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Theorem hlpasch 27117
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 2738 . . . 4 (LineG‘𝐺) = (LineG‘𝐺)
4 hlpasch.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
54adantr 481 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐺 ∈ TarskiG)
6 hlpasch.5 . . . . 5 (𝜑𝐷𝑃)
76adantr 481 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐷𝑃)
8 hlpasch.4 . . . . 5 (𝜑𝑋𝑃)
98adantr 481 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝑋𝑃)
10 hlpasch.3 . . . . 5 (𝜑𝐶𝑃)
1110adantr 481 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶𝑃)
12 hlpasch.2 . . . . 5 (𝜑𝐵𝑃)
1312adantr 481 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐵𝑃)
14 hlpasch.1 . . . . 5 (𝜑𝐴𝑃)
1514adantr 481 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐴𝑃)
16 eqid 2738 . . . . 5 (dist‘𝐺) = (dist‘𝐺)
17 simpr 485 . . . . 5 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐵𝐼𝐷))
181, 16, 2, 5, 13, 11, 7, 17tgbtwncom 26849 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐷𝐼𝐵))
19 hlpasch.8 . . . . 5 (𝜑𝐴 ∈ (𝑋𝐼𝐶))
2019adantr 481 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐴 ∈ (𝑋𝐼𝐶))
211, 2, 3, 5, 7, 9, 11, 13, 15, 18, 20outpasch 27116 . . 3 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → ∃𝑒𝑃 (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒)))
22 hlpasch.k . . . . . . 7 𝐾 = (hlG‘𝐺)
23 simplr 766 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒𝑃)
2413ad2antrr 723 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐵𝑃)
2515ad2antrr 723 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴𝑃)
265ad2antrr 723 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐺 ∈ TarskiG)
27 simprr 770 . . . . . . . 8 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴 ∈ (𝐵𝐼𝑒))
281, 16, 2, 26, 24, 25, 23, 27tgbtwncom 26849 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴 ∈ (𝑒𝐼𝐵))
2926adantr 481 . . . . . . . . . . 11 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐺 ∈ TarskiG)
3024adantr 481 . . . . . . . . . . 11 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐵𝑃)
3125adantr 481 . . . . . . . . . . 11 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴𝑃)
32 simplrr 775 . . . . . . . . . . . 12 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝑒))
33 simpr 485 . . . . . . . . . . . . 13 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝑒 = 𝐵)
3433oveq2d 7291 . . . . . . . . . . . 12 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → (𝐵𝐼𝑒) = (𝐵𝐼𝐵))
3532, 34eleqtrd 2841 . . . . . . . . . . 11 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
361, 16, 2, 29, 30, 31, 35axtgbtwnid 26827 . . . . . . . . . 10 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐵 = 𝐴)
3736eqcomd 2744 . . . . . . . . 9 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 = 𝐵)
38 hlpasch.6 . . . . . . . . . . . 12 (𝜑𝐴𝐵)
3938ad3antrrr 727 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴𝐵)
4039adantr 481 . . . . . . . . . 10 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴𝐵)
4140neneqd 2948 . . . . . . . . 9 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → ¬ 𝐴 = 𝐵)
4237, 41pm2.65da 814 . . . . . . . 8 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → ¬ 𝑒 = 𝐵)
4342neqned 2950 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒𝐵)
441, 2, 22, 23, 24, 25, 26, 25, 28, 43, 39btwnhl2 26974 . . . . . 6 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴(𝐾𝐵)𝑒)
457ad2antrr 723 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐷𝑃)
469ad2antrr 723 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑋𝑃)
47 simprl 768 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒 ∈ (𝐷𝐼𝑋))
481, 16, 2, 26, 45, 23, 46, 47tgbtwncom 26849 . . . . . 6 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒 ∈ (𝑋𝐼𝐷))
4944, 48jca 512 . . . . 5 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
5049ex 413 . . . 4 (((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) → ((𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒)) → (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷))))
5150reximdva 3203 . . 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 723 . . . . . 6 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐷𝑃)
5453adantr 481 . . . . 5 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷𝑃)
