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Theorem hlpasch 26534
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 2819 . . . 4 (LineG‘𝐺) = (LineG‘𝐺)
4 hlpasch.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
54adantr 483 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐺 ∈ TarskiG)
6 hlpasch.5 . . . . 5 (𝜑𝐷𝑃)
76adantr 483 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐷𝑃)
8 hlpasch.4 . . . . 5 (𝜑𝑋𝑃)
98adantr 483 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝑋𝑃)
10 hlpasch.3 . . . . 5 (𝜑𝐶𝑃)
1110adantr 483 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶𝑃)
12 hlpasch.2 . . . . 5 (𝜑𝐵𝑃)
1312adantr 483 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐵𝑃)
14 hlpasch.1 . . . . 5 (𝜑𝐴𝑃)
1514adantr 483 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐴𝑃)
16 eqid 2819 . . . . 5 (dist‘𝐺) = (dist‘𝐺)
17 simpr 487 . . . . 5 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐵𝐼𝐷))
181, 16, 2, 5, 13, 11, 7, 17tgbtwncom 26266 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐷𝐼𝐵))
19 hlpasch.8 . . . . 5 (𝜑𝐴 ∈ (𝑋𝐼𝐶))
2019adantr 483 . . . 4 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → 𝐴 ∈ (𝑋𝐼𝐶))
211, 2, 3, 5, 7, 9, 11, 13, 15, 18, 20outpasch 26533 . . 3 ((𝜑𝐶 ∈ (𝐵𝐼𝐷)) → ∃𝑒𝑃 (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒)))
22 hlpasch.k . . . . . . 7 𝐾 = (hlG‘𝐺)
23 simplr 767 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒𝑃)
2413ad2antrr 724 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐵𝑃)
2515ad2antrr 724 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴𝑃)
265ad2antrr 724 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐺 ∈ TarskiG)
27 simprr 771 . . . . . . . 8 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴 ∈ (𝐵𝐼𝑒))
281, 16, 2, 26, 24, 25, 23, 27tgbtwncom 26266 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴 ∈ (𝑒𝐼𝐵))
2926adantr 483 . . . . . . . . . . 11 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐺 ∈ TarskiG)
3024adantr 483 . . . . . . . . . . 11 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐵𝑃)
3125adantr 483 . . . . . . . . . . 11 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴𝑃)
32 simplrr 776 . . . . . . . . . . . 12 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝑒))
33 simpr 487 . . . . . . . . . . . . 13 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝑒 = 𝐵)
3433oveq2d 7164 . . . . . . . . . . . 12 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → (𝐵𝐼𝑒) = (𝐵𝐼𝐵))
3532, 34eleqtrd 2913 . . . . . . . . . . 11 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐵))
361, 16, 2, 29, 30, 31, 35axtgbtwnid 26244 . . . . . . . . . 10 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐵 = 𝐴)
3736eqcomd 2825 . . . . . . . . 9 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴 = 𝐵)
38 hlpasch.6 . . . . . . . . . . . 12 (𝜑𝐴𝐵)
3938ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴𝐵)
4039adantr 483 . . . . . . . . . 10 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → 𝐴𝐵)
4140neneqd 3019 . . . . . . . . 9 (((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) ∧ 𝑒 = 𝐵) → ¬ 𝐴 = 𝐵)
4237, 41pm2.65da 815 . . . . . . . 8 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → ¬ 𝑒 = 𝐵)
4342neqned 3021 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒𝐵)
441, 2, 22, 23, 24, 25, 26, 25, 28, 43, 39btwnhl2 26391 . . . . . 6 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐴(𝐾𝐵)𝑒)
457ad2antrr 724 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝐷𝑃)
469ad2antrr 724 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑋𝑃)
47 simprl 769 . . . . . . 7 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒 ∈ (𝐷𝐼𝑋))
481, 16, 2, 26, 45, 23, 46, 47tgbtwncom 26266 . . . . . 6 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → 𝑒 ∈ (𝑋𝐼𝐷))
4944, 48jca 514 . . . . 5 ((((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒))) → (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
5049ex 415 . . . 4 (((𝜑𝐶 ∈ (𝐵𝐼𝐷)) ∧ 𝑒𝑃) → ((𝑒 ∈ (𝐷𝐼𝑋) ∧ 𝐴 ∈ (𝐵𝐼𝑒)) → (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷))))
5150reximdva 3272 . . 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 724 . . . . . 6 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐷𝑃)
5453adantr 483 . . . . 5 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷𝑃)
55 simpr 487 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) ∧ 𝑒 = 𝐷) → 𝑒 = 𝐷)
5655breq2d 5069 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) ∧ 𝑒 = 𝐷) → (𝐴(𝐾𝐵)𝑒𝐴(𝐾𝐵)𝐷))
