Metamath Proof Explorer


Theorem rrx2linest

Description: The line passing through the two different points X and Y in a real Euclidean space of dimension 2 in "standard form". (Contributed by AV, 2-Feb-2023)

Ref Expression
Hypotheses rrx2line.i 𝐼 = { 1 , 2 }
rrx2line.e 𝐸 = ( ℝ^ ‘ 𝐼 )
rrx2line.b 𝑃 = ( ℝ ↑m 𝐼 )
rrx2line.l 𝐿 = ( LineM𝐸 )
rrx2linest.a 𝐴 = ( ( 𝑌 ‘ 1 ) − ( 𝑋 ‘ 1 ) )
rrx2linest.b 𝐵 = ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) )
rrx2linest.c 𝐶 = ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑋 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) )
Assertion rrx2linest ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) → ( 𝑋 𝐿 𝑌 ) = { 𝑝𝑃 ∣ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) } )

Proof

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 = ( LineM𝐸 )
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 ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → 𝑋𝑃 )
9 simpl2 ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → 𝑌𝑃 )
10 simpr ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) )
11 simpr ( ( ( 𝑋𝑃𝑌𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) )
12 11 anim1i ( ( ( ( 𝑋𝑃𝑌𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ∧ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) → ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) )
13 1 raleqi ( ∀ 𝑖𝐼 ( 𝑋𝑖 ) = ( 𝑌𝑖 ) ↔ ∀ 𝑖 ∈ { 1 , 2 } ( 𝑋𝑖 ) = ( 𝑌𝑖 ) )
14 1ex 1 ∈ V
15 2ex 2 ∈ V
16 fveq2 ( 𝑖 = 1 → ( 𝑋𝑖 ) = ( 𝑋 ‘ 1 ) )
17 fveq2 ( 𝑖 = 1 → ( 𝑌𝑖 ) = ( 𝑌 ‘ 1 ) )
18 16 17 eqeq12d ( 𝑖 = 1 → ( ( 𝑋𝑖 ) = ( 𝑌𝑖 ) ↔ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) )
19 fveq2 ( 𝑖 = 2 → ( 𝑋𝑖 ) = ( 𝑋 ‘ 2 ) )
20 fveq2 ( 𝑖 = 2 → ( 𝑌𝑖 ) = ( 𝑌 ‘ 2 ) )
21 19 20 eqeq12d ( 𝑖 = 2 → ( ( 𝑋𝑖 ) = ( 𝑌𝑖 ) ↔ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) )
22 14 15 18 21 ralpr ( ∀ 𝑖 ∈ { 1 , 2 } ( 𝑋𝑖 ) = ( 𝑌𝑖 ) ↔ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) )
23 13 22 bitri ( ∀ 𝑖𝐼 ( 𝑋𝑖 ) = ( 𝑌𝑖 ) ↔ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) )
24 12 23 sylibr ( ( ( ( 𝑋𝑃𝑌𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ∧ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) → ∀ 𝑖𝐼 ( 𝑋𝑖 ) = ( 𝑌𝑖 ) )
25 elmapfn ( 𝑋 ∈ ( ℝ ↑m 𝐼 ) → 𝑋 Fn 𝐼 )
26 25 3 eleq2s ( 𝑋𝑃𝑋 Fn 𝐼 )
27 elmapfn ( 𝑌 ∈ ( ℝ ↑m 𝐼 ) → 𝑌 Fn 𝐼 )
28 27 3 eleq2s ( 𝑌𝑃𝑌 Fn 𝐼 )
29 26 28 anim12i ( ( 𝑋𝑃𝑌𝑃 ) → ( 𝑋 Fn 𝐼𝑌 Fn 𝐼 ) )
30 29 ad2antrr ( ( ( ( 𝑋𝑃𝑌𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ∧ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) → ( 𝑋 Fn 𝐼𝑌 Fn 𝐼 ) )
31 eqfnfv ( ( 𝑋 Fn 𝐼𝑌 Fn 𝐼 ) → ( 𝑋 = 𝑌 ↔ ∀ 𝑖𝐼 ( 𝑋𝑖 ) = ( 𝑌𝑖 ) ) )
32 30 31 syl ( ( ( ( 𝑋𝑃𝑌𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ∧ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) → ( 𝑋 = 𝑌 ↔ ∀ 𝑖𝐼 ( 𝑋𝑖 ) = ( 𝑌𝑖 ) ) )
33 24 32 mpbird ( ( ( ( 𝑋𝑃𝑌𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ∧ ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) ) → 𝑋 = 𝑌 )
34 33 ex ( ( ( 𝑋𝑃𝑌𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( ( 𝑋 ‘ 2 ) = ( 𝑌 ‘ 2 ) → 𝑋 = 𝑌 ) )
35 34 necon3d ( ( ( 𝑋𝑃𝑌𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋𝑌 → ( 𝑋 ‘ 2 ) ≠ ( 𝑌 ‘ 2 ) ) )
36 35 ex ( ( 𝑋𝑃𝑌𝑃 ) → ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) → ( 𝑋𝑌 → ( 𝑋 ‘ 2 ) ≠ ( 𝑌 ‘ 2 ) ) ) )
37 36 com23 ( ( 𝑋𝑃𝑌𝑃 ) → ( 𝑋𝑌 → ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) → ( 𝑋 ‘ 2 ) ≠ ( 𝑌 ‘ 2 ) ) ) )
38 37 3impia ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) → ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) → ( 𝑋 ‘ 2 ) ≠ ( 𝑌 ‘ 2 ) ) )
39 38 imp ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 ‘ 2 ) ≠ ( 𝑌 ‘ 2 ) )
