Step |
Hyp |
Ref |
Expression |
1 |
|
rrx2line.i |
⊢ 𝐼 = { 1 , 2 } |
2 |
|
rrx2line.e |
⊢ 𝐸 = ( ℝ^ ‘ 𝐼 ) |
3 |
|
rrx2line.b |
⊢ 𝑃 = ( ℝ ↑m 𝐼 ) |
4 |
|
rrx2line.l |
⊢ 𝐿 = ( LineM ‘ 𝐸 ) |
5 |
|
rrx2linest.a |
⊢ 𝐴 = ( ( 𝑌 ‘ 1 ) − ( 𝑋 ‘ 1 ) ) |
6 |
|
rrx2linest.b |
⊢ 𝐵 = ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) |
7 |
|
rrx2linest.c |
⊢ 𝐶 = ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑋 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) |
8 |
|
simpl1 |
⊢ ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → 𝑋 ∈ 𝑃 ) |
9 |
|
simpl2 |
⊢ ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → 𝑌 ∈ 𝑃 ) |
10 |
|
simpr |
⊢ ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) |
11 |
|
simpr |
⊢ ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) |
12 |
11
|
anim1i |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ∧ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) → ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) ) |
13 |
1
|
raleqi |
⊢ ( ∀ 𝑖 ∈ 𝐼 ( 𝑋 ‘ 𝑖 ) = ( 𝑌 ‘ 𝑖 ) ↔ ∀ 𝑖 ∈ { 1 , 2 } ( 𝑋 ‘ 𝑖 ) = ( 𝑌 ‘ 𝑖 ) ) |
14 |
|
1ex |
⊢ 1 ∈ V |
15 |
|
2ex |
⊢ 2 ∈ V |
16 |
|
fveq2 |
⊢ ( 𝑖 = 1 → ( 𝑋 ‘ 𝑖 ) = ( 𝑋 ‘ 1 ) ) |
17 |
|
fveq2 |
⊢ ( 𝑖 = 1 → ( 𝑌 ‘ 𝑖 ) = ( 𝑌 ‘ 1 ) ) |
18 |
16 17
|
eqeq12d |
⊢ ( 𝑖 = 1 → ( ( 𝑋 ‘ 𝑖 ) = ( 𝑌 ‘ 𝑖 ) ↔ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ) |
19 |
|
fveq2 |
⊢ ( 𝑖 = 2 → ( 𝑋 ‘ 𝑖 ) = ( 𝑋 ‘ 2 ) ) |
20 |
|
fveq2 |
⊢ ( 𝑖 = 2 → ( 𝑌 ‘ 𝑖 ) = ( 𝑌 ‘ 2 ) ) |
21 |
19 20
|
eqeq12d |
⊢ ( 𝑖 = 2 → ( ( 𝑋 ‘ 𝑖 ) = ( 𝑌 ‘ 𝑖 ) ↔ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) ) |
22 |
14 15 18 21
|
ralpr |
⊢ ( ∀ 𝑖 ∈ { 1 , 2 } ( 𝑋 ‘ 𝑖 ) = ( 𝑌 ‘ 𝑖 ) ↔ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) ) |
23 |
13 22
|
bitri |
⊢ ( ∀ 𝑖 ∈ 𝐼 ( 𝑋 ‘ 𝑖 ) = ( 𝑌 ‘ 𝑖 ) ↔ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) ) |
24 |
12 23
|
sylibr |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ∧ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) → ∀ 𝑖 ∈ 𝐼 ( 𝑋 ‘ 𝑖 ) = ( 𝑌 ‘ 𝑖 ) ) |
25 |
|
elmapfn |
⊢ ( 𝑋 ∈ ( ℝ ↑m 𝐼 ) → 𝑋 Fn 𝐼 ) |
26 |
25 3
|
eleq2s |
⊢ ( 𝑋 ∈ 𝑃 → 𝑋 Fn 𝐼 ) |
27 |
|
elmapfn |
⊢ ( 𝑌 ∈ ( ℝ ↑m 𝐼 ) → 𝑌 Fn 𝐼 ) |
28 |
27 3
|
eleq2s |
⊢ ( 𝑌 ∈ 𝑃 → 𝑌 Fn 𝐼 ) |
29 |
26 28
|
anim12i |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) → ( 𝑋 Fn 𝐼 ∧ 𝑌 Fn 𝐼 ) ) |
30 |
29
|
ad2antrr |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ∧ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) → ( 𝑋 Fn 𝐼 ∧ 𝑌 Fn 𝐼 ) ) |
31 |
|
eqfnfv |
⊢ ( ( 𝑋 Fn 𝐼 ∧ 𝑌 Fn 𝐼 ) → ( 𝑋 = 𝑌 ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑋 ‘ 𝑖 ) = ( 𝑌 ‘ 𝑖 ) ) ) |
32 |
30 31
|
syl |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ∧ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) → ( 𝑋 = 𝑌 ↔ ∀ 𝑖 ∈ 𝐼 ( 𝑋 ‘ 𝑖 ) = ( 𝑌 ‘ 𝑖 ) ) ) |
33 |
24 32
|
mpbird |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ∧ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) → 𝑋 = 𝑌 ) |
34 |
33
|
ex |
⊢ ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) → 𝑋 = 𝑌 ) ) |
35 |
34
|
necon3d |
⊢ ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 ≠ 𝑌 → ( 𝑋 ‘ 2 ) ≠ ( 𝑌 ‘ 2 ) ) ) |
36 |
35
|
ex |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) → ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) → ( 𝑋 ≠ 𝑌 → ( 𝑋 ‘ 2 ) ≠ ( 𝑌 ‘ 2 ) ) ) ) |
37 |
36
|
com23 |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ) → ( 𝑋 ≠ 𝑌 → ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) → ( 𝑋 ‘ 2 ) ≠ ( 𝑌 ‘ 2 ) ) ) ) |
