MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  acopyeu Structured version   Visualization version   GIF version

Theorem acopyeu 25770
Description: Angle construction. Theorem 11.15 of [Schwabhauser] p. 98. This is Hilbert's axiom III.4 for geometry. Akin to a uniqueness theorem, this states that if two points 𝑋 and 𝑌 both fulfill the conditions, then they are on the same half-line. (Contributed by Thierry Arnoux, 9-Aug-2020.)
Hypotheses
Ref Expression
dfcgra2.p 𝑃 = (Base‘𝐺)
dfcgra2.i 𝐼 = (Itv‘𝐺)
dfcgra2.m = (dist‘𝐺)
dfcgra2.g (𝜑𝐺 ∈ TarskiG)
dfcgra2.a (𝜑𝐴𝑃)
dfcgra2.b (𝜑𝐵𝑃)
dfcgra2.c (𝜑𝐶𝑃)
dfcgra2.d (𝜑𝐷𝑃)
dfcgra2.e (𝜑𝐸𝑃)
dfcgra2.f (𝜑𝐹𝑃)
acopy.l 𝐿 = (LineG‘𝐺)
acopy.1 (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
acopy.2 (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
acopyeu.x (𝜑𝑋𝑃)
acopyeu.y (𝜑𝑌𝑃)
acopyeu.k 𝐾 = (hlG‘𝐺)
acopyeu.1 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑋”⟩)
acopyeu.2 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑌”⟩)
acopyeu.3 (𝜑𝑋((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
acopyeu.4 (𝜑𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
Assertion
Ref Expression
acopyeu (𝜑𝑋(𝐾𝐸)𝑌)

Proof of Theorem acopyeu
Dummy variables 𝑎 𝑑 𝑡 𝑥 𝑦 𝑏 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfcgra2.p . . . 4 𝑃 = (Base‘𝐺)
2 dfcgra2.i . . . 4 𝐼 = (Itv‘𝐺)
3 acopyeu.k . . . 4 𝐾 = (hlG‘𝐺)
4 acopyeu.x . . . . . 6 (𝜑𝑋𝑃)
54ad2antrr 762 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑋𝑃)
65ad3antrrr 766 . . . 4 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑋𝑃)
7 simplr 807 . . . 4 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑦𝑃)
8 acopyeu.y . . . . . 6 (𝜑𝑌𝑃)
98ad2antrr 762 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑌𝑃)
109ad3antrrr 766 . . . 4 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑌𝑃)
11 dfcgra2.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
1211ad2antrr 762 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐺 ∈ TarskiG)
1312ad3antrrr 766 . . . 4 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝐺 ∈ TarskiG)
14 dfcgra2.e . . . . . 6 (𝜑𝐸𝑃)
1514ad2antrr 762 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐸𝑃)
1615ad3antrrr 766 . . . 4 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝐸𝑃)
17 dfcgra2.m . . . . . . 7 = (dist‘𝐺)
18 acopy.l . . . . . . 7 𝐿 = (LineG‘𝐺)
19 dfcgra2.a . . . . . . . . 9 (𝜑𝐴𝑃)
2019ad2antrr 762 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐴𝑃)
2120ad3antrrr 766 . . . . . . 7 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝐴𝑃)
22 dfcgra2.b . . . . . . . . 9 (𝜑𝐵𝑃)
2322ad2antrr 762 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐵𝑃)
2423ad3antrrr 766 . . . . . . 7 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝐵𝑃)
25 dfcgra2.c . . . . . . . . 9 (𝜑𝐶𝑃)
2625ad2antrr 762 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐶𝑃)
2726ad3antrrr 766 . . . . . . 7 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝐶𝑃)
28 simplr 807 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑𝑃)
2928ad3antrrr 766 . . . . . . 7 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑑𝑃)
30 dfcgra2.f . . . . . . . . 9 (𝜑𝐹𝑃)
3130ad2antrr 762 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐹𝑃)
3231ad3antrrr 766 . . . . . . 7 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝐹𝑃)
33 acopy.1 . . . . . . . . 9 (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
