Step |
Hyp |
Ref |
Expression |
1 |
|
cncph.6 |
⊢ 𝑈 = 〈 〈 + , · 〉 , abs 〉 |
2 |
|
eqid |
⊢ 〈 〈 + , · 〉 , abs 〉 = 〈 〈 + , · 〉 , abs 〉 |
3 |
2
|
cnnv |
⊢ 〈 〈 + , · 〉 , abs 〉 ∈ NrmCVec |
4 |
|
mulm1 |
⊢ ( 𝑦 ∈ ℂ → ( - 1 · 𝑦 ) = - 𝑦 ) |
5 |
4
|
adantl |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( - 1 · 𝑦 ) = - 𝑦 ) |
6 |
5
|
oveq2d |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 + ( - 1 · 𝑦 ) ) = ( 𝑥 + - 𝑦 ) ) |
7 |
|
negsub |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 + - 𝑦 ) = ( 𝑥 − 𝑦 ) ) |
8 |
6 7
|
eqtrd |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 + ( - 1 · 𝑦 ) ) = ( 𝑥 − 𝑦 ) ) |
9 |
8
|
fveq2d |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( abs ‘ ( 𝑥 + ( - 1 · 𝑦 ) ) ) = ( abs ‘ ( 𝑥 − 𝑦 ) ) ) |
10 |
9
|
oveq1d |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( abs ‘ ( 𝑥 + ( - 1 · 𝑦 ) ) ) ↑ 2 ) = ( ( abs ‘ ( 𝑥 − 𝑦 ) ) ↑ 2 ) ) |
11 |
10
|
oveq2d |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( ( abs ‘ ( 𝑥 + 𝑦 ) ) ↑ 2 ) + ( ( abs ‘ ( 𝑥 + ( - 1 · 𝑦 ) ) ) ↑ 2 ) ) = ( ( ( abs ‘ ( 𝑥 + 𝑦 ) ) ↑ 2 ) + ( ( abs ‘ ( 𝑥 − 𝑦 ) ) ↑ 2 ) ) ) |
12 |
|
sqabsadd |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( abs ‘ ( 𝑥 + 𝑦 ) ) ↑ 2 ) = ( ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) + ( 2 · ( ℜ ‘ ( 𝑥 · ( ∗ ‘ 𝑦 ) ) ) ) ) ) |
13 |
|
sqabssub |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( abs ‘ ( 𝑥 − 𝑦 ) ) ↑ 2 ) = ( ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) − ( 2 · ( ℜ ‘ ( 𝑥 · ( ∗ ‘ 𝑦 ) ) ) ) ) ) |
14 |
12 13
|
oveq12d |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( ( abs ‘ ( 𝑥 + 𝑦 ) ) ↑ 2 ) + ( ( abs ‘ ( 𝑥 − 𝑦 ) ) ↑ 2 ) ) = ( ( ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) + ( 2 · ( ℜ ‘ ( 𝑥 · ( ∗ ‘ 𝑦 ) ) ) ) ) + ( ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) − ( 2 · ( ℜ ‘ ( 𝑥 · ( ∗ ‘ 𝑦 ) ) ) ) ) ) ) |
15 |
|
abscl |
⊢ ( 𝑥 ∈ ℂ → ( abs ‘ 𝑥 ) ∈ ℝ ) |
16 |
15
|
recnd |
⊢ ( 𝑥 ∈ ℂ → ( abs ‘ 𝑥 ) ∈ ℂ ) |
17 |
16
|
sqcld |
⊢ ( 𝑥 ∈ ℂ → ( ( abs ‘ 𝑥 ) ↑ 2 ) ∈ ℂ ) |
18 |
|
abscl |
⊢ ( 𝑦 ∈ ℂ → ( abs ‘ 𝑦 ) ∈ ℝ ) |
19 |
18
|
recnd |
⊢ ( 𝑦 ∈ ℂ → ( abs ‘ 𝑦 ) ∈ ℂ ) |
20 |
19
|
sqcld |
⊢ ( 𝑦 ∈ ℂ → ( ( abs ‘ 𝑦 ) ↑ 2 ) ∈ ℂ ) |
21 |
|
addcl |
⊢ ( ( ( ( abs ‘ 𝑥 ) ↑ 2 ) ∈ ℂ ∧ ( ( abs ‘ 𝑦 ) ↑ 2 ) ∈ ℂ ) → ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) ∈ ℂ ) |
22 |
17 20 21
|
syl2an |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) ∈ ℂ ) |
23 |
|
2cn |
⊢ 2 ∈ ℂ |
24 |
|
cjcl |
⊢ ( 𝑦 ∈ ℂ → ( ∗ ‘ 𝑦 ) ∈ ℂ ) |
25 |
|
mulcl |
⊢ ( ( 𝑥 ∈ ℂ ∧ ( ∗ ‘ 𝑦 ) ∈ ℂ ) → ( 𝑥 · ( ∗ ‘ 𝑦 ) ) ∈ ℂ ) |
26 |
24 25
|
sylan2 |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 · ( ∗ ‘ 𝑦 ) ) ∈ ℂ ) |
27 |
|
recl |
⊢ ( ( 𝑥 · ( ∗ ‘ 𝑦 ) ) ∈ ℂ → ( ℜ ‘ ( 𝑥 · ( ∗ ‘ 𝑦 ) ) ) ∈ ℝ ) |
28 |
27
|
recnd |
⊢ ( ( 𝑥 · ( ∗ ‘ 𝑦 ) ) ∈ ℂ → ( ℜ ‘ ( 𝑥 · ( ∗ ‘ 𝑦 ) ) ) ∈ ℂ ) |
29 |
26 28
|
syl |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ℜ ‘ ( 𝑥 · ( ∗ ‘ 𝑦 ) ) ) ∈ ℂ ) |
30 |
|
mulcl |
⊢ ( ( 2 ∈ ℂ ∧ ( ℜ ‘ ( 𝑥 · ( ∗ ‘ 𝑦 ) ) ) ∈ ℂ ) → ( 2 · ( ℜ ‘ ( 𝑥 · ( ∗ ‘ 𝑦 ) ) ) ) ∈ ℂ ) |
31 |
23 29 30
|
sylancr |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 2 · ( ℜ ‘ ( 𝑥 · ( ∗ ‘ 𝑦 ) ) ) ) ∈ ℂ ) |
