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Theorem acopyeu 28857
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 726 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑋𝑃)
65ad3antrrr 730 . . . 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 769 . . . 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 726 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑌𝑃)
109ad3antrrr 730 . . . 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 726 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐺 ∈ TarskiG)
1312ad3antrrr 730 . . . 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 726 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐸𝑃)
1615ad3antrrr 730 . . . 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 726 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐴𝑃)
2120ad3antrrr 730 . . . . . . 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 726 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐵𝑃)
2423ad3antrrr 730 . . . . . . 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 726 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐶𝑃)
2726ad3antrrr 730 . . . . . . 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 769 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑𝑃)
2928ad3antrrr 730 . . . . . . 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 726 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐹𝑃)
3231ad3antrrr 730 . . . . . . 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 726 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
3534ad3antrrr 730 . . . . . . 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 726 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐷𝑃)
38 acopy.2 . . . . . . . . . 10 (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
3938ad2antrr 726 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
40 simprl 771 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑(𝐾𝐸)𝐷)
411, 2, 3, 28, 37, 15, 12, 18, 40hlln 28630 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑 ∈ (𝐷𝐿𝐸))
421, 2, 3, 28, 37, 15, 12, 40hlne1 28628 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑𝐸)
431, 2, 18, 12, 37, 15, 31, 28, 39, 41, 42ncolncol 28669 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ¬ (𝑑 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
4443ad3antrrr 730 . . . . . . 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 773 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐸 𝑑) = (𝐵 𝐴))
461, 17, 2, 12, 15, 28, 23, 20, 45tgcgrcomlr 28503 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝑑 𝐸) = (𝐴 𝐵))
4746eqcomd 2741 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐴 𝐵) = (𝑑 𝐸))
4847ad3antrrr 730 . . . . . . 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 482 . . . . . . . . . . 11 ((𝑢 = 𝑎𝑣 = 𝑏) → 𝑢 = 𝑎)
5049eleq1d 2824 . . . . . . . . . 10 ((𝑢 = 𝑎𝑣 = 𝑏) → (𝑢 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ↔ 𝑎 ∈ (𝑃 ∖ (𝑑𝐿𝐸))))
51 simpr 484 . . . . . . . . . . 11 ((𝑢 = 𝑎𝑣 = 𝑏) → 𝑣 = 𝑏)
5251eleq1d 2824 . . . . . . . . . 10 ((𝑢 = 𝑎𝑣 = 𝑏) → (𝑣 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ↔ 𝑏 ∈ (𝑃 ∖ (𝑑𝐿𝐸))))
5350, 52anbi12d 632 . . . . . . . . 9 ((𝑢 = 𝑎𝑣 = 𝑏) → ((𝑢 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ∧ 𝑣 ∈ (𝑃 ∖ (𝑑𝐿𝐸))) ↔ (𝑎 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑑𝐿𝐸)))))
54 simpr 484 . . . . . . . . . . 11 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑤 = 𝑡)
55 simpll 767 . . . . . . . . . . . 12 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑢 = 𝑎)
56 simplr 769 . . . . . . . . . . . 12 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑣 = 𝑏)
5755, 56oveq12d 7449 . . . . . . . . . . 11 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → (𝑢𝐼𝑣) = (𝑎𝐼𝑏))
5854, 57eleq12d 2833 . . . . . . . . . 10 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → (𝑤 ∈ (𝑢𝐼𝑣) ↔ 𝑡 ∈ (𝑎𝐼𝑏)))
5958cbvrexdva 3238 . . . . . . . . 9 ((𝑢 = 𝑎𝑣 = 𝑏) → (∃𝑤 ∈ (𝑑𝐿𝐸)𝑤 ∈ (𝑢𝐼𝑣) ↔ ∃𝑡 ∈ (𝑑𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏)))
6053, 59anbi12d 632 . . . . . . . 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 5221 . . . . . . 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 776 . . . . . . 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 779 . . . . . . 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 781 . . . . . . 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, 12, 28, 15, 42tgelrnln 28653 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝑑𝐿𝐸) ∈ ran 𝐿)
