Step |
Hyp |
Ref |
Expression |
1 |
|
normlem1.1 |
⊢ 𝑆 ∈ ℂ |
2 |
|
normlem1.2 |
⊢ 𝐹 ∈ ℋ |
3 |
|
normlem1.3 |
⊢ 𝐺 ∈ ℋ |
4 |
1 3
|
hvmulcli |
⊢ ( 𝑆 ·ℎ 𝐺 ) ∈ ℋ |
5 |
2 4
|
hvsubvali |
⊢ ( 𝐹 −ℎ ( 𝑆 ·ℎ 𝐺 ) ) = ( 𝐹 +ℎ ( - 1 ·ℎ ( 𝑆 ·ℎ 𝐺 ) ) ) |
6 |
1
|
mulm1i |
⊢ ( - 1 · 𝑆 ) = - 𝑆 |
7 |
6
|
oveq1i |
⊢ ( ( - 1 · 𝑆 ) ·ℎ 𝐺 ) = ( - 𝑆 ·ℎ 𝐺 ) |
8 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
9 |
8 1 3
|
hvmulassi |
⊢ ( ( - 1 · 𝑆 ) ·ℎ 𝐺 ) = ( - 1 ·ℎ ( 𝑆 ·ℎ 𝐺 ) ) |
10 |
7 9
|
eqtr3i |
⊢ ( - 𝑆 ·ℎ 𝐺 ) = ( - 1 ·ℎ ( 𝑆 ·ℎ 𝐺 ) ) |
11 |
10
|
oveq2i |
⊢ ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) = ( 𝐹 +ℎ ( - 1 ·ℎ ( 𝑆 ·ℎ 𝐺 ) ) ) |
12 |
5 11
|
eqtr4i |
⊢ ( 𝐹 −ℎ ( 𝑆 ·ℎ 𝐺 ) ) = ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) |
13 |
12 12
|
oveq12i |
⊢ ( ( 𝐹 −ℎ ( 𝑆 ·ℎ 𝐺 ) ) ·ih ( 𝐹 −ℎ ( 𝑆 ·ℎ 𝐺 ) ) ) = ( ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) |
14 |
1
|
negcli |
⊢ - 𝑆 ∈ ℂ |
15 |
14 3
|
hvmulcli |
⊢ ( - 𝑆 ·ℎ 𝐺 ) ∈ ℋ |
16 |
2 15
|
hvaddcli |
⊢ ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ∈ ℋ |
17 |
|
ax-his2 |
⊢ ( ( 𝐹 ∈ ℋ ∧ ( - 𝑆 ·ℎ 𝐺 ) ∈ ℋ ∧ ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ∈ ℋ ) → ( ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) = ( ( 𝐹 ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) + ( ( - 𝑆 ·ℎ 𝐺 ) ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) ) ) |
18 |
2 15 16 17
|
mp3an |
⊢ ( ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) = ( ( 𝐹 ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) + ( ( - 𝑆 ·ℎ 𝐺 ) ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) ) |
19 |
|
his7 |
⊢ ( ( 𝐹 ∈ ℋ ∧ 𝐹 ∈ ℋ ∧ ( - 𝑆 ·ℎ 𝐺 ) ∈ ℋ ) → ( 𝐹 ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) = ( ( 𝐹 ·ih 𝐹 ) + ( 𝐹 ·ih ( - 𝑆 ·ℎ 𝐺 ) ) ) ) |
20 |
2 2 15 19
|
mp3an |
⊢ ( 𝐹 ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) = ( ( 𝐹 ·ih 𝐹 ) + ( 𝐹 ·ih ( - 𝑆 ·ℎ 𝐺 ) ) ) |
21 |
|
his5 |