3433ad2antrr 762 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
3534ad3antrrr 766 . . . . . . 7 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
36 dfcgra2.d . . . . . . . . . 10 (𝜑𝐷𝑃)
3736ad2antrr 762 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐷𝑃)
38 acopy.2 . . . . . . . . . 10 (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
3938ad2antrr 762 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
40 simprl 809 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑(𝐾𝐸)𝐷)
411, 2, 3, 28, 37, 15, 12, 18, 40hlln 25547 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑 ∈ (𝐷𝐿𝐸))
421, 2, 3, 28, 37, 15, 12, 40hlne1 25545 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑𝐸)
431, 2, 18, 12, 37, 15, 31, 28, 39, 41, 42ncolncol 25586 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ¬ (𝑑 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
4443ad3antrrr 766 . . . . . . 7 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → ¬ (𝑑 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
45 simprr 811 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐸 𝑑) = (𝐵 𝐴))
461, 17, 2, 12, 15, 28, 23, 20, 45tgcgrcomlr 25420 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝑑 𝐸) = (𝐴 𝐵))
4746eqcomd 2657 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐴 𝐵) = (𝑑 𝐸))
4847ad3antrrr 766 . . . . . . 7 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → (𝐴 𝐵) = (𝑑 𝐸))
49 simpl 472 . . . . . . . . . . 11 ((𝑢 = 𝑎𝑣 = 𝑏) → 𝑢 = 𝑎)
5049eleq1d 2715 . . . . . . . . . 10 ((𝑢 = 𝑎𝑣 = 𝑏) → (𝑢 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ↔ 𝑎 ∈ (𝑃 ∖ (𝑑𝐿𝐸))))
51 simpr 476 . . . . . . . . . . 11 ((𝑢 = 𝑎𝑣 = 𝑏) → 𝑣 = 𝑏)
5251eleq1d 2715 . . . . . . . . . 10 ((𝑢 = 𝑎𝑣 = 𝑏) → (𝑣 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ↔ 𝑏 ∈ (𝑃 ∖ (𝑑𝐿𝐸))))
5350, 52anbi12d 747 . . . . . . . . 9 ((𝑢 = 𝑎𝑣 = 𝑏) → ((𝑢 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ∧ 𝑣 ∈ (𝑃 ∖ (𝑑𝐿𝐸))) ↔ (𝑎 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑑𝐿𝐸)))))
54 simpr 476 . . . . . . . . . . 11 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑤 = 𝑡)
55 simpll 805 . . . . . . . . . . . 12 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑢 = 𝑎)
56 simplr 807 . . . . . . . . . . . 12 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑣 = 𝑏)
5755, 56oveq12d 6708 . . . . . . . . . . 11 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → (𝑢𝐼𝑣) = (𝑎𝐼𝑏))
5854, 57eleq12d 2724 . . . . . . . . . 10 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → (𝑤 ∈ (𝑢𝐼𝑣) ↔ 𝑡 ∈ (𝑎𝐼𝑏)))
5958cbvrexdva 3208 . . . . . . . . 9 ((𝑢 = 𝑎𝑣 = 𝑏) → (∃𝑤 ∈ (𝑑𝐿𝐸)𝑤 ∈ (𝑢𝐼𝑣) ↔ ∃𝑡 ∈ (𝑑𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏)))
6053, 59anbi12d 747 . . . . . . . 8 ((𝑢 = 𝑎𝑣 = 𝑏) → (((𝑢 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ∧ 𝑣 ∈ (𝑃 ∖ (𝑑𝐿𝐸))) ∧ ∃𝑤 ∈ (𝑑𝐿𝐸)𝑤 ∈ (𝑢𝐼𝑣)) ↔ ((𝑎 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑑𝐿𝐸))) ∧ ∃𝑡 ∈ (𝑑𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏))))
6160cbvopabv 4755 . . . . . . 7 {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ∧ 𝑣 ∈ (𝑃 ∖ (𝑑𝐿𝐸))) ∧ ∃𝑤 ∈ (𝑑𝐿𝐸)𝑤 ∈ (𝑢𝐼𝑣))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑑𝐿𝐸))) ∧ ∃𝑡 ∈ (𝑑𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏))}
62 simpllr 815 . . . . . . 7 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑥𝑃)
63 simprll 819 . . . . . . 7 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩)