40 1 2 3 4 rrx2vlinest ( ( 𝑋𝑃𝑌𝑃 ∧ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋 ‘ 2 ) ≠ ( 𝑌 ‘ 2 ) ) ) → ( 𝑋 𝐿 𝑌 ) = { 𝑝𝑃 ∣ ( 𝑝 ‘ 1 ) = ( 𝑋 ‘ 1 ) } )
41 8 9 10 39 40 syl112anc ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 𝐿 𝑌 ) = { 𝑝𝑃 ∣ ( 𝑝 ‘ 1 ) = ( 𝑋 ‘ 1 ) } )
42 ancom ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ↔ ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) )
43 simplr ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) )
44 simpr ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → 𝑝𝑃 )
45 simpll ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) )
46 5 oveq1i ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( ( 𝑌 ‘ 1 ) − ( 𝑋 ‘ 1 ) ) · ( 𝑝 ‘ 2 ) )
47 46 a1i ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 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 ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( ( 𝑌 ‘ 1 ) − ( 𝑋 ‘ 1 ) ) = ( ( 𝑌 ‘ 1 ) − ( 𝑌 ‘ 1 ) ) )
50 1 3 rrx2pxel ( 𝑌𝑃 → ( 𝑌 ‘ 1 ) ∈ ℝ )
51 50 recnd ( 𝑌𝑃 → ( 𝑌 ‘ 1 ) ∈ ℂ )
52 51 3ad2ant2 ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) → ( 𝑌 ‘ 1 ) ∈ ℂ )
53 52 ad2antrr ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑌 ‘ 1 ) ∈ ℂ )
54 53 subidd ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( ( 𝑌 ‘ 1 ) − ( 𝑌 ‘ 1 ) ) = 0 )
55 49 54 eqtrd ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( ( 𝑌 ‘ 1 ) − ( 𝑋 ‘ 1 ) ) = 0 )
56 55 oveq1d ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( ( ( 𝑌 ‘ 1 ) − ( 𝑋 ‘ 1 ) ) · ( 𝑝 ‘ 2 ) ) = ( 0 · ( 𝑝 ‘ 2 ) ) )
57 1 3 rrx2pyel ( 𝑝𝑃 → ( 𝑝 ‘ 2 ) ∈ ℝ )
58 57 recnd ( 𝑝𝑃 → ( 𝑝 ‘ 2 ) ∈ ℂ )
59 58 ad2antlr ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑝 ‘ 2 ) ∈ ℂ )
60 59 mul02d ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 0 · ( 𝑝 ‘ 2 ) ) = 0 )
61 47 56 60 3eqtrd ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝐴 · ( 𝑝 ‘ 2 ) ) = 0 )
62 6 oveq1i ( 𝐵 · ( 𝑝 ‘ 1 ) ) = ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) )
63 62 a1i ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 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 ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → 𝐶 = ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) )
68 63 67 oveq12d ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 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 ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 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 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( ( 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 ( 𝑌𝑃 → ( 𝑌 ‘ 2 ) ∈ ℝ )
72 71 recnd ( 𝑌𝑃 → ( 𝑌 ‘ 2 ) ∈ ℂ )
73 72 3ad2ant2 ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) → ( 𝑌 ‘ 2 ) ∈ ℂ )
74 52 73 mulcomd ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) → ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) = ( ( 𝑌 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) )
75 74 oveq2d ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) → ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) = ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) ) )
76 1 3 rrx2pyel ( 𝑋𝑃 → ( 𝑋 ‘ 2 ) ∈ ℝ )
77 76 recnd ( 𝑋𝑃 → ( 𝑋 ‘ 2 ) ∈ ℂ )
78 77 3ad2ant1 ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) → ( 𝑋 ‘ 2 ) ∈ ℂ )
79 78 73 52 subdird ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) → ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) = ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) ) )
80 75 79 eqtr4d ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) → ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) = ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) )
81 80 ad2antlr ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( ( ( 𝑋 ‘ 2 ) · ( 𝑌 ‘ 1 ) ) − ( ( 𝑌 ‘ 1 ) · ( 𝑌 ‘ 2 ) ) ) = ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) )