55 simpr 485 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) ∧ 𝑒 = 𝐷) → 𝑒 = 𝐷)
5655breq2d 5086 . . . . . 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 631 . . . . 5 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) ∧ 𝑒 = 𝐷) → ((𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)) ↔ (𝐴(𝐾𝐵)𝐷𝐷 ∈ (𝑋𝐼𝐷))))
5914ad2antrr 723 . . . . . . . 8 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐴𝑃)
6059adantr 481 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴𝑃)
6112ad2antrr 723 . . . . . . . 8 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐵𝑃)
6261adantr 481 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐵𝑃)
634ad2antrr 723 . . . . . . . 8 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐺 ∈ TarskiG)
6463adantr 481 . . . . . . 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 26965 . . . . . . . . 9 (𝜑𝐷(𝐾𝐵)𝐶)
6766ad3antrrr 727 . . . . . . . 8 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷(𝐾𝐵)𝐶)
6810adantr 481 . . . . . . . . . 10 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐶𝑃)
6968ad2antrr 723 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐶𝑃)
7019adantr 481 . . . . . . . . . . 11 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐴 ∈ (𝑋𝐼𝐶))
7170ad2antrr 723 . . . . . . . . . 10 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴 ∈ (𝑋𝐼𝐶))
72 simpr 485 . . . . . . . . . . 11 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝑋 = 𝐵)
7372oveq1d 7290 . . . . . . . . . 10 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝑋𝐼𝐶) = (𝐵𝐼𝐶))
7471, 73eleqtrd 2841 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐶))
751, 2, 22, 10, 6, 12, 4ishlg 26963 . . . . . . . . . . . 12 (𝜑 → (𝐶(𝐾𝐵)𝐷 ↔ (𝐶𝐵𝐷𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶)))))
7665, 75mpbid 231 . . . . . . . . . . 11 (𝜑 → (𝐶𝐵𝐷𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶))))
7776simp1d 1141 . . . . . . . . . 10 (𝜑𝐶𝐵)
7877ad3antrrr 727 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐶𝐵)
7938ad2antrr 723 . . . . . . . . . 10 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐴𝐵)
8079adantr 481 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴𝐵)
811, 2, 22, 54, 69, 62, 64, 60, 74, 78, 80hlbtwn 26972 . . . . . . . 8 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝐷(𝐾𝐵)𝐶𝐷(𝐾𝐵)𝐴))
8267, 81mpbid 231 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷(𝐾𝐵)𝐴)
831, 2, 22, 54, 60, 62, 64, 82hlcomd 26965 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴(𝐾𝐵)𝐷)
848ad2antrr 723 . . . . . . . 8 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝑋𝑃)
8584adantr 481 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝑋𝑃)
861, 16, 2, 64, 85, 54tgbtwntriv2 26848 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷 ∈ (𝑋𝐼𝐷))
8783, 86jca 512 . . . . 5 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝐴(𝐾𝐵)𝐷𝐷 ∈ (𝑋𝐼𝐷)))
8854, 58, 87rspcedvd 3563 . . . 4 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
8984ad2antrr 723 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) → 𝑋𝑃)
90 simpr 485 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋)
9190breq2d 5086 . . . . . . . 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 631 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 = 𝑋) → ((𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)) ↔ (𝐴(𝐾𝐵)𝑋𝑋 ∈ (𝑋𝐼𝐷))))
9493ad4ant14 749 . . . . . 6 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) ∧ 𝑒 = 𝑋) → ((𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)) ↔ (𝐴(𝐾𝐵)𝑋𝑋 ∈ (𝑋𝐼𝐷))))
95 simpr 485 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) → 𝐴(𝐾𝐵)𝑋)
961, 16, 2, 63, 84, 53tgbtwntriv1 26852 . . . . . . . 8 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝑋 ∈ (𝑋𝐼𝐷))
9796ad2antrr 723 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) → 𝑋 ∈ (𝑋𝐼𝐷))
9895, 97jca 512 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) → (𝐴(𝐾𝐵)𝑋𝑋 ∈ (𝑋𝐼𝐷)))
9989, 94, 98rspcedvd 3563 . . . . 5 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
10053ad2antrr 723 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷𝑃)
101 simpr 485 . . . . . . . 8 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) ∧ 𝑒 = 𝐷) → 𝑒 = 𝐷)
