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Theorem outpasch 26552
Description: Axiom of Pasch, outer form. This was proven by Gupta from other axioms and is therefore presented as Theorem 9.6 in [Schwabhauser] p. 70. (Contributed by Thierry Arnoux, 16-Aug-2020.)
Hypotheses
Ref Expression
outpasch.p 𝑃 = (Base‘𝐺)
outpasch.i 𝐼 = (Itv‘𝐺)
outpasch.l 𝐿 = (LineG‘𝐺)
outpasch.g (𝜑𝐺 ∈ TarskiG)
outpasch.a (𝜑𝐴𝑃)
outpasch.b (𝜑𝐵𝑃)
outpasch.c (𝜑𝐶𝑃)
outpasch.r (𝜑𝑅𝑃)
outpasch.q (𝜑𝑄𝑃)
outpasch.1 (𝜑𝐶 ∈ (𝐴𝐼𝑅))
outpasch.2 (𝜑𝑄 ∈ (𝐵𝐼𝐶))
Assertion
Ref Expression
outpasch (𝜑 → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐺   𝑥,𝐼   𝑥,𝐿   𝑥,𝑃   𝑥,𝑄   𝑥,𝑅   𝜑,𝑥

Proof of Theorem outpasch
Dummy variables 𝑡 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 outpasch.a . . . . . 6 (𝜑𝐴𝑃)
21adantr 484 . . . . 5 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝐴𝑃)
3 simpr 488 . . . . . . 7 (((𝜑𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
43eleq1d 2900 . . . . . 6 (((𝜑𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → (𝑥 ∈ (𝐴𝐼𝐵) ↔ 𝐴 ∈ (𝐴𝐼𝐵)))
53oveq2d 7165 . . . . . . 7 (((𝜑𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → (𝑅𝐼𝑥) = (𝑅𝐼𝐴))
65eleq2d 2901 . . . . . 6 (((𝜑𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → (𝑄 ∈ (𝑅𝐼𝑥) ↔ 𝑄 ∈ (𝑅𝐼𝐴)))
74, 6anbi12d 633 . . . . 5 (((𝜑𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → ((𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)) ↔ (𝐴 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐴))))
8 outpasch.p . . . . . . . 8 𝑃 = (Base‘𝐺)
9 eqid 2824 . . . . . . . 8 (dist‘𝐺) = (dist‘𝐺)
10 outpasch.i . . . . . . . 8 𝐼 = (Itv‘𝐺)
11 outpasch.g . . . . . . . 8 (𝜑𝐺 ∈ TarskiG)
12 outpasch.b . . . . . . . 8 (𝜑𝐵𝑃)
138, 9, 10, 11, 1, 12tgbtwntriv1 26288 . . . . . . 7 (𝜑𝐴 ∈ (𝐴𝐼𝐵))
1413adantr 484 . . . . . 6 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝐴 ∈ (𝐴𝐼𝐵))
1511adantr 484 . . . . . . 7 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝐺 ∈ TarskiG)
16 outpasch.r . . . . . . . 8 (𝜑𝑅𝑃)
1716adantr 484 . . . . . . 7 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝑅𝑃)
18 outpasch.q . . . . . . . 8 (𝜑𝑄𝑃)
1918adantr 484 . . . . . . 7 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝑄𝑃)
20 outpasch.c . . . . . . . 8 (𝜑𝐶𝑃)
2120adantr 484 . . . . . . 7 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝐶𝑃)
22 simpr 488 . . . . . . 7 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝑄 ∈ (𝑅𝐼𝐶))
23 outpasch.1 . . . . . . . . 9 (𝜑𝐶 ∈ (𝐴𝐼𝑅))
248, 9, 10, 11, 1, 20, 16, 23tgbtwncom 26285 . . . . . . . 8 (𝜑𝐶 ∈ (𝑅𝐼𝐴))
2524adantr 484 . . . . . . 7 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝐶 ∈ (𝑅𝐼𝐴))
268, 9, 10, 15, 17, 19, 21, 2, 22, 25tgbtwnexch 26295 . . . . . 6 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝑄 ∈ (𝑅𝐼𝐴))
2714, 26jca 515 . . . . 5 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → (𝐴 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐴)))
282, 7, 27rspcedvd 3612 . . . 4 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
2928adantlr 714 . . 3 (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
3012ad2antrr 725 . . . 4 (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐵𝑃)
31 eleq1 2903 . . . . . 6 (𝑥 = 𝐵 → (𝑥 ∈ (𝐴𝐼𝐵) ↔ 𝐵 ∈ (𝐴𝐼𝐵)))
32 oveq2 7157 . . . . . . 7 (𝑥 = 𝐵 → (𝑅𝐼𝑥) = (𝑅𝐼𝐵))
3332eleq2d 2901 . . . . . 6 (𝑥 = 𝐵 → (𝑄 ∈ (𝑅𝐼𝑥) ↔ 𝑄 ∈ (𝑅𝐼𝐵)))
3431, 33anbi12d 633 . . . . 5 (𝑥 = 𝐵 → ((𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)) ↔ (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵))))
3534adantl 485 . . . 4 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐵) → ((𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)) ↔ (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵))))
368, 9, 10, 11, 1, 12tgbtwntriv2 26284 . . . . . 6 (𝜑𝐵 ∈ (𝐴𝐼𝐵))
3736ad2antrr 725 . . . . 5 (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐵 ∈ (𝐴𝐼𝐵))
3811ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝐺 ∈ TarskiG)
3920ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝐶𝑃)
4016ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑅𝑃)
