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Theorem acopyeu 25443
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 757 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑋𝑃)
65ad3antrrr 761 . . . 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 787 . . . 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 757 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑌𝑃)
109ad3antrrr 761 . . . 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 757 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐺 ∈ TarskiG)
1312ad3antrrr 761 . . . 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 757 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐸𝑃)
1615ad3antrrr 761 . . . 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 757 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐴𝑃)
2120ad3antrrr 761 . . . . . . 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 757 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐵𝑃)
2423ad3antrrr 761 . . . . . . 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 757 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐶𝑃)
2726ad3antrrr 761 . . . . . . 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 787 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑𝑃)
2928ad3antrrr 761 . . . . . . 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 757 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐹𝑃)
3231ad3antrrr 761 . . . . . . 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 757 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
3534ad3antrrr 761 . . . . . . 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 757 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐷𝑃)
38 acopy.2 . . . . . . . . . 10 (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
3938ad2antrr 757 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
40 simprl 789 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑(𝐾𝐸)𝐷)
411, 2, 3, 28, 37, 15, 12, 18, 40hlln 25220 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑 ∈ (𝐷𝐿𝐸))
421, 2, 3, 28, 37, 15, 12, 40hlne1 25218 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑𝐸)
431, 2, 18, 12, 37, 15, 31, 28, 39, 41, 42ncolncol 25259 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ¬ (𝑑 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
4443ad3antrrr 761 . . . . . . 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 791 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐸 𝑑) = (𝐵 𝐴))
461, 17, 2, 12, 15, 28, 23, 20, 45tgcgrcomlr 25092 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝑑 𝐸) = (𝐴 𝐵))
4746eqcomd 2615 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐴 𝐵) = (𝑑 𝐸))
4847ad3antrrr 761 . . . . . . 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 471 . . . . . . . . . . 11 ((𝑢 = 𝑎𝑣 = 𝑏) → 𝑢 = 𝑎)
5049eleq1d 2671 . . . . . . . . . 10 ((𝑢 = 𝑎𝑣 = 𝑏) → (𝑢 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ↔ 𝑎 ∈ (𝑃 ∖ (𝑑𝐿𝐸))))
51 simpr 475 . . . . . . . . . . 11 ((𝑢 = 𝑎𝑣 = 𝑏) → 𝑣 = 𝑏)
5251eleq1d 2671 . . . . . . . . . 10 ((𝑢 = 𝑎𝑣 = 𝑏) → (𝑣 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ↔ 𝑏 ∈ (𝑃 ∖ (𝑑𝐿𝐸))))
5350, 52anbi12d 742 . . . . . . . . 9 ((𝑢 = 𝑎𝑣 = 𝑏) → ((𝑢 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ∧ 𝑣 ∈ (𝑃 ∖ (𝑑𝐿𝐸))) ↔ (𝑎 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑑𝐿𝐸)))))
54 simpr 475 . . . . . . . . . . 11 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑤 = 𝑡)
55 simpll 785 . . . . . . . . . . . 12 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑢 = 𝑎)
56 simplr 787 . . . . . . . . . . . 12 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑣 = 𝑏)
5755, 56oveq12d 6545 . . . . . . . . . . 11 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → (𝑢𝐼𝑣) = (𝑎𝐼𝑏))
5854, 57eleq12d 2681 . . . . . . . . . 10 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → (𝑤 ∈ (𝑢𝐼𝑣) ↔ 𝑡 ∈ (𝑎𝐼𝑏)))
5958cbvrexdva 3153 . . . . . . . . 9 ((𝑢 = 𝑎𝑣 = 𝑏) → (∃𝑤 ∈ (𝑑𝐿𝐸)𝑤 ∈ (𝑢𝐼𝑣) ↔ ∃𝑡 ∈ (𝑑𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏)))
6053, 59anbi12d 742 . . . . . . . 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 4648 . . . . . . 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 794 . . . . . . 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 797 . . . . . . 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 799 . . . . . . 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 25238 . . . . . . . . . . . . 13 (𝜑𝐷𝐸)
661, 2, 18, 11, 36, 14, 65tgelrnln 25243 . . . . . . . . . . . 12 (𝜑 → (𝐷𝐿𝐸) ∈ ran 𝐿)
6766ad2antrr 757 . . . . . . . . . . 11 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐷𝐿𝐸) ∈ ran 𝐿)
681, 2, 18, 11, 36, 14, 65tglinerflx2 25247 . . . . . . . . . . . 12 (𝜑𝐸 ∈ (𝐷𝐿𝐸))
6968ad2antrr 757 . . . . . . . . . . 11 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐸 ∈ (𝐷𝐿𝐸))
701, 2, 18, 12, 28, 15, 42, 42, 67, 41, 69tglinethru 25249 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐷𝐿𝐸) = (𝑑𝐿𝐸))
7170, 67eqeltrrd 2688 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝑑𝐿𝐸) ∈ ran 𝐿)
7271ad3antrrr 761 . . . . . . . 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 2618 . . . . . . . 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 2689 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐸 ∈ (𝑑𝐿𝐸))
7574ad3antrrr 761 . . . . . . . . 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 761 . . . . . . . . . . . 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 25176 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