102101breq2d 5086 . . . . . . 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 631 . . . . . 6 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) ∧ 𝑒 = 𝐷) → ((𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)) ↔ (𝐴(𝐾𝐵)𝐷𝐷 ∈ (𝑋𝐼𝐷))))
10579ad2antrr 723 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴𝐵)
1061, 2, 22, 10, 6, 12, 4, 65hlne2 26967 . . . . . . . . 9 (𝜑𝐷𝐵)
107106ad4antr 729 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷𝐵)
10863ad2antrr 723 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐺 ∈ TarskiG)
10961ad2antrr 723 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐵𝑃)
11059ad2antrr 723 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴𝑃)
11168ad2antrr 723 . . . . . . . . . 10 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐶𝑃)
112111adantr 481 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐶𝑃)
11384ad2antrr 723 . . . . . . . . . 10 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝑋𝑃)
114 simpr 485 . . . . . . . . . 10 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐵 ∈ (𝑋𝐼𝐴))
11570ad2antrr 723 . . . . . . . . . . 11 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐴 ∈ (𝑋𝐼𝐶))
116115adantr 481 . . . . . . . . . 10 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴 ∈ (𝑋𝐼𝐶))
1171, 16, 2, 108, 113, 109, 110, 112, 114, 116tgbtwnexch3 26855 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴 ∈ (𝐵𝐼𝐶))
118 simp-4r 781 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷 ∈ (𝐵𝐼𝐶))
1191, 2, 108, 109, 110, 100, 112, 117, 118tgbtwnconn3 26938 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → (𝐴 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐴)))
1201, 2, 22, 14, 6, 12, 4ishlg 26963 . . . . . . . . 9 (𝜑 → (𝐴(𝐾𝐵)𝐷 ↔ (𝐴𝐵𝐷𝐵 ∧ (𝐴 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐴)))))
121120ad4antr 729 . . . . . . . 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 26848 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷 ∈ (𝑋𝐼𝐷))
124122, 123jca 512 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → (𝐴(𝐾𝐵)𝐷𝐷 ∈ (𝑋𝐼𝐷)))
125100, 104, 124rspcedvd 3563 . . . . 5 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
1268ad3antrrr 727 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝑋𝑃)
12712ad3antrrr 727 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐵𝑃)
12814ad3antrrr 727 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐴𝑃)
1294ad3antrrr 727 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐺 ∈ TarskiG)
130 simpr 485 . . . . . . . 8 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝑋𝐵)
131130neneqd 2948 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → ¬ 𝑋 = 𝐵)
13263adantr 481 . . . . . . . . . 10 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐺 ∈ TarskiG)
133132adantr 481 . . . . . . . . . . . . . 14 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝐺 ∈ TarskiG)
134126adantr 481 . . . . . . . . . . . . . 14 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝑋𝑃)
135128adantr 481 . . . . . . . . . . . . . 14 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝐴𝑃)
136115adantr 481 . . . . . . . . . . . . . . 15 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝐴 ∈ (𝑋𝐼𝐶))
137 simpr 485 . . . . . . . . . . . . . . . 16 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝑋 = 𝐶)
138137oveq2d 7291 . . . . . . . . . . . . . . 15 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → (𝑋𝐼𝑋) = (𝑋𝐼𝐶))
139136, 138eleqtrrd 2842 . . . . . . . . . . . . . 14 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝐴 ∈ (𝑋𝐼𝑋))
1401, 16, 2, 133, 134, 135, 139axtgbtwnid 26827 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝑋 = 𝐴)
141140olcd 871 . . . . . . . . . . . 12 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → (𝐵 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
142132adantr 481 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝐺 ∈ TarskiG)
143127adantr 481 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝐵𝑃)
144111adantr 481 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝐶𝑃)
145126adantr 481 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝑋𝑃)
146128adantr 481 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝐴𝑃)
147 simpr 485 . . . . . . . . . . . . . . . 16 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝑋𝐶)
148147necomd 2999 . . . . . . . . . . . . . . 15 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝐶𝑋)