82 81 oveq2d ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( ( ( ( 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 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( 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 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( 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 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( 𝑌 ‘ 2 ) ∈ ℂ )
87 78 ad2antlr ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( 𝑋 ‘ 2 ) ∈ ℂ )
88 86 87 subcld ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) ∈ ℂ )
89 1 3 rrx2pxel ( 𝑝𝑃 → ( 𝑝 ‘ 1 ) ∈ ℝ )
90 89 recnd ( 𝑝𝑃 → ( 𝑝 ‘ 1 ) ∈ ℂ )
91 90 adantl ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( 𝑝 ‘ 1 ) ∈ ℂ )
92 88 91 mulcld ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) ∈ ℂ )
93 87 86 subcld ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) ∈ ℂ )
94 52 ad2antlr ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( 𝑌 ‘ 1 ) ∈ ℂ )
95 93 94 mulcld ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( ( ( 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 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( ( ( ( ( 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 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( - ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) = - ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) )
99 87 86 negsubdi2d ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → - ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) = ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) )
100 99 oveq1d ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( - ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) = ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) )
101 98 100 eqtr3d ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → - ( ( ( 𝑋 ‘ 2 ) − ( 𝑌 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) = ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) )
102 101 eqeq2d ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( ( ( ( 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 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) → ( 𝑌 ‘ 2 ) ≠ ( 𝑋 ‘ 2 ) )
105 104 adantr ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( 𝑌 ‘ 2 ) ≠ ( 𝑋 ‘ 2 ) )
106 86 87 105 subne0d ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) ≠ 0 )
107 91 94 88 106 mulcand ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑝 ‘ 1 ) ) = ( ( ( 𝑌 ‘ 2 ) − ( 𝑋 ‘ 2 ) ) · ( 𝑌 ‘ 1 ) ) ↔ ( 𝑝 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) )
108 102 107 bitrd ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( ( ( ( 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 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( 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 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( 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 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) → ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) )
112 111 eqcomd ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) → ( 𝑌 ‘ 1 ) = ( 𝑋 ‘ 1 ) )
113 112 adantr ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( 𝑌 ‘ 1 ) = ( 𝑋 ‘ 1 ) )
114 113 eqeq2d ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( ( 𝑝 ‘ 1 ) = ( 𝑌 ‘ 1 ) ↔ ( 𝑝 ‘ 1 ) = ( 𝑋 ‘ 1 ) ) )
115 70 110 114 3bitrrd ( ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) ∧ 𝑝𝑃 ) → ( ( 𝑝 ‘ 1 ) = ( 𝑋 ‘ 1 ) ↔ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) ) )