38 |
37
|
3impia |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) → ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) → ( 𝑋 ‘ 2 ) ≠ ( 𝑌 ‘ 2 ) ) ) |
39 |
38
|
imp |
⊢ ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 ‘ 2 ) ≠ ( 𝑌 ‘ 2 ) ) |
40 |
1 2 3 4
|
rrx2vlinest |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) ≠ ( 𝑌 ‘ 2 ) ) ) → ( 𝑋 𝐿 𝑌 ) = { 𝑝 ∈ 𝑃 ∣ ( 𝑝 ‘ 1 ) = ( 𝑋 ‘ 1 ) } ) |
41 |
8 9 10 39 40
|
syl112anc |
⊢ ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 𝐿 𝑌 ) = { 𝑝 ∈ 𝑃 ∣ ( 𝑝 ‘ 1 ) = ( 𝑋 ‘ 1 ) } ) |
42 |
|
ancom |
⊢ ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ↔ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ) |
43 |
|
simplr |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) |
44 |
|
simpr |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → 𝑝 ∈ 𝑃 ) |
45 |
|
simpll |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) |
46 |
5
|
oveq1i |
⊢ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( ( 𝑌 ‘ 1 ) − ( 𝑋 ‘ 1 ) ) · ( 𝑝 ‘ 2 ) ) |
47 |
46
|
a1i |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( ( 𝑌 ‘ 1 ) − ( 𝑋 ‘ 1 ) ) · ( 𝑝 ‘ 2 ) ) ) |
48 |
|
oveq2 |
⊢ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) → ( ( 𝑌 ‘ 1 ) − ( 𝑋 ‘ 1 ) ) = ( ( 𝑌 ‘ 1 ) − ( 𝑌 ‘ 1 ) ) ) |
49 |
48
|
adantl |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( ( 𝑌 ‘ 1 ) − ( 𝑋 ‘ 1 ) ) = ( ( 𝑌 ‘ 1 ) − ( 𝑌 ‘ 1 ) ) ) |
50 |
1 3
|
rrx2pxel |
⊢ ( 𝑌 ∈ 𝑃 → ( 𝑌 ‘ 1 ) ∈ ℝ ) |
51 |
50
|
recnd |
⊢ ( 𝑌 ∈ 𝑃 → ( 𝑌 ‘ 1 ) ∈ ℂ ) |
52 |
51
|
3ad2ant2 |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) → ( 𝑌 ‘ 1 ) ∈ ℂ ) |
53 |
52
|
ad2antrr |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑌 ‘ 1 ) ∈ ℂ ) |
54 |
53
|
subidd |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( ( 𝑌 ‘ 1 ) − ( 𝑌 ‘ 1 ) ) = 0 ) |
55 |
49 54
|
eqtrd |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( ( 𝑌 ‘ 1 ) − ( 𝑋 ‘ 1 ) ) = 0 ) |
56 |
55
|
oveq1d |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( ( ( 𝑌 ‘ 1 ) − ( 𝑋 ‘ 1 ) ) · ( 𝑝 ‘ 2 ) ) = ( 0 · ( 𝑝 ‘ 2 ) ) ) |
57 |
1 3
|
rrx2pyel |
⊢ ( 𝑝 ∈ 𝑃 → ( 𝑝 ‘ 2 ) ∈ ℝ ) |
58 |
57
|
recnd |
⊢ ( 𝑝 ∈ 𝑃 → ( 𝑝 ‘ 2 ) ∈ ℂ ) |
59 |
58
|
ad2antlr |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑝 ‘ 2 ) ∈ ℂ ) |
60 |
59
|
mul02d |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 0 · ( 𝑝 ‘ 2 ) ) = 0 ) |
61 |
47 56 60
|
3eqtrd |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝐴 · ( 𝑝 ‘ 2 ) ) = 0 ) |
62 |
6
|
oveq1i |
⊢ ( 𝐵 · ( 𝑝 ‘ 1 ) ) = ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) |
63 |
62
|
a1i |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝐵 · ( 𝑝 ‘ 1 ) ) = ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) ) |
64 |
|
oveq1 |
⊢ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) → ( ( 𝑋 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) = ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) |
65 |
64
|
oveq2d |
⊢ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) → ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑋 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) = ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) ) |
66 |
7 65
|
syl5eq |
⊢ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) → 𝐶 = ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) ) |
67 |
66
|
adantl |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → 𝐶 = ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) ) |
68 |
63 67
|
