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Theorem hlpasch 28778
Description: An application of the axiom of Pasch for half-lines. (Contributed by Thierry Arnoux, 15-Sep-2020.)
Hypotheses
Ref Expression
hlpasch.p 𝑃 = (Base‘𝐺)
hlpasch.i 𝐼 = (Itv‘𝐺)
hlpasch.k 𝐾 = (hlG‘𝐺)
hlpasch.g (𝜑𝐺 ∈ TarskiG)
hlpasch.1 (𝜑𝐴𝑃)
hlpasch.2 (𝜑𝐵𝑃)
hlpasch.3 (𝜑𝐶𝑃)
hlpasch.4 (𝜑𝑋𝑃)
hlpasch.5 (𝜑𝐷𝑃)
hlpasch.6 (𝜑𝐴𝐵)
hlpasch.7 (𝜑𝐶(𝐾𝐵)𝐷)
hlpasch.8 (𝜑𝐴 ∈ (𝑋𝐼𝐶))
Assertion
Ref Expression
hlpasch (𝜑 → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
Distinct variable groups:   𝐴,𝑒   𝐵,𝑒   𝐶,𝑒   𝐷,𝑒   𝑒,𝐺   𝑒,𝐼   𝑒,𝐾   𝑃,𝑒   𝑒,𝑋   𝜑,𝑒

Proof of Theorem hlpasch
StepHypRef Expression
1 hlpasch.p . . . 4 𝑃 = (Base‘𝐺)
2 hlpasch.i . . . 4 𝐼 = (Itv‘𝐺)
3 eqid 2734 . . . 4 (LineG‘𝐺) = (LineG‘𝐺)
4 hlpasch.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
54adantr 480 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐺 ∈ TarskiG)
6 hlpasch.5 . . . . 5 (𝜑𝐷𝑃)
76adantr 480 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐷𝑃)
8 hlpasch.4 . . . . 5 (𝜑𝑋𝑃)
98adantr 480 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝑋𝑃)
10 hlpasch.3 . . . . 5 (𝜑𝐶𝑃)
1110adantr 480 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶𝑃)
12 hlpasch.2 . . . . 5 (𝜑𝐵𝑃)
1312adantr 480 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐵𝑃)
14 hlpasch.1 . . . . 5 (𝜑𝐴𝑃)
1514adantr 480 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐴𝑃)
16 eqid 2734 . . . . 5 (dist‘𝐺) = (dist‘𝐺)
17 simpr 484 . . . . 5 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐵𝐼𝐷))
181, 16, 2, 5, 13, 11, 7, 17tgbtwncom 28510 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐷𝐼𝐵))
19 hlpasch.8 . . . . 5 (𝜑𝐴 ∈ (𝑋𝐼𝐶))
2019adantr 480 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐴 ∈ (𝑋𝐼𝐶))
211, 2, 3, 5, 7, 9, 11, 13, 15, 18, 20outpasch 28777 . . 3 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → ∃𝑒𝑃 (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒)))
22 hlpasch.k . . . . . . 7 𝐾 = (hlG‘𝐺)
23 simplr 769 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒𝑃)
2413ad2antrr 726 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐵𝑃)
2515ad2antrr 726 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴𝑃)
265ad2antrr 726 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐺 ∈ TarskiG)
27 simprr 773 . . . . . . . 8 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴 ∈ (𝐵𝐼𝑒))
281, 16, 2, 26, 24, 25, 23, 27tgbtwncom 28510 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴 ∈ (𝑒𝐼𝐵))
2926adantr 480 . . . . . . . . . . 11 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐺 ∈ TarskiG)
3024adantr 480 . . . . . . . . . . 11 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐵𝑃)
3125adantr 480 . . . . . . . . . . 11 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴𝑃)
32 simplrr 778 . . . . . . . . . . . 12 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝑒))
33 simpr 484 . . . . . . . . . . . . 13 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝑒 = 𝐵)
3433oveq2d 7446 . . . . . . . . . . . 12 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → (𝐵𝐼𝑒) = (𝐵𝐼𝐵))
3532, 34eleqtrd 2840 . . . . . . . . . . 11 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
361, 16, 2, 29, 30, 31, 35axtgbtwnid 28488 . . . . . . . . . 10 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐵 = 𝐴)
3736eqcomd 2740 . . . . . . . . 9 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 = 𝐵)
38 hlpasch.6 . . . . . . . . . . . 12 (𝜑𝐴𝐵)
3938ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴𝐵)
4039adantr 480 . . . . . . . . . 10 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴𝐵)
4140neneqd 2942 . . . . . . . . 9 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → ¬ 𝐴 = 𝐵)
4237, 41pm2.65da 817 . . . . . . . 8 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → ¬ 𝑒 = 𝐵)
4342neqned 2944 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒𝐵)
441, 2, 22, 23, 24, 25, 26, 25, 28, 43, 39btwnhl2 28635 . . . . . 6 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴(𝐾𝐵)𝑒)
