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Theorem acopyeu 26548
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 722 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑋𝑃)
65ad3antrrr 726 . . . 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 765 . . . 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 722 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑌𝑃)
109ad3antrrr 726 . . . 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 722 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐺 ∈ TarskiG)
1312ad3antrrr 726 . . . 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 722 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐸𝑃)
1615ad3antrrr 726 . . . 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 722 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐴𝑃)
2120ad3antrrr 726 . . . . . . 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 722 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐵𝑃)
2423ad3antrrr 726 . . . . . . 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 722 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐶𝑃)
2726ad3antrrr 726 . . . . . . 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 765 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑𝑃)
2928ad3antrrr 726 . . . . . . 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 722 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐹𝑃)
3231ad3antrrr 726 . . . . . . 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 722 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶))
3534ad3antrrr 726 . . . . . . 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 722 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐷𝑃)
38 acopy.2 . . . . . . . . . 10 (𝜑 → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
3938ad2antrr 722 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ¬ (𝐷 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
40 simprl 767 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑(𝐾𝐸)𝐷)
411, 2, 3, 28, 37, 15, 12, 18, 40hlln 26321 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑 ∈ (𝐷𝐿𝐸))
421, 2, 3, 28, 37, 15, 12, 40hlne1 26319 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑑𝐸)
431, 2, 18, 12, 37, 15, 31, 28, 39, 41, 42ncolncol 26360 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ¬ (𝑑 ∈ (𝐸𝐿𝐹) ∨ 𝐸 = 𝐹))
4443ad3antrrr 726 . . . . . . 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 769 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐸 𝑑) = (𝐵 𝐴))
461, 17, 2, 12, 15, 28, 23, 20, 45tgcgrcomlr 26194 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝑑 𝐸) = (𝐴 𝐵))
4746eqcomd 2827 . . . . . . . 8 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐴 𝐵) = (𝑑 𝐸))
4847ad3antrrr 726 . . . . . . 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 483 . . . . . . . . . . 11 ((𝑢 = 𝑎𝑣 = 𝑏) → 𝑢 = 𝑎)
5049eleq1d 2897 . . . . . . . . . 10 ((𝑢 = 𝑎𝑣 = 𝑏) → (𝑢 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ↔ 𝑎 ∈ (𝑃 ∖ (𝑑𝐿𝐸))))
51 simpr 485 . . . . . . . . . . 11 ((𝑢 = 𝑎𝑣 = 𝑏) → 𝑣 = 𝑏)
5251eleq1d 2897 . . . . . . . . . 10 ((𝑢 = 𝑎𝑣 = 𝑏) → (𝑣 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ↔ 𝑏 ∈ (𝑃 ∖ (𝑑𝐿𝐸))))
5350, 52anbi12d 630 . . . . . . . . 9 ((𝑢 = 𝑎𝑣 = 𝑏) → ((𝑢 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ∧ 𝑣 ∈ (𝑃 ∖ (𝑑𝐿𝐸))) ↔ (𝑎 ∈ (𝑃 ∖ (𝑑𝐿𝐸)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑑𝐿𝐸)))))
54 simpr 485 . . . . . . . . . . 11 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑤 = 𝑡)
55 simpll 763 . . . . . . . . . . . 12 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑢 = 𝑎)
56 simplr 765 . . . . . . . . . . . 12 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → 𝑣 = 𝑏)
5755, 56oveq12d 7163 . . . . . . . . . . 11 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → (𝑢𝐼𝑣) = (𝑎𝐼𝑏))
5854, 57eleq12d 2907 . . . . . . . . . 10 (((𝑢 = 𝑎𝑣 = 𝑏) ∧ 𝑤 = 𝑡) → (𝑤 ∈ (𝑢𝐼𝑣) ↔ 𝑡 ∈ (𝑎𝐼𝑏)))
5958cbvrexdva 3461 . . . . . . . . 9 ((𝑢 = 𝑎𝑣 = 𝑏) → (∃𝑤 ∈ (𝑑𝐿𝐸)𝑤 ∈ (𝑢𝐼𝑣) ↔ ∃𝑡 ∈ (𝑑𝐿𝐸)𝑡 ∈ (𝑎𝐼𝑏)))
6053, 59anbi12d 630 . . . . . . . 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 5130 . . . . . . 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 772 . . . . . . 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 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))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩)
64 simprrl 777 . . . . . . 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 26344 . . . . . . . . 9 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝑑𝐿𝐸) ∈ ran 𝐿)
6665ad3antrrr 726 . . . . . . . 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 26348 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐸 ∈ (𝑑𝐿𝐸))
6867ad3antrrr 726 . . . . . . . . 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 726 . . . . . . . . . . . 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 26277 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
721, 2, 17, 11, 19, 22, 25, 36, 14, 4, 70, 18, 71cgrancol 26543 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (𝑋 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
731, 18, 2, 11, 36, 14, 4, 72ncolcom 26275 . . . . . . . . . . . . . 14 (𝜑 → ¬ (𝑋 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
