Step |
Hyp |
Ref |
Expression |
1 |
|
cnnvm.6 |
⊢ 𝑈 = 〈 〈 + , · 〉 , abs 〉 |
2 |
|
mulm1 |
⊢ ( 𝑦 ∈ ℂ → ( - 1 · 𝑦 ) = - 𝑦 ) |
3 |
2
|
adantl |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( - 1 · 𝑦 ) = - 𝑦 ) |
4 |
3
|
oveq2d |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 + ( - 1 · 𝑦 ) ) = ( 𝑥 + - 𝑦 ) ) |
5 |
|
negsub |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 + - 𝑦 ) = ( 𝑥 − 𝑦 ) ) |
6 |
4 5
|
eqtr2d |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 − 𝑦 ) = ( 𝑥 + ( - 1 · 𝑦 ) ) ) |
7 |
6
|
mpoeq3ia |
⊢ ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 − 𝑦 ) ) = ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + ( - 1 · 𝑦 ) ) ) |
8 |
|
subf |
⊢ − : ( ℂ × ℂ ) ⟶ ℂ |
9 |
|
ffn |
⊢ ( − : ( ℂ × ℂ ) ⟶ ℂ → − Fn ( ℂ × ℂ ) ) |
10 |
8 9
|
ax-mp |
⊢ − Fn ( ℂ × ℂ ) |
11 |
|
fnov |
⊢ ( − Fn ( ℂ × ℂ ) ↔ − = ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 − 𝑦 ) ) ) |
12 |
10 11
|
mpbi |
⊢ − = ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 − 𝑦 ) ) |
13 |
1
|
cnnv |
⊢ 𝑈 ∈ NrmCVec |
14 |
1
|
cnnvba |
⊢ ℂ = ( BaseSet ‘ 𝑈 ) |
15 |
1
|
cnnvg |
⊢ + = ( +𝑣 ‘ 𝑈 ) |
16 |
1
|
cnnvs |
⊢ · = ( ·𝑠OLD ‘ 𝑈 ) |
17 |
|
eqid |
⊢ ( −𝑣 ‘ 𝑈 ) = ( −𝑣 ‘ 𝑈 ) |
18 |
14 15 16 17
|
nvmfval |
⊢ ( 𝑈 ∈ NrmCVec → ( −𝑣 ‘ 𝑈 ) = ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + ( - 1 · 𝑦 ) ) ) ) |
19 |
13 18
|
ax-mp |
⊢ ( −𝑣 ‘ 𝑈 ) = ( 𝑥 ∈ ℂ , 𝑦 ∈ ℂ ↦ ( 𝑥 + ( - 1 · 𝑦 ) ) ) |
20 |
7 12 19
|
3eqtr4i |
⊢ − = ( −𝑣 ‘ 𝑈 ) |