149148neneqd 2948 . . . . . . . . . . . . . 14 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → ¬ 𝐶 = 𝑋)
15053adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐷𝑃)
151106ad3antrrr 727 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐷𝐵)
152 simplr 766 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷))
1531, 2, 3, 132, 150, 127, 126, 151, 152lncom 26983 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝑋 ∈ (𝐷(LineG‘𝐺)𝐵))
15477necomd 2999 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐵𝐶)
155154ad3antrrr 727 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐵𝐶)
15666ad3antrrr 727 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐷(𝐾𝐵)𝐶)
1571, 2, 22, 150, 111, 127, 132, 3, 156hlln 26968 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐷 ∈ (𝐶(LineG‘𝐺)𝐵))
1581, 2, 3, 132, 127, 111, 150, 155, 157lncom 26983 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐷 ∈ (𝐵(LineG‘𝐺)𝐶))
159158orcd 870 . . . . . . . . . . . . . . . . . . 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 27008 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝑋 ∈ (𝐵(LineG‘𝐺)𝐶) ∨ 𝐵 = 𝐶))
1611, 3, 2, 132, 127, 111, 126, 160colrot1 26920 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐵 ∈ (𝐶(LineG‘𝐺)𝑋) ∨ 𝐶 = 𝑋))
162161orcomd 868 . . . . . . . . . . . . . . . 16 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐶 = 𝑋𝐵 ∈ (𝐶(LineG‘𝐺)𝑋)))
163162adantr 481 . . . . . . . . . . . . . . 15 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → (𝐶 = 𝑋𝐵 ∈ (𝐶(LineG‘𝐺)𝑋)))
164163ord 861 . . . . . . . . . . . . . 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 26918 . . . . . . . . . . . . . 14 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐶 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
167166adantr 481 . . . . . . . . . . . . 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 27008 . . . . . . . . . . . 12 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → (𝐵 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
169141, 168pm2.61dane 3032 . . . . . . . . . . 11 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐵 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
1701, 3, 2, 132, 126, 128, 127, 169colrot2 26921 . . . . . . . . . 10 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐴 ∈ (𝐵(LineG‘𝐺)𝑋) ∨ 𝐵 = 𝑋))
1711, 3, 2, 132, 127, 126, 128, 170colcom 26919 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐴 ∈ (𝑋(LineG‘𝐺)𝐵) ∨ 𝑋 = 𝐵))
172171orcomd 868 . . . . . . . 8 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝑋 = 𝐵𝐴 ∈ (𝑋(LineG‘𝐺)𝐵)))
173172ord 861 . . . . . . 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 26976 . . . . 5 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐴(𝐾𝐵)𝑋𝐵 ∈ (𝑋𝐼𝐴)))
17699, 125, 175mpjaodan 956 . . . 4 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
17788, 176pm2.61dane 3032 . . 3 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
1784adantr 481 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐺 ∈ TarskiG)
1798adantr 481 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝑋𝑃)
18012adantr 481 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐵𝑃)
18114adantr 481 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐴𝑃)
1826adantr 481 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷𝑃)
183 simpr 485 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷 ∈ (𝐵𝐼𝐶))
1841, 16, 2, 178, 179, 180, 68, 181, 182, 70, 183axtgpasch 26828 . . . . 5 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → ∃𝑒𝑃 (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)))
185184adantr 481 . . . 4 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → ∃𝑒𝑃 (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)))
186 simplr 766 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒𝑃)
187181ad3antrrr 727 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐴𝑃)
188180ad3antrrr 727 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐵𝑃)
189178ad3antrrr 727 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐺 ∈ TarskiG)
190 simprl 768 . . . . . . . . . 10 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ (𝐴𝐼𝐵))
1911, 16, 2, 189, 187, 186, 188, 190tgbtwncom 26849 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ (𝐵𝐼𝐴))
19238necomd 2999 . . . . . . . . . 10 (𝜑𝐵𝐴)