5755eleq1d 2895 . . . . . 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 724 . . . . . . . 8 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐴𝑃)
6059adantr 483 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴𝑃)
6112ad2antrr 724 . . . . . . . 8 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐵𝑃)
6261adantr 483 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐵𝑃)
634ad2antrr 724 . . . . . . . 8 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐺 ∈ TarskiG)
6463adantr 483 . . . . . . 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 26382 . . . . . . . . 9 (𝜑𝐷(𝐾𝐵)𝐶)
6766ad3antrrr 728 . . . . . . . 8 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷(𝐾𝐵)𝐶)
6810adantr 483 . . . . . . . . . 10 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐶𝑃)
6968ad2antrr 724 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐶𝑃)
7019adantr 483 . . . . . . . . . . 11 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐴 ∈ (𝑋𝐼𝐶))
7170ad2antrr 724 . . . . . . . . . 10 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴 ∈ (𝑋𝐼𝐶))
72 simpr 487 . . . . . . . . . . 11 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝑋 = 𝐵)
7372oveq1d 7163 . . . . . . . . . 10 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝑋𝐼𝐶) = (𝐵𝐼𝐶))
7471, 73eleqtrd 2913 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴 ∈ (𝐵𝐼𝐶))
751, 2, 22, 10, 6, 12, 4ishlg 26380 . . . . . . . . . . . 12 (𝜑 → (𝐶(𝐾𝐵)𝐷 ↔ (𝐶𝐵𝐷𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶)))))
7665, 75mpbid 234 . . . . . . . . . . 11 (𝜑 → (𝐶𝐵𝐷𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶))))
7776simp1d 1136 . . . . . . . . . 10 (𝜑𝐶𝐵)
7877ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐶𝐵)
7938ad2antrr 724 . . . . . . . . . 10 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝐴𝐵)
8079adantr 483 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴𝐵)
811, 2, 22, 54, 69, 62, 64, 60, 74, 78, 80hlbtwn 26389 . . . . . . . 8 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝐷(𝐾𝐵)𝐶𝐷(𝐾𝐵)𝐴))
8267, 81mpbid 234 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷(𝐾𝐵)𝐴)
831, 2, 22, 54, 60, 62, 64, 82hlcomd 26382 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐴(𝐾𝐵)𝐷)
848ad2antrr 724 . . . . . . . 8 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝑋𝑃)
8584adantr 483 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝑋𝑃)
861, 16, 2, 64, 85, 54tgbtwntriv2 26265 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → 𝐷 ∈ (𝑋𝐼𝐷))
8783, 86jca 514 . . . . 5 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → (𝐴(𝐾𝐵)𝐷𝐷 ∈ (𝑋𝐼𝐷)))
8854, 58, 87rspcedvd 3624 . . . 4 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋 = 𝐵) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
8984ad2antrr 724 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) → 𝑋𝑃)
90 simpr 487 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋)
9190breq2d 5069 . . . . . . . 8 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒 = 𝑋) → (𝐴(𝐾𝐵)𝑒𝐴(𝐾𝐵)𝑋))
9290eleq1d 2895 . . . . . . . 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 750 . . . . . 6 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) ∧ 𝑒 = 𝑋) → ((𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)) ↔ (𝐴(𝐾𝐵)𝑋𝑋 ∈ (𝑋𝐼𝐷))))
95 simpr 487 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) → 𝐴(𝐾𝐵)𝑋)
961, 16, 2, 63, 84, 53tgbtwntriv1 26269 . . . . . . . 8 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → 𝑋 ∈ (𝑋𝐼𝐷))
9796ad2antrr 724 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) → 𝑋 ∈ (𝑋𝐼𝐷))
9895, 97jca 514 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) → (𝐴(𝐾𝐵)𝑋𝑋 ∈ (𝑋𝐼𝐷)))
9989, 94, 98rspcedvd 3624 . . . . 5 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐴(𝐾𝐵)𝑋) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
10053ad2antrr 724 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷𝑃)
101 simpr 487 . . . . . . . 8 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) ∧ 𝑒 = 𝐷) → 𝑒 = 𝐷)
102101breq2d 5069 . . . . . . 7 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) ∧ 𝑒 = 𝐷) → (𝐴(𝐾𝐵)𝑒𝐴(𝐾𝐵)𝐷))
103101eleq1d 2895 . . . . . . 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 724 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴𝐵)
1061, 2, 22, 10, 6, 12, 4, 65hlne2 26384 . . . . . . . . 9 (𝜑𝐷𝐵)
107106ad4antr 730 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷𝐵)
10863ad2antrr 724 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐺 ∈ TarskiG)