116 115 rabbidva ( ( ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ∧ ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ) → { 𝑝𝑃 ∣ ( 𝑝 ‘ 1 ) = ( 𝑋 ‘ 1 ) } = { 𝑝𝑃 ∣ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) } )
117 42 116 sylbi ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → { 𝑝𝑃 ∣ ( 𝑝 ‘ 1 ) = ( 𝑋 ‘ 1 ) } = { 𝑝𝑃 ∣ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) } )
118 41 117 eqtrd ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 𝐿 𝑌 ) = { 𝑝𝑃 ∣ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) } )
119 1 2 3 4 rrx2line ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) → ( 𝑋 𝐿 𝑌 ) = { 𝑝𝑃 ∣ ∃ 𝑡 ∈ ℝ ( ( 𝑝 ‘ 1 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 1 ) ) + ( 𝑡 · ( 𝑌 ‘ 1 ) ) ) ∧ ( 𝑝 ‘ 2 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 2 ) ) + ( 𝑡 · ( 𝑌 ‘ 2 ) ) ) ) } )
120 119 adantr ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ ¬ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 𝐿 𝑌 ) = { 𝑝𝑃 ∣ ∃ 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 ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) → ( 𝑝 ‘ 1 ) ∈ ℝ )
123 1 3 rrx2pxel ( 𝑋𝑃 → ( 𝑋 ‘ 1 ) ∈ ℝ )
124 123 3ad2ant1 ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) → ( 𝑋 ‘ 1 ) ∈ ℝ )
125 124 ad2antrr ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) → ( 𝑋 ‘ 1 ) ∈ ℝ )
126 50 3ad2ant2 ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) → ( 𝑌 ‘ 1 ) ∈ ℝ )
127 126 ad2antrr ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) → ( 𝑌 ‘ 1 ) ∈ ℝ )
128 simpr ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) → ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) )
129 57 ad2antlr ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) → ( 𝑝 ‘ 2 ) ∈ ℝ )
130 76 3ad2ant1 ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) → ( 𝑋 ‘ 2 ) ∈ ℝ )
131 130 ad2antrr ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) → ( 𝑋 ‘ 2 ) ∈ ℝ )
132 71 3ad2ant2 ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) → ( 𝑌 ‘ 2 ) ∈ ℝ )
133 132 ad2antrr ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 𝑋 ‘ 1 ) ≠ ( 𝑌 ‘ 1 ) ) → ( 𝑌 ‘ 2 ) ∈ ℝ )
134 122 125 127 128 129 131 133 affinecomb2 ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 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 ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) ∧ ( 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 ) → ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) → ( ∃ 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 ) → ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ 𝑝𝑃 ) → ( ∃ 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 ) → ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) → ( 𝑝𝑃 → ( ∃ 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 ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ ¬ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑝𝑃 → ( ∃ 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 ( ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ ¬ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) ∧ 𝑝𝑃 ) → ( ∃ 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 ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ ¬ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → { 𝑝𝑃 ∣ ∃ 𝑡 ∈ ℝ ( ( 𝑝 ‘ 1 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 1 ) ) + ( 𝑡 · ( 𝑌 ‘ 1 ) ) ) ∧ ( 𝑝 ‘ 2 ) = ( ( ( 1 − 𝑡 ) · ( 𝑋 ‘ 2 ) ) + ( 𝑡 · ( 𝑌 ‘ 2 ) ) ) ) } = { 𝑝𝑃 ∣ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) } )
149 120 148 eqtrd ( ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) ∧ ¬ ( 𝑋 ‘ 1 ) = ( 𝑌 ‘ 1 ) ) → ( 𝑋 𝐿 𝑌 ) = { 𝑝𝑃 ∣ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) } )
150 118 149 pm2.61dan ( ( 𝑋𝑃𝑌𝑃𝑋𝑌 ) → ( 𝑋 𝐿 𝑌 ) = { 𝑝𝑃 ∣ ( 𝐴 · ( 𝑝 ‘ 2 ) ) = ( ( 𝐵 · ( 𝑝 ‘ 1 ) ) + 𝐶 ) } )