193192ad4antr 729 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐵𝐴)
194189adantr 481 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐺 ∈ TarskiG)
1956ad5antr 731 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷𝑃)
1968ad5antr 731 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑋𝑃)
197188adantr 481 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵𝑃)
198 simp-4r 781 . . . . . . . . . . . . . 14 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷))
199106necomd 2999 . . . . . . . . . . . . . . . 16 (𝜑𝐵𝐷)
200199ad5antr 731 . . . . . . . . . . . . . . 15 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵𝐷)
201200neneqd 2948 . . . . . . . . . . . . . 14 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → ¬ 𝐵 = 𝐷)
202 ioran 981 . . . . . . . . . . . . . 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 26924 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → ¬ (𝐷 ∈ (𝑋(LineG‘𝐺)𝐵) ∨ 𝑋 = 𝐵))
205 simpr 485 . . . . . . . . . . . . 13 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑒 = 𝐵)
206186adantr 481 . . . . . . . . . . . . . 14 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑒𝑃)
2071, 2, 3, 194, 195, 196, 197, 204ncolne1 26986 . . . . . . . . . . . . . 14 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷𝑋)
208 simplrr 775 . . . . . . . . . . . . . 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 26980 . . . . . . . . . . . . 13 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑒 ∈ (𝐷(LineG‘𝐺)𝑋))
210205, 209eqeltrrd 2840 . . . . . . . . . . . 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 26994 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷 ∈ (𝐷(LineG‘𝐺)𝑋))
212106ad5antr 731 . . . . . . . . . . . . . 14 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷𝐵)
213212necomd 2999 . . . . . . . . . . . . 13 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵𝐷)
2141, 2, 3, 194, 197, 195, 213tglinerflx1 26994 . . . . . . . . . . . 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 26995 . . . . . . . . . . . 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 27006 . . . . . . . . . . 11 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵 = 𝐷)
217216, 201pm2.65da 814 . . . . . . . . . 10 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → ¬ 𝑒 = 𝐵)
218217neqned 2950 . . . . . . . . 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 26973 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒(𝐾𝐵)𝐴)
2201, 2, 22, 186, 187, 188, 189, 219hlcomd 26965 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐴(𝐾𝐵)𝑒)
221178ad3antrrr 727 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝐺 ∈ TarskiG)
222182ad3antrrr 727 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝐷𝑃)
223 simplr 766 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑒𝑃)
224179ad3antrrr 727 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑋𝑃)
225 simpr 485 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑒 ∈ (𝐷𝐼𝑋))
2261, 16, 2, 221, 222, 223, 224, 225tgbtwncom 26849 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑒 ∈ (𝑋𝐼𝐷))
227226adantrl 713 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ (𝑋𝐼𝐷))
228220, 227jca 512 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
229228ex 413 . . . . 5 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) → ((𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷))))
230229reximdva 3203 . . . 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 810 . 2 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
23376simp3d 1143 . 2 (𝜑 → (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶)))
23452, 232, 233mpjaodan 956 1 (𝜑 → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844  w3a 1086   = wceq 1539  wcel 2106  wne 2943  wrex 3065   class class class wbr 5074  cfv 6433  (class class class)co 7275  Basecbs 16912  distcds 16971  TarskiGcstrkg 26788  Itvcitv 26794  LineGclng 26795  hlGchlg 26961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-hash 14045  df-word 14218  df-concat 14274  df-s1 14301  df-s2 14561  df-s3 14562  df-trkgc 26809  df-trkgb 26810  df-trkgcb 26811  df-trkgld 26813  df-trkg 26814  df-cgrg 26872  df-leg 26944  df-hlg 26962  df-mir 27014  df-rag 27055  df-perpg 27057
This theorem is referenced by:  inaghl  27206
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