oveq12d |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) = ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) ) ) |
69 |
61 68
|
eqeq12d |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) ↔ 0 = ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) ) ) ) |
70 |
43 44 45 69
|
syl21anc |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) ↔ 0 = ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) ) ) ) |
71 |
1 3
|
rrx2pyel |
⊢ ( 𝑌 ∈ 𝑃 → ( 𝑌 ‘ 2 ) ∈ ℝ ) |
72 |
71
|
recnd |
⊢ ( 𝑌 ∈ 𝑃 → ( 𝑌 ‘ 2 ) ∈ ℂ ) |
73 |
72
|
3ad2ant2 |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) → ( 𝑌 ‘ 2 ) ∈ ℂ ) |
74 |
52 73
|
mulcomd |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) → ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) = ( ( 𝑌 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) ) |
75 |
74
|
oveq2d |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) → ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) = ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) ) ) |
76 |
1 3
|
rrx2pyel |
⊢ ( 𝑋 ∈ 𝑃 → ( 𝑋 ‘ 2 ) ∈ ℝ ) |
77 |
76
|
recnd |
⊢ ( 𝑋 ∈ 𝑃 → ( 𝑋 ‘ 2 ) ∈ ℂ ) |
78 |
77
|
3ad2ant1 |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) → ( 𝑋 ‘ 2 ) ∈ ℂ ) |
79 |
78 73 52
|
subdird |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) → ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) = ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) ) ) |
80 |
75 79
|
eqtr4d |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) → ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) = ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ) |
81 |
80
|
ad2antlr |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) = ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ) |
82 |
81
|
oveq2d |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) ) = ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ) ) |
83 |
82
|
eqeq2d |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( 0 = ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) ) ↔ 0 = ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ) ) ) |
84 |
|
eqcom |
⊢ ( 0 = ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ) ↔ ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ) = 0 ) |
85 |
84
|
a1i |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( 0 = ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ) ↔ ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ) = 0 ) ) |
86 |
73
|
ad2antlr |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( 𝑌 ‘ 2 ) ∈ ℂ ) |
87 |
78
|
ad2antlr |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( 𝑋 ‘ 2 ) ∈ ℂ ) |
88 |
86 87
|
subcld |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) ∈ ℂ ) |
89 |
1 3
|
rrx2pxel |
⊢ ( 𝑝 ∈ 𝑃 → ( 𝑝 ‘ 1 ) ∈ ℝ ) |
90 |
89
|
recnd |
⊢ ( 𝑝 ∈ 𝑃 → ( 𝑝 ‘ 1 ) ∈ ℂ ) |
91 |
90
|
adantl |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( 𝑝 ‘ 1 ) ∈ ℂ ) |
92 |
88 91
|
mulcld |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) ∈ ℂ ) |
93 |
87 86
|
subcld |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) ∈ ℂ ) |
94 |
52
|
ad2antlr |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( 𝑌 ‘ 1 ) ∈ ℂ ) |
95 |
93 94
|
mulcld |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ∈ ℂ ) |
96 |
|
addeq0 |
⊢ ( ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) ∈ ℂ ∧ ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ∈ ℂ ) → ( ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ) = 0 ↔ ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) = - ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ) ) |