32 |
22 31 22
|
ppncand |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) + ( 2 · ( ℜ ‘ ( 𝑥 · ( ∗ ‘ 𝑦 ) ) ) ) ) + ( ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) − ( 2 · ( ℜ ‘ ( 𝑥 · ( ∗ ‘ 𝑦 ) ) ) ) ) ) = ( ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) + ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) ) ) |
33 |
14 32
|
eqtrd |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( ( abs ‘ ( 𝑥 + 𝑦 ) ) ↑ 2 ) + ( ( abs ‘ ( 𝑥 − 𝑦 ) ) ↑ 2 ) ) = ( ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) + ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) ) ) |
34 |
|
2times |
⊢ ( ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) ∈ ℂ → ( 2 · ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) ) = ( ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) + ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) ) ) |
35 |
34
|
eqcomd |
⊢ ( ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) ∈ ℂ → ( ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) + ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) ) = ( 2 · ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) ) ) |
36 |
22 35
|
syl |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) + ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) ) = ( 2 · ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) ) ) |
37 |
33 36
|
eqtrd |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( ( abs ‘ ( 𝑥 + 𝑦 ) ) ↑ 2 ) + ( ( abs ‘ ( 𝑥 − 𝑦 ) ) ↑ 2 ) ) = ( 2 · ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) ) ) |
38 |
11 37
|
eqtrd |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ( ( abs ‘ ( 𝑥 + 𝑦 ) ) ↑ 2 ) + ( ( abs ‘ ( 𝑥 + ( - 1 · 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) ) ) |
39 |
38
|
rgen2 |
⊢ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℂ ( ( ( abs ‘ ( 𝑥 + 𝑦 ) ) ↑ 2 ) + ( ( abs ‘ ( 𝑥 + ( - 1 · 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) ) |
40 |
|
addex |
⊢ + ∈ V |
41 |
|
mulex |
⊢ · ∈ V |
42 |
|
absf |
⊢ abs : ℂ ⟶ ℝ |
43 |
|
cnex |
⊢ ℂ ∈ V |
44 |
|
fex |
⊢ ( ( abs : ℂ ⟶ ℝ ∧ ℂ ∈ V ) → abs ∈ V ) |
45 |
42 43 44
|
mp2an |
⊢ abs ∈ V |
46 |
|
cnaddabloOLD |
⊢ + ∈ AbelOp |
47 |
|
ablogrpo |
⊢ ( + ∈ AbelOp → + ∈ GrpOp ) |
48 |
46 47
|
ax-mp |
⊢ + ∈ GrpOp |
49 |
|
ax-addf |
⊢ + : ( ℂ × ℂ ) ⟶ ℂ |
50 |
49
|
fdmi |
⊢ dom + = ( ℂ × ℂ ) |
51 |
48 50
|
grporn |
⊢ ℂ = ran + |
52 |
51
|
isphg |
⊢ ( ( + ∈ V ∧ · ∈ V ∧ abs ∈ V ) → ( 〈 〈 + , · 〉 , abs 〉 ∈ CPreHilOLD ↔ ( 〈 〈 + , · 〉 , abs 〉 ∈ NrmCVec ∧ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℂ ( ( ( abs ‘ ( 𝑥 + 𝑦 ) ) ↑ 2 ) + ( ( abs ‘ ( 𝑥 + ( - 1 · 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) ) ) ) ) |
53 |
40 41 45 52
|
mp3an |
⊢ ( 〈 〈 + , · 〉 , abs 〉 ∈ CPreHilOLD ↔ ( 〈 〈 + , · 〉 , abs 〉 ∈ NrmCVec ∧ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℂ ( ( ( abs ‘ ( 𝑥 + 𝑦 ) ) ↑ 2 ) + ( ( abs ‘ ( 𝑥 + ( - 1 · 𝑦 ) ) ) ↑ 2 ) ) = ( 2 · ( ( ( abs ‘ 𝑥 ) ↑ 2 ) + ( ( abs ‘ 𝑦 ) ↑ 2 ) ) ) ) ) |
54 |
3 39 53
|
mpbir2an |
⊢ 〈 〈 + , · 〉 , abs 〉 ∈ CPreHilOLD |
55 |
1 54
|
eqeltri |
⊢ 𝑈 ∈ CPreHilOLD |