791, 2, 17, 11, 19, 22, 25, 36, 14, 4, 77, 18, 78cgrancol 25438 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (𝑋 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
801, 18, 2, 11, 36, 14, 4, 79ncolcom 25174 . . . . . . . . . . . . . 14 (𝜑 → ¬ (𝑋 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
8180ad5antr 765 . . . . . . . . . . . . 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 798 . . . . . . . . . . . . . 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 25220 . . . . . . . . . . . . 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 25218 . . . . . . . . . . . . 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 25259 . . . . . . . . . . . 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 25174 . . . . . . . . . . 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 pm2.45 410 . . . . . . . . . . 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 761 . . . . . . . . . . 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 2672 . . . . . . . . . 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 312 . . . . . . . . 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 25381 . . . . . . . 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 765 . . . . . . . . 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 6092 . . . . . . . . . . 11 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ((hpG‘𝐺)‘(𝐷𝐿𝐸)) = ((hpG‘𝐺)‘(𝑑𝐿𝐸)))
9695ad3antrrr 761 . . . . . . . . . 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 4588 . . . . . . . . 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 220 . . . . . . . 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 25378 . . . . . . 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 25438 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (𝑌 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
1021, 18, 2, 11, 36, 14, 8, 101ncolcom 25174 . . . . . . . . . . . . . 14 (𝜑 → ¬ (𝑌 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
103102ad5antr 765 . . . . . . . . . . . . 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 800 . . . . . . . . . . . . . 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 25220 . . . . . . . . . . . . 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 25218 . . . . . . . . . . . . 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 25259 . . . . . . . . . . . 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 25174 . . . . . . . . . . 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 pm2.45 410 . . . . . . . . . . 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 2672 . . . . . . . . . 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 312 . . . . . . . . 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 25381 . . . . . . . 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 765 . . . . . . . . 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 4588 . . . . . . . . 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 220 . . . . . . . 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 25378 . . . . . . 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 25415 . . . . . 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 4601 . . . . 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 25217 . . . 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 25223 . . 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 757 . . . . . 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 25426 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑋”⟩)
1251, 2, 18, 11, 19, 22, 25, 33ncolne1 25238 . . . . . . 7 (𝜑𝐴𝐵)
126125ad2antrr 757 . . . . . 6 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐴𝐵)
1271, 2, 3, 12, 20, 23, 26, 28, 15, 5, 17, 126, 47iscgra1 25420 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑋”⟩ ↔ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋)))
128124, 127mpbid 220 . . . 4 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋))
129100ad2antrr 757 . . . . . 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 25426 . . . . 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 25420 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑌”⟩ ↔ ∃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌)))
132130, 131mpbid 220 . . . 4 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ∃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))
133 reeanv 3085 . . . 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 694 . . 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 3061 . 2 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑋(𝐾𝐸)𝑌)
136125necomd 2836 . . 3 (𝜑𝐵𝐴)
1371, 2, 3, 14, 22, 19, 11, 36, 17, 65, 136hlcgrex 25229 . 2 (𝜑 → ∃𝑑𝑃 (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴)))
138135, 137r19.29a 3059 1 (𝜑𝑋(𝐾𝐸)𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 381  wa 382   = wceq 1474  wcel 1976  wne 2779  wrex 2896  cdif 3536   class class class wbr 4577  {copab 4636  ran crn 5029  cfv 5790  (class class class)co 6527  ⟨“cs3 13384  Basecbs 15641  distcds 15723  TarskiGcstrkg 25046  Itvcitv 25052  LineGclng 25053  cgrGccgrg 25123  hlGchlg 25213  hpGchpg 25367  cgrAccgra 25417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-hash 12935  df-word 13100  df-concat 13102  df-s1 13103  df-s2 13390  df-s3 13391  df-trkgc 25064  df-trkgb 25065  df-trkgcb 25066  df-trkgld 25068  df-trkg 25069  df-cgrg 25124  df-leg 25196  df-hlg 25214  df-mir 25266  df-rag 25307  df-perpg 25309  df-hpg 25368  df-mid 25384  df-lmi 25385  df-cgra 25418
This theorem is referenced by:  tgasa1  25457
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