64 simprrl 821 . . . . . . 7 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩)
651, 2, 18, 11, 36, 14, 30, 38ncolne1 25565 . . . . . . . . . . . . 13 (𝜑𝐷𝐸)
661, 2, 18, 11, 36, 14, 65tgelrnln 25570 . . . . . . . . . . . 12 (𝜑 → (𝐷𝐿𝐸) ∈ ran 𝐿)
6766ad2antrr 762 . . . . . . . . . . 11 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐷𝐿𝐸) ∈ ran 𝐿)
681, 2, 18, 11, 36, 14, 65tglinerflx2 25574 . . . . . . . . . . . 12 (𝜑𝐸 ∈ (𝐷𝐿𝐸))
6968ad2antrr 762 . . . . . . . . . . 11 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐸 ∈ (𝐷𝐿𝐸))
701, 2, 18, 12, 28, 15, 42, 42, 67, 41, 69tglinethru 25576 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐷𝐿𝐸) = (𝑑𝐿𝐸))
7170, 67eqeltrrd 2731 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝑑𝐿𝐸) ∈ ran 𝐿)
7271ad3antrrr 766 . . . . . . . 8 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → (𝑑𝐿𝐸) ∈ ran 𝐿)
7361eqcomi 2660 . . . . . . . 8 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑑𝐿𝐸))) ∧ ∃𝑡 ∈ (𝑑𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏))} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ∧ 𝑣 ∈ (𝑃 ∖ (𝑑𝐿𝐸))) ∧ ∃𝑤 ∈ (𝑑𝐿𝐸)𝑤 ∈ (𝑢𝐼𝑣))}
7469, 70eleqtrd 2732 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐸 ∈ (𝑑𝐿𝐸))
7574ad3antrrr 766 . . . . . . . . 9 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝐸 ∈ (𝑑𝐿𝐸))
7637ad3antrrr 766 . . . . . . . . . . . 12 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝐷𝑃)
77 acopyeu.1 . . . . . . . . . . . . . . . 16 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑋”⟩)
781, 18, 2, 11, 22, 25, 19, 33ncolrot2 25503 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
791, 2, 17, 11, 19, 22, 25, 36, 14, 4, 77, 18, 78cgrancol 25765 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (𝑋 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
801, 18, 2, 11, 36, 14, 4, 79ncolcom 25501 . . . . . . . . . . . . . 14 (𝜑 → ¬ (𝑋 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
8180ad5antr 773 . . . . . . . . . . . . 13 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → ¬ (𝑋 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
82 simprlr 820 . . . . . . . . . . . . . 14 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑥(𝐾𝐸)𝑋)
831, 2, 3, 62, 6, 16, 13, 18, 82hlln 25547 . . . . . . . . . . . . 13 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑥 ∈ (𝑋𝐿𝐸))
841, 2, 3, 62, 6, 16, 13, 82hlne1 25545 . . . . . . . . . . . . 13 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑥𝐸)
851, 2, 18, 13, 6, 16, 76, 62, 81, 83, 84ncolncol 25586 . . . . . . . . . . . 12 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → ¬ (𝑥 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
861, 18, 2, 13, 16, 76, 62, 85ncolcom 25501 . . . . . . . . . . 11 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → ¬ (𝑥 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
87 pm2.45 411 . . . . . . . . . . 11 (¬ (𝑥 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸) → ¬ 𝑥 ∈ (𝐷𝐿𝐸))
8886, 87syl 17 . . . . . . . . . 10 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → ¬ 𝑥 ∈ (𝐷𝐿𝐸))
8970ad3antrrr 766 . . . . . . . . . . 11 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → (𝐷𝐿𝐸) = (𝑑𝐿𝐸))
9089eleq2d 2716 . . . . . . . . . 10 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → (𝑥 ∈ (𝐷𝐿𝐸) ↔ 𝑥 ∈ (𝑑𝐿𝐸)))
9188, 90mtbid 313 . . . . . . . . 9 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → ¬ 𝑥 ∈ (𝑑𝐿𝐸))
