Step |
Hyp |
Ref |
Expression |
1 |
|
normsub.1 |
⊢ 𝐴 ∈ ℋ |
2 |
|
normsub.2 |
⊢ 𝐵 ∈ ℋ |
3 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
4 |
2 1
|
hvsubcli |
⊢ ( 𝐵 −ℎ 𝐴 ) ∈ ℋ |
5 |
3 4
|
norm-iii-i |
⊢ ( normℎ ‘ ( - 1 ·ℎ ( 𝐵 −ℎ 𝐴 ) ) ) = ( ( abs ‘ - 1 ) · ( normℎ ‘ ( 𝐵 −ℎ 𝐴 ) ) ) |
6 |
2 1
|
hvnegdii |
⊢ ( - 1 ·ℎ ( 𝐵 −ℎ 𝐴 ) ) = ( 𝐴 −ℎ 𝐵 ) |
7 |
6
|
fveq2i |
⊢ ( normℎ ‘ ( - 1 ·ℎ ( 𝐵 −ℎ 𝐴 ) ) ) = ( normℎ ‘ ( 𝐴 −ℎ 𝐵 ) ) |
8 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
9 |
8
|
absnegi |
⊢ ( abs ‘ - 1 ) = ( abs ‘ 1 ) |
10 |
|
abs1 |
⊢ ( abs ‘ 1 ) = 1 |
11 |
9 10
|
eqtri |
⊢ ( abs ‘ - 1 ) = 1 |
12 |
11
|
oveq1i |
⊢ ( ( abs ‘ - 1 ) · ( normℎ ‘ ( 𝐵 −ℎ 𝐴 ) ) ) = ( 1 · ( normℎ ‘ ( 𝐵 −ℎ 𝐴 ) ) ) |
13 |
4
|
normcli |
⊢ ( normℎ ‘ ( 𝐵 −ℎ 𝐴 ) ) ∈ ℝ |
14 |
13
|
recni |
⊢ ( normℎ ‘ ( 𝐵 −ℎ 𝐴 ) ) ∈ ℂ |
15 |
14
|
mulid2i |
⊢ ( 1 · ( normℎ ‘ ( 𝐵 −ℎ 𝐴 ) ) ) = ( normℎ ‘ ( 𝐵 −ℎ 𝐴 ) ) |
16 |
12 15
|
eqtri |
⊢ ( ( abs ‘ - 1 ) · ( normℎ ‘ ( 𝐵 −ℎ 𝐴 ) ) ) = ( normℎ ‘ ( 𝐵 −ℎ 𝐴 ) ) |
17 |
5 7 16
|
3eqtr3i |
⊢ ( normℎ ‘ ( 𝐴 −ℎ 𝐵 ) ) = ( normℎ ‘ ( 𝐵 −ℎ 𝐴 ) ) |