⊢ ( 𝜑 → 𝐾 ∈ Grp ) |
4 |
|
nmpropd2.4 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ) |
5 |
|
nmpropd2.5 |
⊢ ( 𝜑 → ( ( dist ‘ 𝐾 ) ↾ ( 𝐵 × 𝐵 ) ) = ( ( dist ‘ 𝐿 ) ↾ ( 𝐵 × 𝐵 ) ) ) |
6 |
1 2
|
eqtr3d |
⊢ ( 𝜑 → ( Base ‘ 𝐾 ) = ( Base ‘ 𝐿 ) ) |
7 |
1
|
sqxpeqd |
⊢ ( 𝜑 → ( 𝐵 × 𝐵 ) = ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) |
8 |
7
|
reseq2d |
⊢ ( 𝜑 → ( ( dist ‘ 𝐾 ) ↾ ( 𝐵 × 𝐵 ) ) = ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ) |
9 |
5 8
|
eqtr3d |
⊢ ( 𝜑 → ( ( dist ‘ 𝐿 ) ↾ ( 𝐵 × 𝐵 ) ) = ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ) |
10 |
2
|
sqxpeqd |
⊢ ( 𝜑 → ( 𝐵 × 𝐵 ) = ( ( Base ‘ 𝐿 ) × ( Base ‘ 𝐿 ) ) ) |
11 |
10
|
reseq2d |
⊢ ( 𝜑 → ( ( dist ‘ 𝐿 ) ↾ ( 𝐵 × 𝐵 ) ) = ( ( dist ‘ 𝐿 ) ↾ ( ( Base ‘ 𝐿 ) × ( Base ‘ 𝐿 ) ) ) ) |
12 |
9 11
|
eqtr3d |
⊢ ( 𝜑 → ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) = ( ( dist ‘ 𝐿 ) ↾ ( ( Base ‘ 𝐿 ) × ( Base ‘ 𝐿 ) ) ) ) |
13 |
|
eqidd |
⊢ ( 𝜑 → 𝑎 = 𝑎 ) |
14 |
1 2 4
|
grpidpropd |
⊢ ( 𝜑 → ( 0g ‘ 𝐾 ) = ( 0g ‘ 𝐿 ) ) |
15 |
12 13 14
|
oveq123d |
⊢ ( 𝜑 → ( 𝑎 ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ( 0g ‘ 𝐾 ) ) = ( 𝑎 ( ( dist ‘ 𝐿 ) ↾ ( ( Base ‘ 𝐿 ) × ( Base ‘ 𝐿 ) ) ) ( 0g ‘ 𝐿 ) ) ) |
16 |
6 15
|
mpteq12dv |
⊢ ( 𝜑 → ( 𝑎 ∈ ( Base ‘ 𝐾 ) ↦ ( 𝑎 ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ( 0g ‘ 𝐾 ) ) ) = ( 𝑎 ∈ ( Base ‘ 𝐿 ) ↦ ( 𝑎 ( ( dist ‘ 𝐿 ) ↾ ( ( Base ‘ 𝐿 ) × ( Base ‘ 𝐿 ) ) ) ( 0g ‘ 𝐿 ) ) ) ) |
17 |
|
eqid |
⊢ ( norm ‘ 𝐾 ) = ( norm ‘ 𝐾 ) |
18 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
19 |
|
eqid |
⊢ ( 0g ‘ 𝐾 ) = ( 0g ‘ 𝐾 ) |
20 |
|
eqid |
⊢ ( dist ‘ 𝐾 ) = ( dist ‘ 𝐾 ) |
21 |
|
eqid |
⊢ ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) = ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) |
22 |
17 18 19 20 21
|
nmfval2 |
⊢ ( 𝐾 ∈ Grp → ( norm ‘ 𝐾 ) = ( 𝑎 ∈ ( Base ‘ 𝐾 ) ↦ ( 𝑎 ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ( 0g ‘ 𝐾 ) ) ) ) |
23 |
3 22
|
syl |
⊢ ( 𝜑 → ( norm ‘ 𝐾 ) = ( 𝑎 ∈ ( Base ‘ 𝐾 ) ↦ ( 𝑎 ( ( dist ‘ 𝐾 ) ↾ ( ( Base ‘ 𝐾 ) × ( Base ‘ 𝐾 ) ) ) ( 0g ‘ 𝐾 ) ) ) ) |
24 |
1 2 4
|
grppropd |
⊢ ( 𝜑 → ( 𝐾 ∈ Grp ↔ 𝐿 ∈ Grp ) ) |
25 |
3 24
|
mpbid |
⊢ ( 𝜑 → 𝐿 ∈ Grp ) |
26 |
|
eqid |
⊢ ( norm ‘ 𝐿 ) = ( norm ‘ 𝐿 ) |
27 |
|
eqid |
⊢ ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 ) |
28 |
|
eqid |
⊢ ( 0g ‘ 𝐿 ) = ( 0g ‘ 𝐿 ) |
29 |
|
eqid |
⊢ ( dist ‘ 𝐿 ) = ( dist ‘ 𝐿 ) |
30 |
|
eqid |
⊢ ( ( dist ‘ 𝐿 ) ↾ ( ( Base ‘ 𝐿 ) × ( Base ‘ 𝐿 ) ) ) = ( ( dist ‘ 𝐿 ) ↾ ( ( Base ‘ 𝐿 ) × ( Base ‘ 𝐿 ) ) ) |
31 |
26 27 28 29 30
|
nmfval2 |
⊢ ( 𝐿 ∈ Grp → ( norm ‘ 𝐿 ) = ( 𝑎 ∈ ( Base ‘ 𝐿 ) ↦ ( 𝑎 ( ( dist ‘ 𝐿 ) ↾ ( ( Base ‘ 𝐿 ) × ( Base ‘ 𝐿 ) ) ) ( 0g ‘ 𝐿 ) ) ) ) |
32 |
25 31
|
syl |
⊢ ( 𝜑 → ( norm ‘ 𝐿 ) = ( 𝑎 ∈ ( Base ‘ 𝐿 ) ↦ ( 𝑎 ( ( dist ‘ 𝐿 ) ↾ ( ( Base ‘ 𝐿 ) × ( Base ‘ 𝐿 ) ) ) ( 0g ‘ 𝐿 ) ) ) ) |
33 |
16 23 32
|
3eqtr4d |
⊢ ( 𝜑 → ( norm ‘ 𝐾 ) = ( norm ‘ 𝐿 ) ) |