6665ad3antrrr 730 . . . . . . . 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)
671, 2, 18, 12, 28, 15, 42tglinerflx2 28657 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐸 ∈ (𝑑𝐿𝐸))
6867ad3antrrr 730 . . . . . . . . 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))
6937ad3antrrr 730 . . . . . . . . . . . 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)
70 acopyeu.1 . . . . . . . . . . . . . . . 16 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑋”⟩)
711, 18, 2, 11, 22, 25, 19, 33ncolrot2 28586 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
721, 2, 17, 11, 19, 22, 25, 36, 14, 4, 70, 18, 71cgrancol 28852 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (𝑋 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
731, 18, 2, 11, 36, 14, 4, 72ncolcom 28584 . . . . . . . . . . . . . 14 (𝜑 → ¬ (𝑋 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
7473ad5antr 734 . . . . . . . . . . . . 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))
75 simprlr 780 . . . . . . . . . . . . . 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)
761, 2, 3, 62, 6, 16, 13, 18, 75hlln 28630 . . . . . . . . . . . . 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))
771, 2, 3, 62, 6, 16, 13, 75hlne1 28628 . . . . . . . . . . . . 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)
781, 2, 18, 13, 6, 16, 69, 62, 74, 76, 77ncolncol 28669 . . . . . . . . . . . 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))
791, 18, 2, 13, 16, 69, 62, 78ncolcom 28584 . . . . . . . . . . 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))
80 pm2.45 881 . . . . . . . . . . 11 (¬ (𝑥 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸) → ¬ 𝑥 ∈ (𝐷𝐿𝐸))
8179, 80syl 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))
821, 2, 18, 11, 36, 14, 30, 38ncolne1 28648 . . . . . . . . . . . . . 14 (𝜑𝐷𝐸)
831, 2, 18, 11, 36, 14, 82tgelrnln 28653 . . . . . . . . . . . . 13 (𝜑 → (𝐷𝐿𝐸) ∈ ran 𝐿)
8483ad2antrr 726 . . . . . . . . . . . 12 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐷𝐿𝐸) ∈ ran 𝐿)
851, 2, 18, 11, 36, 14, 82tglinerflx2 28657 . . . . . . . . . . . . 13 (𝜑𝐸 ∈ (𝐷𝐿𝐸))
8685ad2antrr 726 . . . . . . . . . . . 12 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐸 ∈ (𝐷𝐿𝐸))
871, 2, 18, 12, 28, 15, 42, 42, 84, 41, 86tglinethru 28659 . . . . . . . . . . 11 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐷𝐿𝐸) = (𝑑𝐿𝐸))
8887ad3antrrr 730 . . . . . . . . . 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))) → (𝐷𝐿𝐸) = (𝑑𝐿𝐸))
8981, 88neleqtrd 2861 . . . . . . . . 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))
901, 2, 18, 13, 66, 16, 61, 3, 68, 62, 6, 89, 75hphl 28794 . . . . . . . 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)
9187fveq2d 6911 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ((hpG‘𝐺)‘(𝐷𝐿𝐸)) = ((hpG‘𝐺)‘(𝑑𝐿𝐸)))
9291ad3antrrr 730 . . . . . . . . 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))) → ((hpG‘𝐺)‘(𝐷𝐿𝐸)) = ((hpG‘𝐺)‘(𝑑𝐿𝐸)))
93 acopyeu.3 . . . . . . . . . 10 (𝜑𝑋((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
9493ad5antr 734 . . . . . . . . 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)
9592, 94breqdi 5163 . . . . . . . 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)
961, 2, 18, 13, 66, 62, 61, 6, 90, 32, 95hpgtr 28791 . . . . . . 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)
97 acopyeu.2 . . . . . . . . . . . . . . . 16 (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑌”⟩)
981, 2, 17, 11, 19, 22, 25, 36, 14, 8, 97, 18, 71cgrancol 28852 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (𝑌 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
991, 18, 2, 11, 36, 14, 8, 98ncolcom 28584 . . . . . . . . . . . . . 14 (𝜑 → ¬ (𝑌 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
10099ad5antr 734 . . . . . . . . . . . . 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))
101 simprrr 782 . . . . . . . . . . . . . 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)
1021, 2, 3, 7, 10, 16, 13, 18, 101hlln 28630 . . . . . . . . . . . . 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))