97 |
92 95 96
|
syl2anc |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ) = 0 ↔ ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) = - ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ) ) |
98 |
93 94
|
mulneg1d |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( - ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) = - ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ) |
99 |
87 86
|
negsubdi2d |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → - ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) = ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) ) |
100 |
99
|
oveq1d |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( - ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) = ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ) |
101 |
98 100
|
eqtr3d |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → - ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) = ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ) |
102 |
101
|
eqeq2d |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) = - ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ↔ ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) = ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ) ) |
103 |
|
necom |
⊢ ( ( 𝑋 ‘ 2 ) ≠ ( 𝑌 ‘ 2 ) ↔ ( 𝑌 ‘ 2 ) ≠ ( 𝑋 ‘ 2 ) ) |
104 |
39 42 103
|
3imtr3i |
⊢ ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) → ( 𝑌 ‘ 2 ) ≠ ( 𝑋 ‘ 2 ) ) |
105 |
104
|
adantr |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( 𝑌 ‘ 2 ) ≠ ( 𝑋 ‘ 2 ) ) |
106 |
86 87 105
|
subne0d |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) ≠ 0 ) |
107 |
91 94 88 106
|
mulcand |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) = ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ↔ ( 𝑝 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ) |
108 |
102 107
|
bitrd |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) = - ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ↔ ( 𝑝 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ) |
109 |
85 97 108
|
3bitrd |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( 0 = ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ) ↔ ( 𝑝 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ) |
110 |
83 109
|
bitrd |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( 0 = ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) ) ↔ ( 𝑝 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ) |
111 |
|
simpl |
⊢ ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) → ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) |
112 |
111
|
eqcomd |
⊢ ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) → ( 𝑌 ‘ 1 ) = ( 𝑋 ‘ 1 ) ) |
113 |
112
|
adantr |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( 𝑌 ‘ 1 ) = ( 𝑋 ‘ 1 ) ) |
114 |
113
|
eqeq2d |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( 𝑝 ‘ 1 ) = ( 𝑌 ‘ 1 ) ↔ ( 𝑝 ‘ 1 ) = ( 𝑋 ‘ 1 ) ) ) |
115 |
70 110 114
|
3bitrrd |
⊢ ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ( 𝑝 ‘ 1 ) = ( 𝑋 ‘ 1 ) ↔ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) ) ) |
116 |
115
|
rabbidva |
⊢ ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ) → { 𝑝 ∈ 𝑃 ∣ ( 𝑝 ‘ 1 ) = ( 𝑋 ‘ 1 ) } = { 𝑝 ∈ 𝑃 ∣ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) } ) |
117 |
42 116
|
sylbi |
