Step |
Hyp |
Ref |
Expression |
1 |
|
ax-hv0cl |
⊢ 0ℎ ∈ ℋ |
2 |
|
normsub |
⊢ ( ( 0ℎ ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( normℎ ‘ ( 0ℎ −ℎ 𝐴 ) ) = ( normℎ ‘ ( 𝐴 −ℎ 0ℎ ) ) ) |
3 |
1 2
|
mpan |
⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ ( 0ℎ −ℎ 𝐴 ) ) = ( normℎ ‘ ( 𝐴 −ℎ 0ℎ ) ) ) |
4 |
|
hv2neg |
⊢ ( 𝐴 ∈ ℋ → ( 0ℎ −ℎ 𝐴 ) = ( - 1 ·ℎ 𝐴 ) ) |
5 |
4
|
fveq2d |
⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ ( 0ℎ −ℎ 𝐴 ) ) = ( normℎ ‘ ( - 1 ·ℎ 𝐴 ) ) ) |
6 |
|
hvsub0 |
⊢ ( 𝐴 ∈ ℋ → ( 𝐴 −ℎ 0ℎ ) = 𝐴 ) |
7 |
6
|
fveq2d |
⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ ( 𝐴 −ℎ 0ℎ ) ) = ( normℎ ‘ 𝐴 ) ) |
8 |
3 5 7
|
3eqtr3d |
⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ ( - 1 ·ℎ 𝐴 ) ) = ( normℎ ‘ 𝐴 ) ) |