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Theorem acopyeu 26727
 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 725 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑋𝑃)
65ad3antrrr 729 . . . 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 768 . . . 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 725 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑌𝑃)
109ad3antrrr 729 . . . 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 725 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐺 ∈ TarskiG)
1312ad3antrrr 729 . . . 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 725 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐸𝑃)
1615ad3antrrr 729 . . . 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 725 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐴𝑃)
2120ad3antrrr 729 . . . . . . 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 725 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐵𝑃)
2423ad3antrrr 729 . . . . . . 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 725 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐶𝑃)
2726ad3antrrr 729 . . . . . . 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 768 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑𝑃)
2928ad3antrrr 729 . . . . . . 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 725 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐹𝑃)
3231ad3antrrr 729 . . . . . . 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 725 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
3534ad3antrrr 729 . . . . . . 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 725 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐷𝑃)
38 acopy.2 . . . . . . . . . 10 (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
3938ad2antrr 725 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
40 simprl 770 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑(𝐾𝐸)𝐷)
411, 2, 3, 28, 37, 15, 12, 18, 40hlln 26500 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑 ∈ (𝐷𝐿𝐸))
421, 2, 3, 28, 37, 15, 12, 40hlne1 26498 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑𝐸)
431, 2, 18, 12, 37, 15, 31, 28, 39, 41, 42ncolncol 26539 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ¬ (𝑑 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
4443ad3antrrr 729 . . . . . . 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 772 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐸 𝑑) = (𝐵 𝐴))
461, 17, 2, 12, 15, 28, 23, 20, 45tgcgrcomlr 26373 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝑑 𝐸) = (𝐴 𝐵))
4746eqcomd 2764 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐴 𝐵) = (𝑑 𝐸))
4847ad3antrrr 729 . . . . . . 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 486 . . . . . . . . . . 11 ((𝑢 = 𝑎𝑣 = 𝑏) → 𝑢 = 𝑎)
5049eleq1d 2836 . . . . . . . . . 10 ((𝑢 = 𝑎𝑣 = 𝑏) → (𝑢 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ↔ 𝑎 ∈ (𝑃 ∖ (𝑑𝐿𝐸))))
51 simpr 488 . . . . . . . . . . 11 ((𝑢 = 𝑎𝑣 = 𝑏) → 𝑣 = 𝑏)
5251eleq1d 2836 . . . . . . . . . 10 ((𝑢 = 𝑎𝑣 = 𝑏) → (𝑣 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ↔ 𝑏 ∈ (𝑃 ∖ (𝑑𝐿𝐸))))
5350, 52anbi12d 633 . . . . . . . . 9 ((𝑢 = 𝑎𝑣 = 𝑏) → ((𝑢 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ∧ 𝑣 ∈ (𝑃 ∖ (𝑑𝐿𝐸))) ↔ (𝑎 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑑𝐿𝐸)))))
54 simpr 488 . . . . . . . . . . 11 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑤 = 𝑡)
55 simpll 766 . . . . . . . . . . . 12 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑢 = 𝑎)
56 simplr 768 . . . . . . . . . . . 12 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑣 = 𝑏)
5755, 56oveq12d 7168 . . . . . . . . . . 11 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → (𝑢𝐼𝑣) = (𝑎𝐼𝑏))
5854, 57eleq12d 2846 . . . . . . . . . 10 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → (𝑤 ∈ (𝑢𝐼𝑣) ↔ 𝑡 ∈ (𝑎𝐼𝑏)))
5958cbvrexdva 3372 . . . . . . . . 9 ((𝑢 = 𝑎𝑣 = 𝑏) → (∃𝑤 ∈ (𝑑𝐿𝐸)𝑤 ∈ (𝑢𝐼𝑣) ↔ ∃𝑡 ∈ (𝑑𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏)))
6053, 59anbi12d 633 . . . . . . . 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 5104 . . . . . . 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 775 . . . . . . 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 778 . . . . . . 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 780 . . . . . . 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 26523 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝑑𝐿𝐸) ∈ ran 𝐿)
6665ad3antrrr 729 . . . . . . . 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 26527 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐸 ∈ (𝑑𝐿𝐸))
6867ad3antrrr 729 . . . . . . . . 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 729 . . . . . . . . . . . 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 26456 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
721, 2, 17, 11, 19, 22, 25, 36, 14, 4, 70, 18, 71cgrancol 26722 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (𝑋 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
731, 18, 2, 11, 36, 14, 4, 72ncolcom 26454 . . . . . . . . . . . . . 14 (𝜑 → ¬ (𝑋 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