⊢ ( ( - 𝑆 ∈ ℂ ∧ 𝐹 ∈ ℋ ∧ 𝐺 ∈ ℋ ) → ( 𝐹 ·ih ( - 𝑆 ·ℎ 𝐺 ) ) = ( ( ∗ ‘ - 𝑆 ) · ( 𝐹 ·ih 𝐺 ) ) ) |
22 |
14 2 3 21
|
mp3an |
⊢ ( 𝐹 ·ih ( - 𝑆 ·ℎ 𝐺 ) ) = ( ( ∗ ‘ - 𝑆 ) · ( 𝐹 ·ih 𝐺 ) ) |
23 |
1
|
cjnegi |
⊢ ( ∗ ‘ - 𝑆 ) = - ( ∗ ‘ 𝑆 ) |
24 |
23
|
oveq1i |
⊢ ( ( ∗ ‘ - 𝑆 ) · ( 𝐹 ·ih 𝐺 ) ) = ( - ( ∗ ‘ 𝑆 ) · ( 𝐹 ·ih 𝐺 ) ) |
25 |
22 24
|
eqtri |
⊢ ( 𝐹 ·ih ( - 𝑆 ·ℎ 𝐺 ) ) = ( - ( ∗ ‘ 𝑆 ) · ( 𝐹 ·ih 𝐺 ) ) |
26 |
25
|
oveq2i |
⊢ ( ( 𝐹 ·ih 𝐹 ) + ( 𝐹 ·ih ( - 𝑆 ·ℎ 𝐺 ) ) ) = ( ( 𝐹 ·ih 𝐹 ) + ( - ( ∗ ‘ 𝑆 ) · ( 𝐹 ·ih 𝐺 ) ) ) |
27 |
20 26
|
eqtri |
⊢ ( 𝐹 ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) = ( ( 𝐹 ·ih 𝐹 ) + ( - ( ∗ ‘ 𝑆 ) · ( 𝐹 ·ih 𝐺 ) ) ) |
28 |
|
ax-his3 |
⊢ ( ( - 𝑆 ∈ ℂ ∧ 𝐺 ∈ ℋ ∧ ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ∈ ℋ ) → ( ( - 𝑆 ·ℎ 𝐺 ) ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) = ( - 𝑆 · ( 𝐺 ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) ) ) |
29 |
14 3 16 28
|
mp3an |
⊢ ( ( - 𝑆 ·ℎ 𝐺 ) ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) = ( - 𝑆 · ( 𝐺 ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) ) |
30 |
|
his7 |
⊢ ( ( 𝐺 ∈ ℋ ∧ 𝐹 ∈ ℋ ∧ ( - 𝑆 ·ℎ 𝐺 ) ∈ ℋ ) → ( 𝐺 ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) = ( ( 𝐺 ·ih 𝐹 ) + ( 𝐺 ·ih ( - 𝑆 ·ℎ 𝐺 ) ) ) ) |
31 |
3 2 15 30
|
mp3an |
⊢ ( 𝐺 ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) = ( ( 𝐺 ·ih 𝐹 ) + ( 𝐺 ·ih ( - 𝑆 ·ℎ 𝐺 ) ) ) |
32 |
|
his5 |
⊢ ( ( - 𝑆 ∈ ℂ ∧ 𝐺 ∈ ℋ ∧ 𝐺 ∈ ℋ ) → ( 𝐺 ·ih ( - 𝑆 ·ℎ 𝐺 ) ) = ( ( ∗ ‘ - 𝑆 ) · ( 𝐺 ·ih 𝐺 ) ) ) |
33 |
14 3 3 32
|
mp3an |
⊢ ( 𝐺 ·ih ( - 𝑆 ·ℎ 𝐺 ) ) = ( ( ∗ ‘ - 𝑆 ) · ( 𝐺 ·ih 𝐺 ) ) |
34 |
33
|
oveq2i |
⊢ ( ( 𝐺 ·ih 𝐹 ) + ( 𝐺 ·ih ( - 𝑆 ·ℎ 𝐺 ) ) ) = ( ( 𝐺 ·ih 𝐹 ) + ( ( ∗ ‘ - 𝑆 ) · ( 𝐺 ·ih 𝐺 ) ) ) |
35 |
31 34
|
eqtri |
⊢ ( 𝐺 ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) = ( ( 𝐺 ·ih 𝐹 ) + ( ( ∗ ‘ - 𝑆 ) · ( 𝐺 ·ih 𝐺 ) ) ) |
36 |
35
|
oveq2i |
⊢ ( - 𝑆 · ( 𝐺 ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) ) = ( - 𝑆 · ( ( 𝐺 ·ih 𝐹 ) + ( ( ∗ ‘ - 𝑆 ) · ( 𝐺 ·ih 𝐺 ) ) ) ) |
