Step |
Hyp |
Ref |
Expression |
1 |
|
chordthmlem.angdef |
⊢ 𝐹 = ( 𝑥 ∈ ( ℂ ∖ { 0 } ) , 𝑦 ∈ ( ℂ ∖ { 0 } ) ↦ ( ℑ ‘ ( log ‘ ( 𝑦 / 𝑥 ) ) ) ) |
2 |
|
chordthmlem.A |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
3 |
|
chordthmlem.B |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
4 |
|
chordthmlem.Q |
⊢ ( 𝜑 → 𝑄 ∈ ℂ ) |
5 |
|
chordthmlem.M |
⊢ ( 𝜑 → 𝑀 = ( ( 𝐴 + 𝐵 ) / 2 ) ) |
6 |
|
chordthmlem.ABequidistQ |
⊢ ( 𝜑 → ( abs ‘ ( 𝐴 − 𝑄 ) ) = ( abs ‘ ( 𝐵 − 𝑄 ) ) ) |
7 |
|
chordthmlem.AneB |
⊢ ( 𝜑 → 𝐴 ≠ 𝐵 ) |
8 |
|
chordthmlem.QneM |
⊢ ( 𝜑 → 𝑄 ≠ 𝑀 ) |
9 |
|
negpitopissre |
⊢ ( - π (,] π ) ⊆ ℝ |
10 |
2 3
|
addcld |
⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ∈ ℂ ) |
11 |
10
|
halfcld |
⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) / 2 ) ∈ ℂ ) |
12 |
5 11
|
eqeltrd |
⊢ ( 𝜑 → 𝑀 ∈ ℂ ) |
13 |
4 12
|
subcld |
⊢ ( 𝜑 → ( 𝑄 − 𝑀 ) ∈ ℂ ) |
14 |
4 12 8
|
subne0d |
⊢ ( 𝜑 → ( 𝑄 − 𝑀 ) ≠ 0 ) |
15 |
3 12
|
subcld |
⊢ ( 𝜑 → ( 𝐵 − 𝑀 ) ∈ ℂ ) |
16 |
5
|
oveq1d |
⊢ ( 𝜑 → ( 𝑀 · 2 ) = ( ( ( 𝐴 + 𝐵 ) / 2 ) · 2 ) ) |
17 |
12
|
times2d |
⊢ ( 𝜑 → ( 𝑀 · 2 ) = ( 𝑀 + 𝑀 ) ) |
18 |
|
2cnd |
⊢ ( 𝜑 → 2 ∈ ℂ ) |
19 |
|
2ne0 |
⊢ 2 ≠ 0 |
20 |
19
|
a1i |
⊢ ( 𝜑 → 2 ≠ 0 ) |
21 |
10 18 20
|
divcan1d |
⊢ ( 𝜑 → ( ( ( 𝐴 + 𝐵 ) / 2 ) · 2 ) = ( 𝐴 + 𝐵 ) ) |
22 |
16 17 21
|
3eqtr3d |
⊢ ( 𝜑 → ( 𝑀 + 𝑀 ) = ( 𝐴 + 𝐵 ) ) |
23 |
2 3 3 7
|
addneintr2d |
⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ≠ ( 𝐵 + 𝐵 ) ) |
24 |
22 23
|
eqnetrd |
⊢ ( 𝜑 → ( 𝑀 + 𝑀 ) ≠ ( 𝐵 + 𝐵 ) ) |
25 |
24
|
neneqd |
⊢ ( 𝜑 → ¬ ( 𝑀 + 𝑀 ) = ( 𝐵 + 𝐵 ) ) |
26 |
|
oveq12 |
⊢ ( ( 𝑀 = 𝐵 ∧ 𝑀 = 𝐵 ) → ( 𝑀 + 𝑀 ) = ( 𝐵 + 𝐵 ) ) |
27 |
26
|
anidms |
⊢ ( 𝑀 = 𝐵 → ( 𝑀 + 𝑀 ) = ( 𝐵 + 𝐵 ) ) |
28 |
25 27
|
nsyl |
⊢ ( 𝜑 → ¬ 𝑀 = 𝐵 ) |
29 |
28
|
neqned |
⊢ ( 𝜑 → 𝑀 ≠ 𝐵 ) |
30 |
29
|
necomd |
⊢ ( 𝜑 → 𝐵 ≠ 𝑀 ) |
31 |
3 12 30
|
subne0d |
⊢ ( 𝜑 → ( 𝐵 − 𝑀 ) ≠ 0 ) |
32 |
1 13 14 15 31
|
angcld |
⊢ ( 𝜑 → ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐵 − 𝑀 ) ) ∈ ( - π (,] π ) ) |
33 |
9 32
|
sselid |
⊢ ( 𝜑 → ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐵 − 𝑀 ) ) ∈ ℝ ) |
34 |
33
|
recnd |
⊢ ( 𝜑 → ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐵 − 𝑀 ) ) ∈ ℂ ) |
35 |
34
|
coscld |
⊢ ( 𝜑 → ( cos ‘ ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐵 − 𝑀 ) ) ) ∈ ℂ ) |
36 |
3 12
|
negsubdi2d |
⊢ ( 𝜑 → - ( 𝐵 − 𝑀 ) = ( 𝑀 − 𝐵 ) ) |
37 |
12 12 2 3
|
addsubeq4d |
⊢ ( 𝜑 → ( ( 𝑀 + 𝑀 ) = ( 𝐴 + 𝐵 ) ↔ ( 𝐴 − 𝑀 ) = ( 𝑀 − 𝐵 ) ) ) |
38 |
22 37
|
mpbid |
⊢ ( 𝜑 → ( 𝐴 − 𝑀 ) = ( 𝑀 − 𝐵 ) ) |
39 |
36 38
|
eqtr4d |