10961ad2antrr 724 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐵𝑃)
11059ad2antrr 724 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴𝑃)
11168ad2antrr 724 . . . . . . . . . 10 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐶𝑃)
112111adantr 483 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐶𝑃)
11384ad2antrr 724 . . . . . . . . . 10 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝑋𝑃)
114 simpr 487 . . . . . . . . . 10 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐵 ∈ (𝑋𝐼𝐴))
11570ad2antrr 724 . . . . . . . . . . 11 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐴 ∈ (𝑋𝐼𝐶))
116115adantr 483 . . . . . . . . . 10 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴 ∈ (𝑋𝐼𝐶))
1171, 16, 2, 108, 113, 109, 110, 112, 114, 116tgbtwnexch3 26272 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴 ∈ (𝐵𝐼𝐶))
118 simp-4r 782 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷 ∈ (𝐵𝐼𝐶))
1191, 2, 108, 109, 110, 100, 112, 117, 118tgbtwnconn3 26355 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → (𝐴 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐴)))
1201, 2, 22, 14, 6, 12, 4ishlg 26380 . . . . . . . . 9 (𝜑 → (𝐴(𝐾𝐵)𝐷 ↔ (𝐴𝐵𝐷𝐵 ∧ (𝐴 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐴)))))
121120ad4antr 730 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → (𝐴(𝐾𝐵)𝐷 ↔ (𝐴𝐵𝐷𝐵 ∧ (𝐴 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐴)))))
122105, 107, 119, 121mpbir3and 1336 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐴(𝐾𝐵)𝐷)
1231, 16, 2, 108, 113, 100tgbtwntriv2 26265 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → 𝐷 ∈ (𝑋𝐼𝐷))
124122, 123jca 514 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → (𝐴(𝐾𝐵)𝐷𝐷 ∈ (𝑋𝐼𝐷)))
125100, 104, 124rspcedvd 3624 . . . . 5 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝐵 ∈ (𝑋𝐼𝐴)) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
1268ad3antrrr 728 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝑋𝑃)
12712ad3antrrr 728 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐵𝑃)
12814ad3antrrr 728 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐴𝑃)
1294ad3antrrr 728 . . . . . 6 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐺 ∈ TarskiG)
130 simpr 487 . . . . . . . 8 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝑋𝐵)
131130neneqd 3019 . . . . . . 7 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → ¬ 𝑋 = 𝐵)
13263adantr 483 . . . . . . . . . 10 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐺 ∈ TarskiG)
133132adantr 483 . . . . . . . . . . . . . 14 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝐺 ∈ TarskiG)
134126adantr 483 . . . . . . . . . . . . . 14 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝑋𝑃)
135128adantr 483 . . . . . . . . . . . . . 14 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝐴𝑃)
136115adantr 483 . . . . . . . . . . . . . . 15 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝐴 ∈ (𝑋𝐼𝐶))
137 simpr 487 . . . . . . . . . . . . . . . 16 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝑋 = 𝐶)
138137oveq2d 7164 . . . . . . . . . . . . . . 15 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → (𝑋𝐼𝑋) = (𝑋𝐼𝐶))
139136, 138eleqtrrd 2914 . . . . . . . . . . . . . 14 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝐴 ∈ (𝑋𝐼𝑋))
1401, 16, 2, 133, 134, 135, 139axtgbtwnid 26244 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → 𝑋 = 𝐴)
141140olcd 872 . . . . . . . . . . . 12 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋 = 𝐶) → (𝐵 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
142132adantr 483 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝐺 ∈ TarskiG)
143127adantr 483 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝐵𝑃)
144111adantr 483 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝐶𝑃)
145126adantr 483 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝑋𝑃)
146128adantr 483 . . . . . . . . . . . . 13 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝐴𝑃)
147 simpr 487 . . . . . . . . . . . . . . . 16 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝑋𝐶)
148147necomd 3069 . . . . . . . . . . . . . . 15 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → 𝐶𝑋)
149148neneqd 3019 . . . . . . . . . . . . . 14 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → ¬ 𝐶 = 𝑋)
15053adantr 483 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐷𝑃)
151106ad3antrrr 728 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐷𝐵)