457ad2antrr 726 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐷𝑃)
469ad2antrr 726 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑋𝑃)
47 simprl 771 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒 ∈ (𝐷𝐼𝑋))
481, 16, 2, 26, 45, 23, 46, 47tgbtwncom 28510 . . . . . 6 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒 ∈ (𝑋𝐼𝐷))
4944, 48jca 511 . . . . 5 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
5049ex 412 . . . 4 (((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) → ((𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒)) → (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷))))
5150reximdva 3165 . . 3 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → (∃𝑒𝑃 (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒)) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷))))
5221, 51mpd 15 . 2 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
536ad2antrr 726 . . . . . 6 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐷𝑃)
5453adantr 480 . . . . 5 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷𝑃)
55 simpr 484 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) ∧ 𝑒 = 𝐷) → 𝑒 = 𝐷)
5655breq2d 5159 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) ∧ 𝑒 = 𝐷) → (𝐴(𝐾𝐵)𝑒𝐴(𝐾𝐵)𝐷))
5755eleq1d 2823 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) ∧ 𝑒 = 𝐷) → (𝑒 ∈ (𝑋𝐼𝐷) ↔ 𝐷 ∈ (𝑋𝐼𝐷)))
5856, 57anbi12d 632 . . . . 5 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) ∧ 𝑒 = 𝐷) → ((𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)) ↔ (𝐴(𝐾𝐵)𝐷𝐷 ∈ (𝑋𝐼𝐷))))
5914ad2antrr 726 . . . . . . . 8 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐴𝑃)
6059adantr 480 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴𝑃)
6112ad2antrr 726 . . . . . . . 8 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐵𝑃)
6261adantr 480 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐵𝑃)
634ad2antrr 726 . . . . . . . 8 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐺 ∈ TarskiG)
6463adantr 480 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐺 ∈ TarskiG)
65 hlpasch.7 . . . . . . . . . 10 (𝜑𝐶(𝐾𝐵)𝐷)
661, 2, 22, 10, 6, 12, 4, 65hlcomd 28626 . . . . . . . . 9 (𝜑𝐷(𝐾𝐵)𝐶)
6766ad3antrrr 730 . . . . . . . 8 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷(𝐾𝐵)𝐶)
6810adantr 480 . . . . . . . . . 10 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐶𝑃)
6968ad2antrr 726 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐶𝑃)
7019adantr 480 . . . . . . . . . . 11 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐴 ∈ (𝑋𝐼𝐶))
7170ad2antrr 726 . . . . . . . . . 10 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴 ∈ (𝑋𝐼𝐶))
72 simpr 484 . . . . . . . . . . 11 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝑋 = 𝐵)
7372oveq1d 7445 . . . . . . . . . 10 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝑋𝐼𝐶) = (𝐵𝐼𝐶))
7471, 73eleqtrd 2840 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐶))
751, 2, 22, 10, 6, 12, 4ishlg 28624 . . . . . . . . . . . 12 (𝜑 → (𝐶(𝐾𝐵)𝐷 ↔ (𝐶𝐵𝐷𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶)))))
7665, 75mpbid 232 . . . . . . . . . . 11 (𝜑 → (𝐶𝐵𝐷𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶))))
7776simp1d 1141 . . . . . . . . . 10 (𝜑𝐶𝐵)
7877ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐶𝐵)
7938ad2antrr 726 . . . . . . . . . 10 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐴𝐵)
8079adantr 480 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴𝐵)
811, 2, 22, 54, 69, 62, 64, 60, 74, 78, 80hlbtwn 28633 . . . . . . . 8 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝐷(𝐾𝐵)𝐶𝐷(𝐾𝐵)𝐴))
8267, 81mpbid 232 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷(𝐾𝐵)𝐴)
831, 2, 22, 54, 60, 62, 64, 82hlcomd 28626 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴(𝐾𝐵)𝐷)
848ad2antrr 726 . . . . . . . 8 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝑋𝑃)
8584adantr 480 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝑋𝑃)