37 |
3 2
|
hicli |
⊢ ( 𝐺 ·ih 𝐹 ) ∈ ℂ |
38 |
14
|
cjcli |
⊢ ( ∗ ‘ - 𝑆 ) ∈ ℂ |
39 |
3 3
|
hicli |
⊢ ( 𝐺 ·ih 𝐺 ) ∈ ℂ |
40 |
38 39
|
mulcli |
⊢ ( ( ∗ ‘ - 𝑆 ) · ( 𝐺 ·ih 𝐺 ) ) ∈ ℂ |
41 |
14 37 40
|
adddii |
⊢ ( - 𝑆 · ( ( 𝐺 ·ih 𝐹 ) + ( ( ∗ ‘ - 𝑆 ) · ( 𝐺 ·ih 𝐺 ) ) ) ) = ( ( - 𝑆 · ( 𝐺 ·ih 𝐹 ) ) + ( - 𝑆 · ( ( ∗ ‘ - 𝑆 ) · ( 𝐺 ·ih 𝐺 ) ) ) ) |
42 |
14 38 39
|
mulassi |
⊢ ( ( - 𝑆 · ( ∗ ‘ - 𝑆 ) ) · ( 𝐺 ·ih 𝐺 ) ) = ( - 𝑆 · ( ( ∗ ‘ - 𝑆 ) · ( 𝐺 ·ih 𝐺 ) ) ) |
43 |
23
|
oveq2i |
⊢ ( - 𝑆 · ( ∗ ‘ - 𝑆 ) ) = ( - 𝑆 · - ( ∗ ‘ 𝑆 ) ) |
44 |
1
|
cjcli |
⊢ ( ∗ ‘ 𝑆 ) ∈ ℂ |
45 |
1 44
|
mul2negi |
⊢ ( - 𝑆 · - ( ∗ ‘ 𝑆 ) ) = ( 𝑆 · ( ∗ ‘ 𝑆 ) ) |
46 |
43 45
|
eqtri |
⊢ ( - 𝑆 · ( ∗ ‘ - 𝑆 ) ) = ( 𝑆 · ( ∗ ‘ 𝑆 ) ) |
47 |
46
|
oveq1i |
⊢ ( ( - 𝑆 · ( ∗ ‘ - 𝑆 ) ) · ( 𝐺 ·ih 𝐺 ) ) = ( ( 𝑆 · ( ∗ ‘ 𝑆 ) ) · ( 𝐺 ·ih 𝐺 ) ) |
48 |
42 47
|
eqtr3i |
⊢ ( - 𝑆 · ( ( ∗ ‘ - 𝑆 ) · ( 𝐺 ·ih 𝐺 ) ) ) = ( ( 𝑆 · ( ∗ ‘ 𝑆 ) ) · ( 𝐺 ·ih 𝐺 ) ) |
49 |
48
|
oveq2i |
⊢ ( ( - 𝑆 · ( 𝐺 ·ih 𝐹 ) ) + ( - 𝑆 · ( ( ∗ ‘ - 𝑆 ) · ( 𝐺 ·ih 𝐺 ) ) ) ) = ( ( - 𝑆 · ( 𝐺 ·ih 𝐹 ) ) + ( ( 𝑆 · ( ∗ ‘ 𝑆 ) ) · ( 𝐺 ·ih 𝐺 ) ) ) |
50 |
41 49
|
eqtri |
⊢ ( - 𝑆 · ( ( 𝐺 ·ih 𝐹 ) + ( ( ∗ ‘ - 𝑆 ) · ( 𝐺 ·ih 𝐺 ) ) ) ) = ( ( - 𝑆 · ( 𝐺 ·ih 𝐹 ) ) + ( ( 𝑆 · ( ∗ ‘ 𝑆 ) ) · ( 𝐺 ·ih 𝐺 ) ) ) |
51 |
29 36 50
|
3eqtri |
⊢ ( ( - 𝑆 ·ℎ 𝐺 ) ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) = ( ( - 𝑆 · ( 𝐺 ·ih 𝐹 ) ) + ( ( 𝑆 · ( ∗ ‘ 𝑆 ) ) · ( 𝐺 ·ih 𝐺 ) ) ) |
52 |
27 51
|
oveq12i |
⊢ ( ( 𝐹 ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) + ( ( - 𝑆 ·ℎ 𝐺 ) ·ih ( 𝐹 +ℎ ( - 𝑆 ·ℎ 𝐺 ) ) ) ) = ( ( ( 𝐹 ·ih 𝐹 ) + ( - ( ∗ ‘ 𝑆 ) · ( 𝐹 ·ih 𝐺 ) ) ) + ( ( - 𝑆 · ( 𝐺 ·ih 𝐹 ) ) + ( ( 𝑆 · ( ∗ ‘ 𝑆 ) ) · ( 𝐺 ·ih 𝐺 ) ) ) ) |
53 |
13 18 52
|
3eqtri |
⊢ ( ( 𝐹 −ℎ ( 𝑆 ·ℎ 𝐺 ) ) ·ih ( 𝐹 −ℎ ( 𝑆 ·ℎ 𝐺 ) ) ) = ( ( ( 𝐹 ·ih 𝐹 ) + ( - ( ∗ ‘ 𝑆 ) · ( 𝐹 ·ih 𝐺 ) ) ) + ( ( - 𝑆 · ( 𝐺 ·ih 𝐹 ) ) + ( ( 𝑆 · ( ∗ ‘ 𝑆 ) ) · ( 𝐺 ·ih 𝐺 ) ) ) ) |