⊢ ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → { 𝑝 ∈ 𝑃 ∣ ( 𝑝 ‘ 1 ) = ( 𝑋 ‘ 1 ) } = { 𝑝 ∈ 𝑃 ∣ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) } ) |
118 |
41 117
|
eqtrd |
⊢ ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 𝐿 𝑌 ) = { 𝑝 ∈ 𝑃 ∣ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) } ) |
119 |
1 2 3 4
|
rrx2line |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) → ( 𝑋 𝐿 𝑌 ) = { 𝑝 ∈ 𝑃 ∣ ∃ 𝑡 ∈ ℝ ( ( 𝑝 ‘ 1 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 1 ) ) + ( 𝑡 · ( 𝑌 ‘ 1 ) ) ) ∧ ( 𝑝 ‘ 2 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 2 ) ) + ( 𝑡 · ( 𝑌 ‘ 2 ) ) ) ) } ) |
120 |
119
|
adantr |
⊢ ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ ¬ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 𝐿 𝑌 ) = { 𝑝 ∈ 𝑃 ∣ ∃ 𝑡 ∈ ℝ ( ( 𝑝 ‘ 1 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 1 ) ) + ( 𝑡 · ( 𝑌 ‘ 1 ) ) ) ∧ ( 𝑝 ‘ 2 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 2 ) ) + ( 𝑡 · ( 𝑌 ‘ 2 ) ) ) ) } ) |
121 |
|
df-ne |
⊢ ( ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ↔ ¬ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) |
122 |
89
|
ad2antlr |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) → ( 𝑝 ‘ 1 ) ∈ ℝ ) |
123 |
1 3
|
rrx2pxel |
⊢ ( 𝑋 ∈ 𝑃 → ( 𝑋 ‘ 1 ) ∈ ℝ ) |
124 |
123
|
3ad2ant1 |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) → ( 𝑋 ‘ 1 ) ∈ ℝ ) |
125 |
124
|
ad2antrr |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) → ( 𝑋 ‘ 1 ) ∈ ℝ ) |
126 |
50
|
3ad2ant2 |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) → ( 𝑌 ‘ 1 ) ∈ ℝ ) |
127 |
126
|
ad2antrr |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) → ( 𝑌 ‘ 1 ) ∈ ℝ ) |
128 |
|
simpr |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) → ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) |
129 |
57
|
ad2antlr |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) → ( 𝑝 ‘ 2 ) ∈ ℝ ) |
130 |
76
|
3ad2ant1 |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) → ( 𝑋 ‘ 2 ) ∈ ℝ ) |
131 |
130
|
ad2antrr |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) → ( 𝑋 ‘ 2 ) ∈ ℝ ) |
132 |
71
|
3ad2ant2 |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) → ( 𝑌 ‘ 2 ) ∈ ℝ ) |
133 |
132
|
ad2antrr |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) → ( 𝑌 ‘ 2 ) ∈ ℝ ) |
134 |
122 125 127 128 129 131 133
|
affinecomb2 |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) → ( ∃ 𝑡 ∈ ℝ ( ( 𝑝 ‘ 1 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 1 ) ) + ( 𝑡 · ( 𝑌 ‘ 1 ) ) ) ∧ ( 𝑝 ‘ 2 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 2 ) ) + ( 𝑡 · ( 𝑌 ‘ 2 ) ) ) ) ↔ ( ( ( 𝑌 ‘ 1 ) − ( 𝑋 ‘ 1 ) ) · ( 𝑝 ‘ 2 ) ) = ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑋 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) ) ) ) |
135 |
5
|
eqcomi |
⊢ ( ( 𝑌 ‘ 1 ) − ( 𝑋 ‘ 1 ) ) = 𝐴 |
136 |
135
|
oveq1i |
⊢ ( ( ( 𝑌 ‘ 1 ) − ( 𝑋 ‘ 1 ) ) · ( 𝑝 ‘ 2 ) ) = ( 𝐴 · ( 𝑝 ‘ 2 ) ) |
137 |
6
|
eqcomi |
⊢ ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) = 𝐵 |
138 |
137
|
oveq1i |
⊢ ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) = ( 𝐵 · ( 𝑝 ‘ 1 ) ) |
139 |
7
|
eqcomi |
⊢ ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑋 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) = 𝐶 |
140 |
138 139
|
oveq12i |
⊢ ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑋 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) |
141 |
136 140
|
eqeq12i |