861, 16, 2, 64, 85, 54tgbtwntriv2 28509 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷 ∈ (𝑋𝐼𝐷))
8783, 86jca 511 . . . . 5 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝐴(𝐾𝐵)𝐷𝐷 ∈ (𝑋𝐼𝐷)))
8854, 58, 87rspcedvd 3623 . . . 4 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
8984ad2antrr 726 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) → 𝑋𝑃)
90 simpr 484 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋)
9190breq2d 5159 . . . . . . . 8 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 = 𝑋) → (𝐴(𝐾𝐵)𝑒𝐴(𝐾𝐵)𝑋))
9290eleq1d 2823 . . . . . . . 8 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 = 𝑋) → (𝑒 ∈ (𝑋𝐼𝐷) ↔ 𝑋 ∈ (𝑋𝐼𝐷)))
9391, 92anbi12d 632 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 = 𝑋) → ((𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)) ↔ (𝐴(𝐾𝐵)𝑋𝑋 ∈ (𝑋𝐼𝐷))))
9493ad4ant14 752 . . . . . 6 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) ∧ 𝑒 = 𝑋) → ((𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)) ↔ (𝐴(𝐾𝐵)𝑋𝑋 ∈ (𝑋𝐼𝐷))))
95 simpr 484 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) → 𝐴(𝐾𝐵)𝑋)
961, 16, 2, 63, 84, 53tgbtwntriv1 28513 . . . . . . . 8 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝑋 ∈ (𝑋𝐼𝐷))
9796ad2antrr 726 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) → 𝑋 ∈ (𝑋𝐼𝐷))
9895, 97jca 511 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) → (𝐴(𝐾𝐵)𝑋𝑋 ∈ (𝑋𝐼𝐷)))
9989, 94, 98rspcedvd 3623 . . . . 5 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
10053ad2antrr 726 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷𝑃)
101 simpr 484 . . . . . . . 8 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) ∧ 𝑒 = 𝐷) → 𝑒 = 𝐷)
102101breq2d 5159 . . . . . . 7 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) ∧ 𝑒 = 𝐷) → (𝐴(𝐾𝐵)𝑒𝐴(𝐾𝐵)𝐷))
103101eleq1d 2823 . . . . . . 7 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) ∧ 𝑒 = 𝐷) → (𝑒 ∈ (𝑋𝐼𝐷) ↔ 𝐷 ∈ (𝑋𝐼𝐷)))
104102, 103anbi12d 632 . . . . . 6 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) ∧ 𝑒 = 𝐷) → ((𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)) ↔ (𝐴(𝐾𝐵)𝐷𝐷 ∈ (𝑋𝐼𝐷))))
10579ad2antrr 726 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴𝐵)
1061, 2, 22, 10, 6, 12, 4, 65hlne2 28628 . . . . . . . . 9 (𝜑𝐷𝐵)
107106ad4antr 732 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷𝐵)
10863ad2antrr 726 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐺 ∈ TarskiG)
10961ad2antrr 726 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐵𝑃)
11059ad2antrr 726 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴𝑃)
11168ad2antrr 726 . . . . . . . . . 10 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐶𝑃)
112111adantr 480 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐶𝑃)
11384ad2antrr 726 . . . . . . . . . 10 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝑋𝑃)
114 simpr 484 . . . . . . . . . 10 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐵 ∈ (𝑋𝐼𝐴))
11570ad2antrr 726 . . . . . . . . . . 11 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐴 ∈ (𝑋𝐼𝐶))
116115adantr 480 . . . . . . . . . 10 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴 ∈ (𝑋𝐼𝐶))
1171, 16, 2, 108, 113, 109, 110, 112, 114, 116tgbtwnexch3 28516 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴 ∈ (𝐵𝐼𝐶))
118 simp-4r 784 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷 ∈ (𝐵𝐼𝐶))
1191, 2, 108, 109, 110, 100, 112, 117, 118tgbtwnconn3 28599 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → (𝐴 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐴)))
1201, 2, 22, 14, 6, 12, 4ishlg 28624 . . . . . . . . 9 (𝜑 → (𝐴(𝐾𝐵)𝐷 ↔ (𝐴𝐵𝐷𝐵 ∧ (𝐴 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐴)))))
121120ad4antr 732 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → (𝐴(𝐾𝐵)𝐷 ↔ (𝐴𝐵𝐷𝐵 ∧ (𝐴 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐴)))))
122105, 107, 119, 121mpbir3and 1341 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴(𝐾𝐵)𝐷)
1231, 16, 2, 108, 113, 100tgbtwntriv2 28509 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷 ∈ (𝑋𝐼𝐷))