⊢ ( 𝜑 → - ( 𝐵 − 𝑀 ) = ( 𝐴 − 𝑀 ) ) |
40 |
39
|
oveq2d |
⊢ ( 𝜑 → ( ( 𝑄 − 𝑀 ) 𝐹 - ( 𝐵 − 𝑀 ) ) = ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐴 − 𝑀 ) ) ) |
41 |
40
|
fveq2d |
⊢ ( 𝜑 → ( cos ‘ ( ( 𝑄 − 𝑀 ) 𝐹 - ( 𝐵 − 𝑀 ) ) ) = ( cos ‘ ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐴 − 𝑀 ) ) ) ) |
42 |
1 13 14 15 31
|
cosangneg2d |
⊢ ( 𝜑 → ( cos ‘ ( ( 𝑄 − 𝑀 ) 𝐹 - ( 𝐵 − 𝑀 ) ) ) = - ( cos ‘ ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐵 − 𝑀 ) ) ) ) |
43 |
2 2 3 7
|
addneintrd |
⊢ ( 𝜑 → ( 𝐴 + 𝐴 ) ≠ ( 𝐴 + 𝐵 ) ) |
44 |
43 22
|
neeqtrrd |
⊢ ( 𝜑 → ( 𝐴 + 𝐴 ) ≠ ( 𝑀 + 𝑀 ) ) |
45 |
44
|
necomd |
⊢ ( 𝜑 → ( 𝑀 + 𝑀 ) ≠ ( 𝐴 + 𝐴 ) ) |
46 |
45
|
neneqd |
⊢ ( 𝜑 → ¬ ( 𝑀 + 𝑀 ) = ( 𝐴 + 𝐴 ) ) |
47 |
|
oveq12 |
⊢ ( ( 𝑀 = 𝐴 ∧ 𝑀 = 𝐴 ) → ( 𝑀 + 𝑀 ) = ( 𝐴 + 𝐴 ) ) |
48 |
47
|
anidms |
⊢ ( 𝑀 = 𝐴 → ( 𝑀 + 𝑀 ) = ( 𝐴 + 𝐴 ) ) |
49 |
46 48
|
nsyl |
⊢ ( 𝜑 → ¬ 𝑀 = 𝐴 ) |
50 |
49
|
neqned |
⊢ ( 𝜑 → 𝑀 ≠ 𝐴 ) |
51 |
|
eqidd |
⊢ ( 𝜑 → ( abs ‘ ( 𝑄 − 𝑀 ) ) = ( abs ‘ ( 𝑄 − 𝑀 ) ) ) |
52 |
2 12
|
subcld |
⊢ ( 𝜑 → ( 𝐴 − 𝑀 ) ∈ ℂ ) |
53 |
52
|
absnegd |
⊢ ( 𝜑 → ( abs ‘ - ( 𝐴 − 𝑀 ) ) = ( abs ‘ ( 𝐴 − 𝑀 ) ) ) |
54 |
2 12
|
negsubdi2d |
⊢ ( 𝜑 → - ( 𝐴 − 𝑀 ) = ( 𝑀 − 𝐴 ) ) |
55 |
54
|
fveq2d |
⊢ ( 𝜑 → ( abs ‘ - ( 𝐴 − 𝑀 ) ) = ( abs ‘ ( 𝑀 − 𝐴 ) ) ) |
56 |
38
|
fveq2d |
⊢ ( 𝜑 → ( abs ‘ ( 𝐴 − 𝑀 ) ) = ( abs ‘ ( 𝑀 − 𝐵 ) ) ) |
57 |
53 55 56
|
3eqtr3d |
⊢ ( 𝜑 → ( abs ‘ ( 𝑀 − 𝐴 ) ) = ( abs ‘ ( 𝑀 − 𝐵 ) ) ) |
58 |
1 4 12 2 4 12 3 8 50 8 29 51 57 6
|
ssscongptld |
⊢ ( 𝜑 → ( cos ‘ ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐴 − 𝑀 ) ) ) = ( cos ‘ ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐵 − 𝑀 ) ) ) ) |
59 |
41 42 58
|
3eqtr3rd |
⊢ ( 𝜑 → ( cos ‘ ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐵 − 𝑀 ) ) ) = - ( cos ‘ ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐵 − 𝑀 ) ) ) ) |
60 |
35 59
|
eqnegad |
⊢ ( 𝜑 → ( cos ‘ ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐵 − 𝑀 ) ) ) = 0 ) |
61 |
|
coseq0negpitopi |
⊢ ( ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐵 − 𝑀 ) ) ∈ ( - π (,] π ) → ( ( cos ‘ ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐵 − 𝑀 ) ) ) = 0 ↔ ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐵 − 𝑀 ) ) ∈ { ( π / 2 ) , - ( π / 2 ) } ) ) |
62 |
32 61
|
syl |
⊢ ( 𝜑 → ( ( cos ‘ ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐵 − 𝑀 ) ) ) = 0 ↔ ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐵 − 𝑀 ) ) ∈ { ( π / 2 ) , - ( π / 2 ) } ) ) |
63 |
60 62
|
mpbid |
⊢ ( 𝜑 → ( ( 𝑄 − 𝑀 ) 𝐹 ( 𝐵 − 𝑀 ) ) ∈ { ( π / 2 ) , - ( π / 2 ) } ) |