921, 2, 18, 13, 72, 16, 61, 3, 75, 62, 6, 91, 82hphl 25708 . . . . . . . 8 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑥((hpG‘𝐺)‘(𝑑𝐿𝐸))𝑋)
93 acopyeu.3 . . . . . . . . . 10 (𝜑𝑋((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
9493ad5antr 773 . . . . . . . . 9 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑋((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
9570fveq2d 6233 . . . . . . . . . . 11 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ((hpG‘𝐺)‘(𝐷𝐿𝐸)) = ((hpG‘𝐺)‘(𝑑𝐿𝐸)))
9695ad3antrrr 766 . . . . . . . . . 10 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → ((hpG‘𝐺)‘(𝐷𝐿𝐸)) = ((hpG‘𝐺)‘(𝑑𝐿𝐸)))
9796breqd 4696 . . . . . . . . 9 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → (𝑋((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹𝑋((hpG‘𝐺)‘(𝑑𝐿𝐸))𝐹))
9894, 97mpbid 222 . . . . . . . 8 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑋((hpG‘𝐺)‘(𝑑𝐿𝐸))𝐹)
991, 2, 18, 13, 72, 62, 73, 6, 92, 32, 98hpgtr 25705 . . . . . . 7 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑥((hpG‘𝐺)‘(𝑑𝐿𝐸))𝐹)
100 acopyeu.2 . . . . . . . . . . . . . . . 16 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑌”⟩)
1011, 2, 17, 11, 19, 22, 25, 36, 14, 8, 100, 18, 78cgrancol 25765 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (𝑌 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
1021, 18, 2, 11, 36, 14, 8, 101ncolcom 25501 . . . . . . . . . . . . . 14 (𝜑 → ¬ (𝑌 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
103102ad5antr 773 . . . . . . . . . . . . 13 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → ¬ (𝑌 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
104 simprrr 822 . . . . . . . . . . . . . 14 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑦(𝐾𝐸)𝑌)
1051, 2, 3, 7, 10, 16, 13, 18, 104hlln 25547 . . . . . . . . . . . . 13 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑦 ∈ (𝑌𝐿𝐸))
1061, 2, 3, 7, 10, 16, 13, 104hlne1 25545 . . . . . . . . . . . . 13 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑦𝐸)
1071, 2, 18, 13, 10, 16, 76, 7, 103, 105, 106ncolncol 25586 . . . . . . . . . . . 12 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → ¬ (𝑦 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
1081, 18, 2, 13, 16, 76, 7, 107ncolcom 25501 . . . . . . . . . . 11 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → ¬ (𝑦 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
109 pm2.45 411 . . . . . . . . . . 11 (¬ (𝑦 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸) → ¬ 𝑦 ∈ (𝐷𝐿𝐸))
110108, 109syl 17 . . . . . . . . . 10 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → ¬ 𝑦 ∈ (𝐷𝐿𝐸))
11189eleq2d 2716 . . . . . . . . . 10 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → (𝑦 ∈ (𝐷𝐿𝐸) ↔ 𝑦 ∈ (𝑑𝐿𝐸)))
112110, 111mtbid 313 . . . . . . . . 9 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → ¬ 𝑦 ∈ (𝑑𝐿𝐸))
1131, 2, 18, 13, 72, 16, 61, 3, 75, 7, 10, 112, 104hphl 25708 . . . . . . . 8 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑦((hpG‘𝐺)‘(𝑑𝐿𝐸))𝑌)
114 acopyeu.4 . . . . . . . . . 10 (𝜑𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
115114ad5antr 773 . . . . . . . . 9 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
11696breqd 4696 . . . . . . . . 9 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → (𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹𝑌((hpG‘𝐺)‘(𝑑𝐿𝐸))𝐹))