152 simplr 767 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷))
1531, 2, 3, 132, 150, 127, 126, 151, 152lncom 26400 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝑋 ∈ (𝐷(LineG‘𝐺)𝐵))
15477necomd 3069 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐵𝐶)
155154ad3antrrr 728 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐵𝐶)
15666ad3antrrr 728 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐷(𝐾𝐵)𝐶)
1571, 2, 22, 150, 111, 127, 132, 3, 156hlln 26385 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐷 ∈ (𝐶(LineG‘𝐺)𝐵))
1581, 2, 3, 132, 127, 111, 150, 155, 157lncom 26400 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → 𝐷 ∈ (𝐵(LineG‘𝐺)𝐶))
159158orcd 871 . . . . . . . . . . . . . . . . . . 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 26425 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝑋 ∈ (𝐵(LineG‘𝐺)𝐶) ∨ 𝐵 = 𝐶))
1611, 3, 2, 132, 127, 111, 126, 160colrot1 26337 . . . . . . . . . . . . . . . . 17 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐵 ∈ (𝐶(LineG‘𝐺)𝑋) ∨ 𝐶 = 𝑋))
162161orcomd 867 . . . . . . . . . . . . . . . 16 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐶 = 𝑋𝐵 ∈ (𝐶(LineG‘𝐺)𝑋)))
163162adantr 483 . . . . . . . . . . . . . . 15 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → (𝐶 = 𝑋𝐵 ∈ (𝐶(LineG‘𝐺)𝑋)))
164163ord 860 . . . . . . . . . . . . . 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 26335 . . . . . . . . . . . . . 14 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐶 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
167166adantr 483 . . . . . . . . . . . . 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 26425 . . . . . . . . . . . 12 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) ∧ 𝑋𝐶) → (𝐵 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
169141, 168pm2.61dane 3102 . . . . . . . . . . 11 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐵 ∈ (𝑋(LineG‘𝐺)𝐴) ∨ 𝑋 = 𝐴))
1701, 3, 2, 132, 126, 128, 127, 169colrot2 26338 . . . . . . . . . 10 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐴 ∈ (𝐵(LineG‘𝐺)𝑋) ∨ 𝐵 = 𝑋))
1711, 3, 2, 132, 127, 126, 128, 170colcom 26336 . . . . . . . . 9 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐴 ∈ (𝑋(LineG‘𝐺)𝐵) ∨ 𝑋 = 𝐵))
172171orcomd 867 . . . . . . . 8 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝑋 = 𝐵𝐴 ∈ (𝑋(LineG‘𝐺)𝐵)))
173172ord 860 . . . . . . 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 26393 . . . . 5 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → (𝐴(𝐾𝐵)𝑋𝐵 ∈ (𝑋𝐼𝐴)))
17699, 125, 175mpjaodan 954 . . . 4 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑋𝐵) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
17788, 176pm2.61dane 3102 . . 3 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
1784adantr 483 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐺 ∈ TarskiG)
1798adantr 483 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝑋𝑃)
18012adantr 483 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐵𝑃)
18114adantr 483 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐴𝑃)
1826adantr 483 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷𝑃)
183 simpr 487 . . . . . 6 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷 ∈ (𝐵𝐼𝐶))
1841, 16, 2, 178, 179, 180, 68, 181, 182, 70, 183axtgpasch 26245 . . . . 5 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → ∃𝑒𝑃 (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)))
185184adantr 483 . . . 4 (((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) → ∃𝑒𝑃 (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)))
186 simplr 767 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒𝑃)
187181ad3antrrr 728 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐴𝑃)
188180ad3antrrr 728 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐵𝑃)
189178ad3antrrr 728 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐺 ∈ TarskiG)
190 simprl 769 . . . . . . . . . 10 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ (𝐴𝐼𝐵))
1911, 16, 2, 189, 187, 186, 188, 190tgbtwncom 26266 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ (𝐵𝐼𝐴))
19238necomd 3069 . . . . . . . . . 10 (𝜑𝐵𝐴)
193192ad4antr 730 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐵𝐴)
194189adantr 483 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐺 ∈ TarskiG)
1956ad5antr 732 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷𝑃)
1968ad5antr 732 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑋𝑃)