4118ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑄𝑃)
4212ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝐵𝑃)
43 simpr 488 . . . . . . . 8 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑅 ∈ (𝑄𝐼𝐶))
448, 9, 10, 38, 41, 40, 39, 43tgbtwncom 26285 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑅 ∈ (𝐶𝐼𝑄))
45 outpasch.2 . . . . . . . . 9 (𝜑𝑄 ∈ (𝐵𝐼𝐶))
468, 9, 10, 11, 12, 18, 20, 45tgbtwncom 26285 . . . . . . . 8 (𝜑𝑄 ∈ (𝐶𝐼𝐵))
4746ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑄 ∈ (𝐶𝐼𝐵))
488, 9, 10, 38, 39, 40, 41, 42, 44, 47tgbtwnexch3 26291 . . . . . 6 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑄 ∈ (𝑅𝐼𝐵))
4911ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝐺 ∈ TarskiG)
5012ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝐵𝑃)
5118ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄𝑃)
5216ad3antrrr 729 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑅𝑃)
5320ad3antrrr 729 . . . . . . . 8 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝐶𝑃)
54 simpr 488 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐶) → 𝑄 = 𝐶)
558, 9, 10, 11, 16, 20tgbtwntriv2 26284 . . . . . . . . . . . 12 (𝜑𝐶 ∈ (𝑅𝐼𝐶))
5655ad4antr 731 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐶) → 𝐶 ∈ (𝑅𝐼𝐶))
5754, 56eqeltrd 2916 . . . . . . . . . 10 (((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐶) → 𝑄 ∈ (𝑅𝐼𝐶))
58 simpllr 775 . . . . . . . . . 10 (((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐶) → ¬ 𝑄 ∈ (𝑅𝐼𝐶))
5957, 58pm2.65da 816 . . . . . . . . 9 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → ¬ 𝑄 = 𝐶)
6059neqned 3021 . . . . . . . 8 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄𝐶)
6145ad3antrrr 729 . . . . . . . 8 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄 ∈ (𝐵𝐼𝐶))
62 simpr 488 . . . . . . . 8 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝐶 ∈ (𝑄𝐼𝑅))
638, 9, 10, 49, 50, 51, 53, 52, 60, 61, 62tgbtwnouttr 26294 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄 ∈ (𝐵𝐼𝑅))
648, 9, 10, 49, 50, 51, 52, 63tgbtwncom 26285 . . . . . 6 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄 ∈ (𝑅𝐼𝐵))
65 outpasch.l . . . . . . . . . 10 𝐿 = (LineG‘𝐺)
668, 65, 10, 11, 18, 20, 16tgcolg 26351 . . . . . . . . 9 (𝜑 → ((𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶) ↔ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅))))
6766biimpa 480 . . . . . . . 8 ((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)))
68 3orcoma 1090 . . . . . . . . 9 ((𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)) ↔ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)))
69 3orass 1087 . . . . . . . . 9 ((𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)) ↔ (𝑄 ∈ (𝑅𝐼𝐶) ∨ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅))))
7068, 69bitr3i 280 . . . . . . . 8 ((𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)) ↔ (𝑄 ∈ (𝑅𝐼𝐶) ∨ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅))))
7167, 70sylib 221 . . . . . . 7 ((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → (𝑄 ∈ (𝑅𝐼𝐶) ∨ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅))))
7271orcanai 1000 . . . . . 6 (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)))
7348, 64, 72mpjaodan 956 . . . . 5 (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝑄 ∈ (𝑅𝐼𝐵))
7437, 73jca 515 . . . 4 (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵)))
7530, 35, 74rspcedvd 3612 . . 3 (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
7629, 75pm2.61dan 812 . 2 ((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
7712ad2antrr 725 . . . 4 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵𝑃)
7834adantl 485 . . . 4 ((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 = 𝐵) → ((𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)) ↔ (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵))))
7936ad2antrr 725 . . . . 5 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ (𝐴𝐼𝐵))
8011ad2antrr 725 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐺 ∈ TarskiG)
8116ad2antrr 725 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅𝑃)
8218ad2antrr 725 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄𝑃)