7473ad5antr 730 . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . 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 26321 . . . . . . . . . . . . 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 26319 . . . . . . . . . . . . 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 26360 . . . . . . . . . . . 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 26275 . . . . . . . . . . 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 pm2.45 875 . . . . . . . . . . 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 26339 . . . . . . . . . . . . . 14 (𝜑𝐷𝐸)
831, 2, 18, 11, 36, 14, 82tgelrnln 26344 . . . . . . . . . . . . 13 (𝜑 → (𝐷𝐿𝐸) ∈ ran 𝐿)
8483ad2antrr 722 . . . . . . . . . . . 12 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐷𝐿𝐸) ∈ ran 𝐿)
851, 2, 18, 11, 36, 14, 82tglinerflx2 26348 . . . . . . . . . . . . 13 (𝜑𝐸 ∈ (𝐷𝐿𝐸))
8685ad2antrr 722 . . . . . . . . . . . 12 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐸 ∈ (𝐷𝐿𝐸))
871, 2, 18, 12, 28, 15, 42, 42, 84, 41, 86tglinethru 26350 . . . . . . . . . . 11 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (𝐷𝐿𝐸) = (𝑑𝐿𝐸))
8887ad3antrrr 726 . . . . . . . . . 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 2934 . . . . . . . . 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 26485 . . . . . . . 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 6668 . . . . . . . . . 10 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ((hpG‘𝐺)‘(𝐷𝐿𝐸)) = ((hpG‘𝐺)‘(𝑑𝐿𝐸)))
9291ad3antrrr 726 . . . . . . . . 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 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))) → 𝑋((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
9592, 94breqdi 5073 . . . . . . . 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 26482 . . . . . . 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 26543 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (𝑌 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
991, 18, 2, 11, 36, 14, 8, 98ncolcom 26275 . . . . . . . . . . . . . 14 (𝜑 → ¬ (𝑌 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
10099ad5antr 730 . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . 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 26321 . . . . . . . . . . . . 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 26319 . . . . . . . . . . . . 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 26360 . . . . . . . . . . . 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 26275 . . . . . . . . . . 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 pm2.45 875 . . . . . . . . . . 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 2934 . . . . . . . . 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 26485 . . . . . . . 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 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))) → 𝑌((hpG‘𝐺)‘(𝐷𝐿𝐸))𝐹)
11292, 111breqdi 5073 . . . . . . . 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 26482 . . . . . . 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 26519 . . . . . 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 5082 . . . . 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 26318 . . . 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 26324 . . 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 722 . . . . . 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 26530 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑋”⟩)
1201, 2, 18, 11, 19, 22, 25, 33ncolne1 26339 . . . . . . 7 (𝜑𝐴𝐵)
121120ad2antrr 722 . . . . . 6 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝐴𝐵)
1221, 2, 3, 12, 20, 23, 26, 28, 15, 5, 17, 121, 47iscgra1 26524 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑋”⟩ ↔ ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋)))
123119, 122mpbid 233 . . . 4 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ∃𝑥𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑥”⟩ ∧ 𝑥(𝐾𝐸)𝑋))
12497ad2antrr 722 . . . . . 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 26530 . . . . 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 26524 . . . . 5 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → (⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝑑𝐸𝑌”⟩ ↔ ∃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌)))
127125, 126mpbid 233 . . . 4 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → ∃𝑦𝑃 (⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑑𝐸𝑦”⟩ ∧ 𝑦(𝐾𝐸)𝑌))
128 reeanv 3368 . . . 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 3336 . 2 (((𝜑𝑑𝑃) ∧ (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴))) → 𝑋(𝐾𝐸)𝑌)
131120necomd 3071 . . 3 (𝜑𝐵𝐴)
1321, 2, 3, 14, 22, 19, 11, 36, 17, 82, 131hlcgrex 26330 . 2 (𝜑 → ∃𝑑𝑃 (𝑑(𝐾𝐸)𝐷 ∧ (𝐸 𝑑) = (𝐵 𝐴)))
133130, 132r19.29a 3289 1 (𝜑𝑋(𝐾𝐸)𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  wne 3016  wrex 3139  cdif 3932   class class class wbr 5058  {copab 5120  ran crn 5550  cfv 6349  (class class class)co 7145  ⟨“cs3 14194  Basecbs 16473  distcds 16564  TarskiGcstrkg 26144  Itvcitv 26150  LineGclng 26151  cgrGccgrg 26224  hlGchlg 26314  hpGchpg 26471  cgrAccgra 26521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-pm 8399  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-dju 9319  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-xnn0 11957  df-z 11971  df-uz 12233  df-fz 12883  df-fzo 13024  df-hash 13681  df-word 13852  df-concat 13913  df-s1 13940  df-s2 14200  df-s3 14201  df-trkgc 26162  df-trkgb 26163  df-trkgcb 26164  df-trkgld 26166  df-trkg 26167  df-cgrg 26225  df-leg 26297  df-hlg 26315  df-mir 26367  df-rag 26408  df-perpg 26410  df-hpg 26472  df-mid 26488  df-lmi 26489  df-cgra 26522
This theorem is referenced by:  tgasa1  26572
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