Step |
Hyp |
Ref |
Expression |
1 |
|
normlem1.1 |
⊢ 𝑆 ∈ ℂ |
2 |
|
normlem1.2 |
⊢ 𝐹 ∈ ℋ |
3 |
|
normlem1.3 |
⊢ 𝐺 ∈ ℋ |
4 |
|
normlem2.4 |
⊢ 𝐵 = - ( ( ( ∗ ‘ 𝑆 ) · ( 𝐹 ·ih 𝐺 ) ) + ( 𝑆 · ( 𝐺 ·ih 𝐹 ) ) ) |
5 |
|
normlem3.5 |
⊢ 𝐴 = ( 𝐺 ·ih 𝐺 ) |
6 |
|
normlem3.6 |
⊢ 𝐶 = ( 𝐹 ·ih 𝐹 ) |
7 |
|
normlem4.7 |
⊢ 𝑅 ∈ ℝ |
8 |
|
normlem4.8 |
⊢ ( abs ‘ 𝑆 ) = 1 |
9 |
1 2 3 7 8
|
normlem1 |
⊢ ( ( 𝐹 −ℎ ( ( 𝑆 · 𝑅 ) ·ℎ 𝐺 ) ) ·ih ( 𝐹 −ℎ ( ( 𝑆 · 𝑅 ) ·ℎ 𝐺 ) ) ) = ( ( ( 𝐹 ·ih 𝐹 ) + ( ( ( ∗ ‘ 𝑆 ) · - 𝑅 ) · ( 𝐹 ·ih 𝐺 ) ) ) + ( ( ( 𝑆 · - 𝑅 ) · ( 𝐺 ·ih 𝐹 ) ) + ( ( 𝑅 ↑ 2 ) · ( 𝐺 ·ih 𝐺 ) ) ) ) |
10 |
1 2 3 4 5 6 7
|
normlem3 |
⊢ ( ( ( 𝐴 · ( 𝑅 ↑ 2 ) ) + ( 𝐵 · 𝑅 ) ) + 𝐶 ) = ( ( ( 𝐹 ·ih 𝐹 ) + ( ( ( ∗ ‘ 𝑆 ) · - 𝑅 ) · ( 𝐹 ·ih 𝐺 ) ) ) + ( ( ( 𝑆 · - 𝑅 ) · ( 𝐺 ·ih 𝐹 ) ) + ( ( 𝑅 ↑ 2 ) · ( 𝐺 ·ih 𝐺 ) ) ) ) |
11 |
9 10
|
eqtr4i |
⊢ ( ( 𝐹 −ℎ ( ( 𝑆 · 𝑅 ) ·ℎ 𝐺 ) ) ·ih ( 𝐹 −ℎ ( ( 𝑆 · 𝑅 ) ·ℎ 𝐺 ) ) ) = ( ( ( 𝐴 · ( 𝑅 ↑ 2 ) ) + ( 𝐵 · 𝑅 ) ) + 𝐶 ) |