124122, 123jca 511 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → (𝐴(𝐾𝐵)𝐷𝐷 ∈ (𝑋𝐼𝐷)))
125100, 104, 124rspcedvd 3623 . . . . 5 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
1268ad3antrrr 730 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝑋𝑃)
12712ad3antrrr 730 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐵𝑃)
12814ad3antrrr 730 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐴𝑃)
1294ad3antrrr 730 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐺 ∈ TarskiG)
130 simpr 484 . . . . . . . 8 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝑋𝐵)
131130neneqd 2942 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → ¬ 𝑋 = 𝐵)
13263adantr 480 . . . . . . . . . 10 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐺 ∈ TarskiG)
133132adantr 480 . . . . . . . . . . . . . 14 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝐺 ∈ TarskiG)
134126adantr 480 . . . . . . . . . . . . . 14 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝑋𝑃)
135128adantr 480 . . . . . . . . . . . . . 14 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝐴𝑃)
136115adantr 480 . . . . . . . . . . . . . . 15 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝐴 ∈ (𝑋𝐼𝐶))
137 simpr 484 . . . . . . . . . . . . . . . 16 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝑋 = 𝐶)
138137oveq2d 7446 . . . . . . . . . . . . . . 15 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → (𝑋𝐼𝑋) = (𝑋𝐼𝐶))
139136, 138eleqtrrd 2841 . . . . . . . . . . . . . 14 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝐴 ∈ (𝑋𝐼𝑋))
1401, 16, 2, 133, 134, 135, 139axtgbtwnid 28488 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝑋 = 𝐴)
141140olcd 874 . . . . . . . . . . . 12 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → (𝐵 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
142132adantr 480 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝐺 ∈ TarskiG)
143127adantr 480 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝐵𝑃)
144111adantr 480 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝐶𝑃)
145126adantr 480 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝑋𝑃)
146128adantr 480 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝐴𝑃)
147 simpr 484 . . . . . . . . . . . . . . . 16 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝑋𝐶)
148147necomd 2993 . . . . . . . . . . . . . . 15 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝐶𝑋)
149148neneqd 2942 . . . . . . . . . . . . . 14 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → ¬ 𝐶 = 𝑋)
15053adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐷𝑃)
151106ad3antrrr 730 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐷𝐵)
152 simplr 769 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷))
1531, 2, 3, 132, 150, 127, 126, 151, 152lncom 28644 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝑋 ∈ (𝐷(LineG‘𝐺)𝐵))
15477necomd 2993 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐵𝐶)
155154ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐵𝐶)
15666ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐷(𝐾𝐵)𝐶)
1571, 2, 22, 150, 111, 127, 132, 3, 156hlln 28629 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐷 ∈ (𝐶(LineG‘𝐺)𝐵))
1581, 2, 3, 132, 127, 111, 150, 155, 157lncom 28644 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐷 ∈ (𝐵(LineG‘𝐺)𝐶))
159158orcd 873 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐷 ∈ (𝐵(LineG‘𝐺)𝐶) ∨ 𝐵 = 𝐶))
1601, 2, 3, 132, 126, 150, 127, 111, 153, 159coltr 28669 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝑋 ∈ (𝐵(LineG‘𝐺)𝐶) ∨ 𝐵 = 𝐶))
1611, 3, 2, 132, 127, 111, 126, 160colrot1 28581 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐵 ∈ (𝐶(LineG‘𝐺)𝑋) ∨ 𝐶 = 𝑋))
162161orcomd 871 . . . . . . . . . . . . . . . 16 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐶 = 𝑋𝐵 ∈ (𝐶(LineG‘𝐺)𝑋)))
163162adantr 480 . . . . . . . . . . . . . . 15 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → (𝐶 = 𝑋𝐵 ∈ (𝐶(LineG‘𝐺)𝑋)))
164163ord 864 . . . . . . . . . . . . . 14 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → (¬ 𝐶 = 𝑋𝐵 ∈ (𝐶(LineG‘𝐺)𝑋)))