8320ad2antrr 725 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶𝑃)
84 simplr 768 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶))
85 simpr 488 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ (𝑅𝐿𝑄))
8611adantr 484 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐺 ∈ TarskiG)
8716adantr 484 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑅𝑃)
8818adantr 484 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄𝑃)
8920adantr 484 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐶𝑃)
90 simpr 488 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶))
918, 10, 65, 86, 87, 88, 89, 90ncolne1 26422 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑅𝑄)
928, 10, 65, 86, 87, 88, 91tglinerflx2 26431 . . . . . . . 8 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ∈ (𝑅𝐿𝑄))
9392adantr 484 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝑅𝐿𝑄))
948, 65, 10, 86, 88, 89, 87, 90ncolcom 26358 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ (𝑅 ∈ (𝐶𝐿𝑄) ∨ 𝐶 = 𝑄))
958, 65, 10, 86, 89, 88, 87, 94ncolrot1 26359 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ (𝐶 ∈ (𝑄𝐿𝑅) ∨ 𝑄 = 𝑅))
968, 10, 65, 86, 89, 88, 87, 95ncolne1 26422 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐶𝑄)
9796adantr 484 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶𝑄)
9846ad2antrr 725 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝐶𝐼𝐵))
998, 10, 65, 80, 83, 82, 77, 97, 98btwnlng3 26418 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ (𝐶𝐿𝑄))
1008, 10, 65, 80, 83, 82, 97tglinerflx2 26431 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝐶𝐿𝑄))
1018, 10, 65, 80, 81, 82, 83, 82, 84, 85, 93, 99, 100tglineinteq 26442 . . . . . 6 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 = 𝑄)
1028, 9, 10, 11, 16, 12tgbtwntriv2 26284 . . . . . . 7 (𝜑𝐵 ∈ (𝑅𝐼𝐵))
103102ad2antrr 725 . . . . . 6 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ (𝑅𝐼𝐵))
104101, 103eqeltrrd 2917 . . . . 5 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝑅𝐼𝐵))
10579, 104jca 515 . . . 4 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵)))
10677, 78, 105rspcedvd 3612 . . 3 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
107 eleq1 2903 . . . . . . . . . 10 (𝑡 = 𝑥 → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑥 ∈ (𝑎𝐼𝑏)))
108107cbvrexvw 3435 . . . . . . . . 9 (∃𝑡 ∈ (𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝑎𝐼𝑏))
109108anbi2i 625 . . . . . . . 8 (((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑡 ∈ (𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏)) ↔ ((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝑎𝐼𝑏)))
110109opabbii 5119 . . . . . . 7 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑡 ∈ (𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏))} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝑎𝐼𝑏))}
11111ad2antrr 725 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐺 ∈ TarskiG)
11216ad2antrr 725 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅𝑃)
11318ad2antrr 725 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄𝑃)
11491adantr 484 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅𝑄)
1158, 10, 65, 111, 112, 113, 114tgelrnln 26427 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → (𝑅𝐿𝑄) ∈ ran 𝐿)
116 eqid 2824 . . . . . . 7 (hlG‘𝐺) = (hlG‘𝐺)
11720ad2antrr 725 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶𝑃)
1181ad2antrr 725 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐴𝑃)
11912ad2antrr 725 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵𝑃)
12092adantr 484 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝑅𝐿𝑄))
1218, 65, 10, 86, 88, 89, 87, 90ncolrot2 26360 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ (𝐶 ∈ (𝑅𝐿𝑄) ∨ 𝑅 = 𝑄))
122 pm2.45 879 . . . . . . . . . 10 (¬ (𝐶 ∈ (𝑅𝐿𝑄) ∨ 𝑅 = 𝑄) → ¬ 𝐶 ∈ (𝑅𝐿𝑄))
123121, 122syl 17 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ 𝐶 ∈ (𝑅𝐿𝑄))
124123adantr 484 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ¬ 𝐶 ∈ (𝑅𝐿𝑄))
125 simpr 488 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ¬ 𝐵 ∈ (𝑅𝐿𝑄))