⊢ ( ( ( ( 𝑌 ‘ 1 ) − ( 𝑋 ‘ 1 ) ) · ( 𝑝 ‘ 2 ) ) = ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) + ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑋 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) ) ↔ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) ) |
142 |
134 141
|
bitrdi |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) → ( ∃ 𝑡 ∈ ℝ ( ( 𝑝 ‘ 1 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 1 ) ) + ( 𝑡 · ( 𝑌 ‘ 1 ) ) ) ∧ ( 𝑝 ‘ 2 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 2 ) ) + ( 𝑡 · ( 𝑌 ‘ 2 ) ) ) ) ↔ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) ) ) |
143 |
142
|
expcom |
⊢ ( ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) → ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) → ( ∃ 𝑡 ∈ ℝ ( ( 𝑝 ‘ 1 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 1 ) ) + ( 𝑡 · ( 𝑌 ‘ 1 ) ) ) ∧ ( 𝑝 ‘ 2 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 2 ) ) + ( 𝑡 · ( 𝑌 ‘ 2 ) ) ) ) ↔ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) ) ) ) |
144 |
121 143
|
sylbir |
⊢ ( ¬ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) → ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ 𝑝 ∈ 𝑃 ) → ( ∃ 𝑡 ∈ ℝ ( ( 𝑝 ‘ 1 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 1 ) ) + ( 𝑡 · ( 𝑌 ‘ 1 ) ) ) ∧ ( 𝑝 ‘ 2 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 2 ) ) + ( 𝑡 · ( 𝑌 ‘ 2 ) ) ) ) ↔ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) ) ) ) |
145 |
144
|
expd |
⊢ ( ¬ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) → ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) → ( 𝑝 ∈ 𝑃 → ( ∃ 𝑡 ∈ ℝ ( ( 𝑝 ‘ 1 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 1 ) ) + ( 𝑡 · ( 𝑌 ‘ 1 ) ) ) ∧ ( 𝑝 ‘ 2 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 2 ) ) + ( 𝑡 · ( 𝑌 ‘ 2 ) ) ) ) ↔ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) ) ) ) ) |
146 |
145
|
impcom |
⊢ ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ ¬ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑝 ∈ 𝑃 → ( ∃ 𝑡 ∈ ℝ ( ( 𝑝 ‘ 1 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 1 ) ) + ( 𝑡 · ( 𝑌 ‘ 1 ) ) ) ∧ ( 𝑝 ‘ 2 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 2 ) ) + ( 𝑡 · ( 𝑌 ‘ 2 ) ) ) ) ↔ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) ) ) ) |
147 |
146
|
imp |
⊢ ( ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ ¬ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ∧ 𝑝 ∈ 𝑃 ) → ( ∃ 𝑡 ∈ ℝ ( ( 𝑝 ‘ 1 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 1 ) ) + ( 𝑡 · ( 𝑌 ‘ 1 ) ) ) ∧ ( 𝑝 ‘ 2 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 2 ) ) + ( 𝑡 · ( 𝑌 ‘ 2 ) ) ) ) ↔ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) ) ) |
148 |
147
|
rabbidva |
⊢ ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ ¬ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → { 𝑝 ∈ 𝑃 ∣ ∃ 𝑡 ∈ ℝ ( ( 𝑝 ‘ 1 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 1 ) ) + ( 𝑡 · ( 𝑌 ‘ 1 ) ) ) ∧ ( 𝑝 ‘ 2 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 2 ) ) + ( 𝑡 · ( 𝑌 ‘ 2 ) ) ) ) } = { 𝑝 ∈ 𝑃 ∣ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) } ) |
149 |
120 148
|
eqtrd |
⊢ ( ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) ∧ ¬ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 𝐿 𝑌 ) = { 𝑝 ∈ 𝑃 ∣ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) } ) |
150 |
118 149
|
pm2.61dan |
⊢ ( ( 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌 ) → ( 𝑋 𝐿 𝑌 ) = { 𝑝 ∈ 𝑃 ∣ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) } ) |