165149, 164mpd 15 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝐵 ∈ (𝐶(LineG‘𝐺)𝑋))
1661, 3, 2, 132, 126, 128, 111, 115btwncolg3 28579 . . . . . . . . . . . . . 14 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐶 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
167166adantr 480 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → (𝐶 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
1681, 2, 3, 142, 143, 144, 145, 146, 165, 167coltr 28669 . . . . . . . . . . . 12 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → (𝐵 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
169141, 168pm2.61dane 3026 . . . . . . . . . . 11 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐵 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
1701, 3, 2, 132, 126, 128, 127, 169colrot2 28582 . . . . . . . . . 10 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐴 ∈ (𝐵(LineG‘𝐺)𝑋) ∨ 𝐵 = 𝑋))
1711, 3, 2, 132, 127, 126, 128, 170colcom 28580 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐴 ∈ (𝑋(LineG‘𝐺)𝐵) ∨ 𝑋 = 𝐵))
172171orcomd 871 . . . . . . . 8 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝑋 = 𝐵𝐴 ∈ (𝑋(LineG‘𝐺)𝐵)))
173172ord 864 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (¬ 𝑋 = 𝐵𝐴 ∈ (𝑋(LineG‘𝐺)𝐵)))
174131, 173mpd 15 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐴 ∈ (𝑋(LineG‘𝐺)𝐵))
1751, 2, 22, 126, 127, 128, 129, 128, 3, 174lnhl 28637 . . . . 5 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐴(𝐾𝐵)𝑋𝐵 ∈ (𝑋𝐼𝐴)))
17699, 125, 175mpjaodan 960 . . . 4 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
17788, 176pm2.61dane 3026 . . 3 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
1784adantr 480 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐺 ∈ TarskiG)
1798adantr 480 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝑋𝑃)
18012adantr 480 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐵𝑃)
18114adantr 480 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐴𝑃)
1826adantr 480 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷𝑃)
183 simpr 484 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷 ∈ (𝐵𝐼𝐶))
1841, 16, 2, 178, 179, 180, 68, 181, 182, 70, 183axtgpasch 28489 . . . . 5 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → ∃𝑒𝑃 (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)))
185184adantr 480 . . . 4 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → ∃𝑒𝑃 (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)))
186 simplr 769 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒𝑃)
187181ad3antrrr 730 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐴𝑃)
188180ad3antrrr 730 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐵𝑃)
189178ad3antrrr 730 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐺 ∈ TarskiG)
190 simprl 771 . . . . . . . . . 10 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ (𝐴𝐼𝐵))
1911, 16, 2, 189, 187, 186, 188, 190tgbtwncom 28510 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ (𝐵𝐼𝐴))
19238necomd 2993 . . . . . . . . . 10 (𝜑𝐵𝐴)
193192ad4antr 732 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐵𝐴)
194189adantr 480 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐺 ∈ TarskiG)
1956ad5antr 734 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷𝑃)
1968ad5antr 734 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑋𝑃)
197188adantr 480 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵𝑃)
198 simp-4r 784 . . . . . . . . . . . . . 14 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷))
199106necomd 2993 . . . . . . . . . . . . . . . 16 (𝜑𝐵𝐷)
200199ad5antr 734 . . . . . . . . . . . . . . 15 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵𝐷)
201200neneqd 2942 . . . . . . . . . . . . . 14 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → ¬ 𝐵 = 𝐷)
202 ioran 985 . . . . . . . . . . . . . 14 (¬ (𝑋 ∈ (𝐵(LineG‘𝐺)𝐷) ∨ 𝐵 = 𝐷) ↔ (¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷) ∧ ¬ 𝐵 = 𝐷))