12646ad2antrr 725 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝐶𝐼𝐵))
1278, 9, 10, 110, 117, 119, 120, 124, 125, 126islnoppd 26537 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑡 ∈ (𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏))}𝐵)
1288, 10, 65, 86, 87, 88, 91tglinerflx1 26430 . . . . . . . 8 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑅 ∈ (𝑅𝐿𝑄))
129128adantr 484 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅 ∈ (𝑅𝐿𝑄))
13024ad2antrr 725 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶 ∈ (𝑅𝐼𝐴))
1318, 10, 65, 86, 89, 87, 88, 121ncolne1 26422 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐶𝑅)
132131adantr 484 . . . . . . . . 9 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶𝑅)
1338, 9, 10, 111, 112, 117, 118, 130, 132tgbtwnne 26287 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅𝐴)
1348, 10, 116, 112, 118, 117, 111, 118, 130, 133, 132btwnhl1 26409 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶((hlG‘𝐺)‘𝑅)𝐴)
1358, 9, 10, 110, 65, 115, 111, 116, 117, 118, 119, 127, 129, 134opphl 26551 . . . . . 6 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑡 ∈ (𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏))}𝐵)
1368, 9, 10, 110, 118, 119islnopp 26536 . . . . . 6 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → (𝐴{⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑡 ∈ (𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏))}𝐵 ↔ ((¬ 𝐴 ∈ (𝑅𝐿𝑄) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝐴𝐼𝐵))))
137135, 136mpbid 235 . . . . 5 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ((¬ 𝐴 ∈ (𝑅𝐿𝑄) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝐴𝐼𝐵)))
138137simprd 499 . . . 4 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝐴𝐼𝐵))
139111ad2antrr 725 . . . . . . . 8 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
140115ad2antrr 725 . . . . . . . 8 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → (𝑅𝐿𝑄) ∈ ran 𝐿)
141 simplr 768 . . . . . . . 8 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑥 ∈ (𝑅𝐿𝑄))
1428, 65, 10, 139, 140, 141tglnpt 26346 . . . . . . 7 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑥𝑃)
143 simpr 488 . . . . . . 7 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑥 ∈ (𝐴𝐼𝐵))
144139ad2antrr 725 . . . . . . . . . 10 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐺 ∈ TarskiG)
14587ad3antrrr 729 . . . . . . . . . . 11 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑅𝑃)
146145ad2antrr 725 . . . . . . . . . 10 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑅𝑃)
14788ad5antr 733 . . . . . . . . . 10 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄𝑃)
148117ad2antrr 725 . . . . . . . . . . 11 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐶𝑃)
149148ad2antrr 725 . . . . . . . . . 10 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐶𝑃)
15090ad5antr 733 . . . . . . . . . 10 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶))
151 simplr 768 . . . . . . . . . . 11 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡𝑃)
152114ad4antr 731 . . . . . . . . . . 11 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑅𝑄)
153142ad2antrr 725 . . . . . . . . . . . 12 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑥𝑃)
15491necomd 3069 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄𝑅)
155154ad5antr 733 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄𝑅)
156141ad2antrr 725 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑥 ∈ (𝑅𝐿𝑄))
1578, 10, 65, 144, 147, 146, 153, 155, 156lncom 26419 . . . . . . . . . . . 12 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑥 ∈ (𝑄𝐿𝑅))
158 simprl 770 . . . . . . . . . . . 12 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝑥𝐼𝑅))
1598, 10, 65, 144, 153, 147, 146, 151, 157, 158coltr3 26445 . . . . . . . . . . 11 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝑄𝐿𝑅))
1608, 10, 65, 144, 146, 147, 151, 152, 159lncom 26419 . . . . . . . . . 10 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝑅𝐿𝑄))
16192ad5antr 733 . . . . . . . . . 10 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄 ∈ (𝑅𝐿𝑄))