197188adantr 483 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵𝑃)
198 simp-4r 782 . . . . . . . . . . . . . 14 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷))
199106necomd 3069 . . . . . . . . . . . . . . . 16 (𝜑𝐵𝐷)
200199ad5antr 732 . . . . . . . . . . . . . . 15 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵𝐷)
201200neneqd 3019 . . . . . . . . . . . . . 14 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → ¬ 𝐵 = 𝐷)
202 ioran 979 . . . . . . . . . . . . . 14 (¬ (𝑋 ∈ (𝐵(LineG‘𝐺)𝐷) ∨ 𝐵 = 𝐷) ↔ (¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷) ∧ ¬ 𝐵 = 𝐷))
203198, 201, 202sylanbrc 585 . . . . . . . . . . . . 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 26341 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → ¬ (𝐷 ∈ (𝑋(LineG‘𝐺)𝐵) ∨ 𝑋 = 𝐵))
205 simpr 487 . . . . . . . . . . . . 13 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑒 = 𝐵)
206186adantr 483 . . . . . . . . . . . . . 14 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑒𝑃)
2071, 2, 3, 194, 195, 196, 197, 204ncolne1 26403 . . . . . . . . . . . . . 14 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷𝑋)
208 simplrr 776 . . . . . . . . . . . . . 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 26397 . . . . . . . . . . . . 13 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝑒 ∈ (𝐷(LineG‘𝐺)𝑋))
210205, 209eqeltrrd 2912 . . . . . . . . . . . 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 26411 . . . . . . . . . . . 12 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷 ∈ (𝐷(LineG‘𝐺)𝑋))
212106ad5antr 732 . . . . . . . . . . . . . 14 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐷𝐵)
213212necomd 3069 . . . . . . . . . . . . 13 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵𝐷)
2141, 2, 3, 194, 197, 195, 213tglinerflx1 26411 . . . . . . . . . . . 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 26412 . . . . . . . . . . . 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 26423 . . . . . . . . . . 11 ((((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) ∧ 𝑒 = 𝐵) → 𝐵 = 𝐷)
217216, 201pm2.65da 815 . . . . . . . . . 10 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → ¬ 𝑒 = 𝐵)
218217neqned 3021 . . . . . . . . 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 26390 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒(𝐾𝐵)𝐴)
2201, 2, 22, 186, 187, 188, 189, 219hlcomd 26382 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝐴(𝐾𝐵)𝑒)
221178ad3antrrr 728 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝐺 ∈ TarskiG)
222182ad3antrrr 728 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝐷𝑃)
223 simplr 767 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑒𝑃)
224179ad3antrrr 728 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑋𝑃)
225 simpr 487 . . . . . . . . 9 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑒 ∈ (𝐷𝐼𝑋))
2261, 16, 2, 221, 222, 223, 224, 225tgbtwncom 26266 . . . . . . . 8 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → 𝑒 ∈ (𝑋𝐼𝐷))
227226adantrl 714 . . . . . . 7 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → 𝑒 ∈ (𝑋𝐼𝐷))
228220, 227jca 514 . . . . . 6 (((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) ∧ (𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋))) → (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
229228ex 415 . . . . 5 ((((𝜑𝐷 ∈ (𝐵𝐼𝐶)) ∧ ¬ 𝑋 ∈ (𝐵(LineG‘𝐺)𝐷)) ∧ 𝑒𝑃) → ((𝑒 ∈ (𝐴𝐼𝐵) ∧ 𝑒 ∈ (𝐷𝐼𝑋)) → (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷))))
230229reximdva 3272 . . . 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 811 . 2 ((𝜑𝐷 ∈ (𝐵𝐼𝐶)) → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
23376simp3d 1138 . 2 (𝜑 → (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶)))
23452, 232, 233mpjaodan 954 1 (𝜑 → ∃𝑒𝑃 (𝐴(𝐾𝐵)𝑒𝑒 ∈ (𝑋𝐼𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843  w3a 1081   = wceq 1530  wcel 2107  wne 3014  wrex 3137   class class class wbr 5057  cfv 6348  (class class class)co 7148  Basecbs 16475  distcds 16566  TarskiGcstrkg 26208  Itvcitv 26214  LineGclng 26215  hlGchlg 26378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12885  df-fzo 13026  df-hash 13683  df-word 13854  df-concat 13915  df-s1 13942  df-s2 14202  df-s3 14203  df-trkgc 26226  df-trkgb 26227  df-trkgcb 26228  df-trkgld 26230  df-trkg 26231  df-cgrg 26289  df-leg 26361  df-hlg 26379  df-mir 26431  df-rag 26472  df-perpg 26474
This theorem is referenced by:  inaghl  26623
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