1031, 2, 3, 7, 10, 16, 13, 101hlne1 28628 . . . . . . . . . . . . 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)
1041, 2, 18, 13, 10, 16, 69, 7, 100, 102, 103ncolncol 28669 . . . . . . . . . . . 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))
1051, 18, 2, 13, 16, 69, 7, 104ncolcom 28584 . . . . . . . . . . 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))
106 pm2.45 881 . . . . . . . . . . 11 (¬ (𝑦 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸) → ¬ 𝑦 ∈ (𝐷𝐿𝐸))
107105, 106syl 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))
108107, 88neleqtrd 2861 . . . . . . . . 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))
1091, 2, 18, 13, 66, 16, 61, 3, 68, 7, 10, 108, 101hphl 28794 . . . . . . . 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)
110 acopyeu.4 . . . . . . . . . 10 (𝜑𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
111110ad5antr 734 . . . . . . . . 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)
11292, 111breqdi 5163 . . . . . . . 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)
1131, 2, 18, 13, 66, 7, 61, 10, 109, 32, 112hpgtr 28791 . . . . . . 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)
1141, 17, 2, 18, 3, 13, 21, 24, 27, 29, 16, 32, 35, 44, 48, 61, 62, 7, 63, 64, 96, 113trgcopyeulem 28828 . . . . . 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)
115114, 75eqbrtrrd 5172 . . . . 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)
1161, 2, 3, 7, 6, 16, 13, 115hlcomd 28627 . . . 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)
1171, 2, 3, 6, 7, 10, 13, 16, 116, 101hltr 28633 . . 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)
11870ad2antrr 726 . . . . . 6 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑋”⟩)
1191, 2, 3, 12, 20, 23, 26, 37, 15, 5, 118, 28, 40cgrahl1 28839 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑋”⟩)
1201, 2, 18, 11, 19, 22, 25, 33ncolne1 28648 . . . . . . 7 (𝜑𝐴𝐵)
121120ad2antrr 726 . . . . . 6 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐴𝐵)
1221, 2, 3, 12, 20, 23, 26, 28, 15, 5, 17, 121, 47iscgra1 28833 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑋”⟩ ↔ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋)))
123119, 122mpbid 232 . . . 4 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋))
12497ad2antrr 726 . . . . . 6 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑌”⟩)
1251, 2, 3, 12, 20, 23, 26, 37, 15, 9, 124, 28, 40cgrahl1 28839 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑌”⟩)
1261, 2, 3, 12, 20, 23, 26, 28, 15, 9, 17, 121, 47iscgra1 28833 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑌”⟩ ↔ ∃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌)))
127125, 126mpbid 232 . . . 4 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ∃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))
128 reeanv 3227 . . . 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)))
129123, 127, 128sylanbrc 583 . . 3 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ∃𝑥𝑃𝑦𝑃 ((⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋) ∧ (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌)))
130117, 129r19.29vva 3214 . 2 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑋(𝐾𝐸)𝑌)
131120necomd 2994 . . 3 (𝜑𝐵𝐴)
1321, 2, 3, 14, 22, 19, 11, 36, 17, 82, 131hlcgrex 28639 . 2 (𝜑 → ∃𝑑𝑃 (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴)))
133130, 132r19.29a 3160 1 (𝜑𝑋(𝐾𝐸)𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847   = wceq 1537  wcel 2106  wne 2938  wrex 3068  cdif 3960   class class class wbr 5148  {copab 5210  ran crn 5690  cfv 6563  (class class class)co 7431  ⟨“cs3 14878  Basecbs 17245  distcds 17307  TarskiGcstrkg 28450  Itvcitv 28456  LineGclng 28457  cgrGccgrg 28533  hlGchlg 28623  hpGchpg 28780  cgrAccgra 28830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-concat 14606  df-s1 14631  df-s2 14884  df-s3 14885  df-trkgc 28471  df-trkgb 28472  df-trkgcb 28473  df-trkgld 28475  df-trkg 28476  df-cgrg 28534  df-leg 28606  df-hlg 28624  df-mir 28676  df-rag 28717  df-perpg 28719  df-hpg 28781  df-mid 28797  df-lmi 28798  df-cgra 28831
This theorem is referenced by:  tgasa1  28881
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