16296ad5antr 733 . . . . . . . . . . 11 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐶𝑄)
163119ad2antrr 725 . . . . . . . . . . . . 13 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐵𝑃)
164163ad2antrr 725 . . . . . . . . . . . 12 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐵𝑃)
16512adantr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐵𝑃)
16696necomd 3069 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄𝐶)
16745adantr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ∈ (𝐵𝐼𝐶))
1688, 10, 65, 86, 88, 89, 165, 166, 167btwnlng2 26417 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐵 ∈ (𝑄𝐿𝐶))
169168ad5antr 733 . . . . . . . . . . . 12 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐵 ∈ (𝑄𝐿𝐶))
170 simprr 772 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝐶𝐼𝐵))
1718, 9, 10, 144, 149, 151, 164, 170tgbtwncom 26285 . . . . . . . . . . . 12 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝐵𝐼𝐶))
1728, 10, 65, 144, 164, 147, 149, 151, 169, 171coltr3 26445 . . . . . . . . . . 11 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝑄𝐿𝐶))
1738, 10, 65, 144, 149, 147, 151, 162, 172lncom 26419 . . . . . . . . . 10 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝐶𝐿𝑄))
1748, 10, 65, 86, 89, 88, 96tglinerflx2 26431 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ∈ (𝐶𝐿𝑄))
175174ad5antr 733 . . . . . . . . . 10 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄 ∈ (𝐶𝐿𝑄))
1768, 10, 65, 144, 146, 147, 149, 147, 150, 160, 161, 173, 175tglineinteq 26442 . . . . . . . . 9 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 = 𝑄)
1778, 9, 10, 144, 153, 151, 146, 158tgbtwncom 26285 . . . . . . . . 9 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝑅𝐼𝑥))
178176, 177eqeltrrd 2917 . . . . . . . 8 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄 ∈ (𝑅𝐼𝑥))
179118ad2antrr 725 . . . . . . . . 9 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐴𝑃)
1808, 9, 10, 139, 179, 142, 163, 143tgbtwncom 26285 . . . . . . . . 9 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑥 ∈ (𝐵𝐼𝐴))
18124ad4antr 731 . . . . . . . . 9 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ (𝑅𝐼𝐴))
1828, 9, 10, 139, 163, 145, 179, 142, 148, 180, 181axtgpasch 26264 . . . . . . . 8 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → ∃𝑡𝑃 (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵)))
183178, 182r19.29a 3281 . . . . . . 7 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑄 ∈ (𝑅𝐼𝑥))
184142, 143, 183jca32 519 . . . . . 6 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → (𝑥𝑃 ∧ (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))))
185184expl 461 . . . . 5 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ((𝑥 ∈ (𝑅𝐿𝑄) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → (𝑥𝑃 ∧ (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))))
186185reximdv2 3263 . . . 4 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → (∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝐴𝐼𝐵) → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))))
187138, 186mpd 15 . . 3 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
188106, 187pm2.61dan 812 . 2 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
18976, 188pm2.61dan 812 1 (𝜑 → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3o 1083   = wceq 1538  wcel 2115  wne 3014  wrex 3134  cdif 3916   class class class wbr 5052  {copab 5114  ran crn 5543  cfv 6343  (class class class)co 7149  Basecbs 16483  distcds 16574  TarskiGcstrkg 26227  Itvcitv 26233  LineGclng 26234  hlGchlg 26397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-dju 9327  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-xnn0 11965  df-z 11979  df-uz 12241  df-fz 12895  df-fzo 13038  df-hash 13696  df-word 13867  df-concat 13923  df-s1 13950  df-s2 14210  df-s3 14211  df-trkgc 26245  df-trkgb 26246  df-trkgcb 26247  df-trkgld 26249  df-trkg 26250  df-cgrg 26308  df-leg 26380  df-hlg 26398  df-mir 26450  df-rag 26491  df-perpg 26493
This theorem is referenced by:  hlpasch  26553
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