203198, 201, 202sylanbrc 583 . . . . . . . . . . . . 13 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → ¬ (𝑋 ∈ (𝐵(LineG‘𝐺)𝐷) ∨ 𝐵 = 𝐷))
2041, 3, 2, 194, 197, 195, 196, 203ncolrot2 28585 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → ¬ (𝐷 ∈ (𝑋(LineG‘𝐺)𝐵) ∨ 𝑋 = 𝐵))
205 simpr 484 . . . . . . . . . . . . 13 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑒 = 𝐵)
206186adantr 480 . . . . . . . . . . . . . 14 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑒𝑃)
2071, 2, 3, 194, 195, 196, 197, 204ncolne1 28647 . . . . . . . . . . . . . 14 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷𝑋)
208 simplrr 778 . . . . . . . . . . . . . 14 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑒 ∈ (𝐷𝐼𝑋))
2091, 2, 3, 194, 195, 196, 206, 207, 208btwnlng1 28641 . . . . . . . . . . . . 13 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑒 ∈ (𝐷(LineG‘𝐺)𝑋))
210205, 209eqeltrrd 2839 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵 ∈ (𝐷(LineG‘𝐺)𝑋))
2111, 2, 3, 194, 195, 196, 207tglinerflx1 28655 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷 ∈ (𝐷(LineG‘𝐺)𝑋))
212106ad5antr 734 . . . . . . . . . . . . . 14 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷𝐵)
213212necomd 2993 . . . . . . . . . . . . 13 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵𝐷)
2141, 2, 3, 194, 197, 195, 213tglinerflx1 28655 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵 ∈ (𝐵(LineG‘𝐺)𝐷))
2151, 2, 3, 194, 197, 195, 213tglinerflx2 28656 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷 ∈ (𝐵(LineG‘𝐺)𝐷))
2161, 2, 3, 194, 195, 196, 197, 195, 204, 210, 211, 214, 215tglineinteq 28667 . . . . . . . . . . 11 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵 = 𝐷)
217216, 201pm2.65da 817 . . . . . . . . . 10 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → ¬ 𝑒 = 𝐵)
218217neqned 2944 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒𝐵)
2191, 2, 22, 188, 187, 186, 189, 187, 191, 193, 218btwnhl1 28634 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒(𝐾𝐵)𝐴)
2201, 2, 22, 186, 187, 188, 189, 219hlcomd 28626 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐴(𝐾𝐵)𝑒)
221178ad3antrrr 730 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝐺 ∈ TarskiG)
222182ad3antrrr 730 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝐷𝑃)
223 simplr 769 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑒𝑃)
224179ad3antrrr 730 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑋𝑃)
225 simpr 484 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑒 ∈ (𝐷𝐼𝑋))
2261, 16, 2, 221, 222, 223, 224, 225tgbtwncom 28510 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑒 ∈ (𝑋𝐼𝐷))
227226adantrl 716 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ (𝑋𝐼𝐷))
228220, 227jca 511 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
229228ex 412 . . . . 5 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) → ((𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷))))
230229reximdva 3165 . . . 4 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → (∃𝑒𝑃 (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷))))
231185, 230mpd 15 . . 3 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
232177, 231pm2.61dan 813 . 2 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
23376simp3d 1143 . 2 (𝜑 → (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶)))
23452, 232, 233mpjaodan 960 1 (𝜑 → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wrex 3067   class class class wbr 5147  cfv 6562  (class class class)co 7430  Basecbs 17244  distcds 17306  TarskiGcstrkg 28449  Itvcitv 28455  LineGclng 28456  hlGchlg 28622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-hash 14366  df-word 14549  df-concat 14605  df-s1 14630  df-s2 14883  df-s3 14884  df-trkgc 28470  df-trkgb 28471  df-trkgcb 28472  df-trkgld 28474  df-trkg 28475  df-cgrg 28533  df-leg 28605  df-hlg 28623  df-mir 28675  df-rag 28716  df-perpg 28718
This theorem is referenced by:  inaghl  28867
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