Step |
Hyp |
Ref |
Expression |
1 |
|
nmfval2.n |
⊢ 𝑁 = ( norm ‘ 𝑊 ) |
2 |
|
nmfval2.x |
⊢ 𝑋 = ( Base ‘ 𝑊 ) |
3 |
|
nmfval2.z |
⊢ 0 = ( 0g ‘ 𝑊 ) |
4 |
|
nmfval2.d |
⊢ 𝐷 = ( dist ‘ 𝑊 ) |
5 |
|
nmfval2.e |
⊢ 𝐸 = ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) |
6 |
1 2 3 4
|
nmval |
⊢ ( 𝐴 ∈ 𝑋 → ( 𝑁 ‘ 𝐴 ) = ( 𝐴 𝐷 0 ) ) |
7 |
6
|
adantl |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) = ( 𝐴 𝐷 0 ) ) |
8 |
5
|
oveqi |
⊢ ( 𝐴 𝐸 0 ) = ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 0 ) |
9 |
|
id |
⊢ ( 𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑋 ) |
10 |
2 3
|
grpidcl |
⊢ ( 𝑊 ∈ Grp → 0 ∈ 𝑋 ) |
11 |
|
ovres |
⊢ ( ( 𝐴 ∈ 𝑋 ∧ 0 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 0 ) = ( 𝐴 𝐷 0 ) ) |
12 |
9 10 11
|
syl2anr |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) 0 ) = ( 𝐴 𝐷 0 ) ) |
13 |
8 12
|
eqtr2id |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐷 0 ) = ( 𝐴 𝐸 0 ) ) |
14 |
7 13
|
eqtrd |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) = ( 𝐴 𝐸 0 ) ) |