7473ad5antr 733 . . . . . . . . . . . . 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 779 . . . . . . . . . . . . . 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 26500 . . . . . . . . . . . . 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 26498 . . . . . . . . . . . . 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 26539 . . . . . . . . . . . 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 26454 . . . . . . . . . . 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 pm2.45 879 . . . . . . . . . . 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 26518 . . . . . . . . . . . . . 14 (𝜑𝐷𝐸)
831, 2, 18, 11, 36, 14, 82tgelrnln 26523 . . . . . . . . . . . . 13 (𝜑 → (𝐷𝐿𝐸) ∈ ran 𝐿)
8483ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐷𝐿𝐸) ∈ ran 𝐿)
851, 2, 18, 11, 36, 14, 82tglinerflx2 26527 . . . . . . . . . . . . 13 (𝜑𝐸 ∈ (𝐷𝐿𝐸))
8685ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐸 ∈ (𝐷𝐿𝐸))
871, 2, 18, 12, 28, 15, 42, 42, 84, 41, 86tglinethru 26529 . . . . . . . . . . 11 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐷𝐿𝐸) = (𝑑𝐿𝐸))
8887ad3antrrr 729 . . . . . . . . . 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 2873 . . . . . . . . 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 26664 . . . . . . . 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 6662 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ((hpG‘𝐺)‘(𝐷𝐿𝐸)) = ((hpG‘𝐺)‘(𝑑𝐿𝐸)))
9291ad3antrrr 729 . . . . . . . . 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 733 . . . . . . . . 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 5047 . . . . . . . 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 26661 . . . . . . 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 26722 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (𝑌 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
991, 18, 2, 11, 36, 14, 8, 98ncolcom 26454 . . . . . . . . . . . . . 14 (𝜑 → ¬ (𝑌 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
10099ad5antr 733 . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . 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 26500 . . . . . . . . . . . . 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 26498 . . . . . . . . . . . . 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 26539 . . . . . . . . . . . 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 26454 . . . . . . . . . . 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 pm2.45 879 . . . . . . . . . . 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 2873 . . . . . . . . 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 26664 . . . . . . . 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 733 . . . . . . . . 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 5047 . . . . . . . 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 26661 . . . . . . 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 26698 . . . . . 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 5056 . . . . 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 26497 . . . 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 26503 . . 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 725 . . . . . 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 26709 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑋”⟩)
1201, 2, 18, 11, 19, 22, 25, 33ncolne1 26518 . . . . . . 7 (𝜑𝐴𝐵)
121120ad2antrr 725 . . . . . 6 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐴𝐵)
1221, 2, 3, 12, 20, 23, 26, 28, 15, 5, 17, 121, 47iscgra1 26703 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑋”⟩ ↔ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋)))
123119, 122mpbid 235 . . . 4 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋))
12497ad2antrr 725 . . . . . 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 26709 . . . . 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 26703 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑌”⟩ ↔ ∃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌)))
127125, 126mpbid 235 . . . 4 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ∃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))
128 reeanv 3285 . . . 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 586 . . 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 3257 . 2 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑋(𝐾𝐸)𝑌)
131120necomd 3006 . . 3 (𝜑𝐵𝐴)
1321, 2, 3, 14, 22, 19, 11, 36, 17, 82, 131hlcgrex 26509 . 2 (𝜑 → ∃𝑑𝑃 (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴)))
133130, 132r19.29a 3213 1 (𝜑𝑋(𝐾𝐸)𝑌)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∃wrex 3071   ∖ cdif 3855   class class class wbr 5032  {copab 5094  ran crn 5525  ‘cfv 6335  (class class class)co 7150  ⟨“cs3 14251  Basecbs 16541  distcds 16632  TarskiGcstrkg 26323  Itvcitv 26329  LineGclng 26330  cgrGccgrg 26403  hlGchlg 26493  hpGchpg 26650  cgrAccgra 26700 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-oadd 8116  df-er 8299  df-map 8418  df-pm 8419  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-dju 9363  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-3 11738  df-n0 11935  df-xnn0 12007  df-z 12021  df-uz 12283  df-fz 12940  df-fzo 13083  df-hash 13741  df-word 13914  df-concat 13970  df-s1 13997  df-s2 14257  df-s3 14258  df-trkgc 26341  df-trkgb 26342  df-trkgcb 26343  df-trkgld 26345  df-trkg 26346  df-cgrg 26404  df-leg 26476  df-hlg 26494  df-mir 26546  df-rag 26587  df-perpg 26589  df-hpg 26651  df-mid 26667  df-lmi 26668  df-cgra 26701 This theorem is referenced by:  tgasa1  26751
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