117115, 116mpbid 222 . . . . . . . 8 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑌((hpG‘𝐺)‘(𝑑𝐿𝐸))𝐹)
1181, 2, 18, 13, 72, 7, 73, 10, 113, 32, 117hpgtr 25705 . . . . . . 7 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑦((hpG‘𝐺)‘(𝑑𝐿𝐸))𝐹)
1191, 17, 2, 18, 3, 13, 21, 24, 27, 29, 16, 32, 35, 44, 48, 61, 62, 7, 63, 64, 99, 118trgcopyeulem 25742 . . . . . 6 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑥 = 𝑦)
120119, 82eqbrtrrd 4709 . . . . 5 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑦(𝐾𝐸)𝑋)
1211, 2, 3, 7, 6, 16, 13, 120hlcomd 25544 . . . 4 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑋(𝐾𝐸)𝑦)
1221, 2, 3, 6, 7, 10, 13, 16, 121, 104hltr 25550 . . 3 ((((((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) ∧ 𝑥𝑃) ∧ 𝑦𝑃) ∧ ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))) → 𝑋(𝐾𝐸)𝑌)
12377ad2antrr 762 . . . . . 6 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑋”⟩)
1241, 2, 3, 12, 20, 23, 26, 37, 15, 5, 123, 28, 40cgrahl1 25753 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑋”⟩)
1251, 2, 18, 11, 19, 22, 25, 33ncolne1 25565 . . . . . . 7 (𝜑𝐴𝐵)
126125ad2antrr 762 . . . . . 6 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐴𝐵)
1271, 2, 3, 12, 20, 23, 26, 28, 15, 5, 17, 126, 47iscgra1 25747 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑋”⟩ ↔ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋)))
128124, 127mpbid 222 . . . 4 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋))
129100ad2antrr 762 . . . . . 6 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑌”⟩)
1301, 2, 3, 12, 20, 23, 26, 37, 15, 9, 129, 28, 40cgrahl1 25753 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑌”⟩)
1311, 2, 3, 12, 20, 23, 26, 28, 15, 9, 17, 126, 47iscgra1 25747 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑌”⟩ ↔ ∃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌)))
132130, 131mpbid 222 . . . 4 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ∃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))
133 reeanv 3136 . . . 4 (∃𝑥𝑃𝑦𝑃 ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌)) ↔ (∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ ∃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌)))
134128, 132, 133sylanbrc 699 . . 3 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ∃𝑥𝑃𝑦𝑃 ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌)))
135122, 134r19.29vva 3110 . 2 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑋(𝐾𝐸)𝑌)
136125necomd 2878 . . 3 (𝜑𝐵𝐴)
1371, 2, 3, 14, 22, 19, 11, 36, 17, 65, 136hlcgrex 25556 . 2 (𝜑 → ∃𝑑𝑃 (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴)))
138135, 137r19.29a 3107 1 (𝜑𝑋(𝐾𝐸)𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  wrex 2942  cdif 3604   class class class wbr 4685  {copab 4745  ran crn 5144  cfv 5926  (class class class)co 6690  ⟨“cs3 13633  Basecbs 15904  distcds 15997  TarskiGcstrkg 25374  Itvcitv 25380  LineGclng 25381  cgrGccgrg 25450  hlGchlg 25540  hpGchpg 25694  cgrAccgra 25744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-s2 13639  df-s3 13640  df-trkgc 25392  df-trkgb 25393  df-trkgcb 25394  df-trkgld 25396  df-trkg 25397  df-cgrg 25451  df-leg 25523  df-hlg 25541  df-mir 25593  df-rag 25634  df-perpg 25636  df-hpg 25695  df-mid 25711  df-lmi 25712  df-cgra 25745
This theorem is referenced by:  tgasa1  25784
  Copyright terms: Public domain W3C validator