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Theorem outpasch 26469
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 481 . . . . 5 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝐴𝑃)
3 simpr 485 . . . . . . 7 (((𝜑𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
43eleq1d 2897 . . . . . 6 (((𝜑𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → (𝑥 ∈ (𝐴𝐼𝐵) ↔ 𝐴 ∈ (𝐴𝐼𝐵)))
53oveq2d 7161 . . . . . . 7 (((𝜑𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → (𝑅𝐼𝑥) = (𝑅𝐼𝐴))
65eleq2d 2898 . . . . . 6 (((𝜑𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → (𝑄 ∈ (𝑅𝐼𝑥) ↔ 𝑄 ∈ (𝑅𝐼𝐴)))
74, 6anbi12d 630 . . . . 5 (((𝜑𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐴) → ((𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)) ↔ (𝐴 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐴))))
8 outpasch.p . . . . . . . 8 𝑃 = (Base‘𝐺)
9 eqid 2821 . . . . . . . 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 26205 . . . . . . 7 (𝜑𝐴 ∈ (𝐴𝐼𝐵))
1413adantr 481 . . . . . 6 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝐴 ∈ (𝐴𝐼𝐵))
1511adantr 481 . . . . . . 7 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝐺 ∈ TarskiG)
16 outpasch.r . . . . . . . 8 (𝜑𝑅𝑃)
1716adantr 481 . . . . . . 7 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝑅𝑃)
18 outpasch.q . . . . . . . 8 (𝜑𝑄𝑃)
1918adantr 481 . . . . . . 7 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝑄𝑃)
20 outpasch.c . . . . . . . 8 (𝜑𝐶𝑃)
2120adantr 481 . . . . . . 7 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝐶𝑃)
22 simpr 485 . . . . . . 7 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝑄 ∈ (𝑅𝐼𝐶))
23 outpasch.1 . . . . . . . . 9 (𝜑𝐶 ∈ (𝐴𝐼𝑅))
248, 9, 10, 11, 1, 20, 16, 23tgbtwncom 26202 . . . . . . . 8 (𝜑𝐶 ∈ (𝑅𝐼𝐴))
2524adantr 481 . . . . . . 7 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝐶 ∈ (𝑅𝐼𝐴))
268, 9, 10, 15, 17, 19, 21, 2, 22, 25tgbtwnexch 26212 . . . . . 6 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → 𝑄 ∈ (𝑅𝐼𝐴))
2714, 26jca 512 . . . . 5 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → (𝐴 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐴)))
282, 7, 27rspcedvd 3625 . . . 4 ((𝜑𝑄 ∈ (𝑅𝐼𝐶)) → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
2928adantlr 711 . . 3 (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝑄 ∈ (𝑅𝐼𝐶)) → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
3012ad2antrr 722 . . . 4 (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐵𝑃)
31 eleq1 2900 . . . . . 6 (𝑥 = 𝐵 → (𝑥 ∈ (𝐴𝐼𝐵) ↔ 𝐵 ∈ (𝐴𝐼𝐵)))
32 oveq2 7153 . . . . . . 7 (𝑥 = 𝐵 → (𝑅𝐼𝑥) = (𝑅𝐼𝐵))
3332eleq2d 2898 . . . . . 6 (𝑥 = 𝐵 → (𝑄 ∈ (𝑅𝐼𝑥) ↔ 𝑄 ∈ (𝑅𝐼𝐵)))
3431, 33anbi12d 630 . . . . 5 (𝑥 = 𝐵 → ((𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)) ↔ (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵))))
3534adantl 482 . . . 4 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑥 = 𝐵) → ((𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)) ↔ (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵))))
368, 9, 10, 11, 1, 12tgbtwntriv2 26201 . . . . . 6 (𝜑𝐵 ∈ (𝐴𝐼𝐵))
3736ad2antrr 722 . . . . 5 (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝐵 ∈ (𝐴𝐼𝐵))
3811ad3antrrr 726 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝐺 ∈ TarskiG)
3920ad3antrrr 726 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝐶𝑃)
4016ad3antrrr 726 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑅𝑃)
4118ad3antrrr 726 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑄𝑃)
4212ad3antrrr 726 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝐵𝑃)
43 simpr 485 . . . . . . . 8 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑅 ∈ (𝑄𝐼𝐶))
448, 9, 10, 38, 41, 40, 39, 43tgbtwncom 26202 . . . . . . 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 26202 . . . . . . . 8 (𝜑𝑄 ∈ (𝐶𝐼𝐵))
4746ad3antrrr 726 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑄 ∈ (𝐶𝐼𝐵))
488, 9, 10, 38, 39, 40, 41, 42, 44, 47tgbtwnexch3 26208 . . . . . 6 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝑅 ∈ (𝑄𝐼𝐶)) → 𝑄 ∈ (𝑅𝐼𝐵))
4911ad3antrrr 726 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝐺 ∈ TarskiG)
5012ad3antrrr 726 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝐵𝑃)
5118ad3antrrr 726 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄𝑃)
5216ad3antrrr 726 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑅𝑃)
5320ad3antrrr 726 . . . . . . . 8 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝐶𝑃)
54 simpr 485 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐶) → 𝑄 = 𝐶)
558, 9, 10, 11, 16, 20tgbtwntriv2 26201 . . . . . . . . . . . 12 (𝜑𝐶 ∈ (𝑅𝐼𝐶))
5655ad4antr 728 . . . . . . . . . . 11 (((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐶) → 𝐶 ∈ (𝑅𝐼𝐶))
5754, 56eqeltrd 2913 . . . . . . . . . 10 (((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐶) → 𝑄 ∈ (𝑅𝐼𝐶))
58 simpllr 772 . . . . . . . . . 10 (((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐶) → ¬ 𝑄 ∈ (𝑅𝐼𝐶))
5957, 58pm2.65da 813 . . . . . . . . 9 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → ¬ 𝑄 = 𝐶)
6059neqned 3023 . . . . . . . 8 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄𝐶)
6145ad3antrrr 726 . . . . . . . 8 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄 ∈ (𝐵𝐼𝐶))
62 simpr 485 . . . . . . . 8 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝐶 ∈ (𝑄𝐼𝑅))
638, 9, 10, 49, 50, 51, 53, 52, 60, 61, 62tgbtwnouttr 26211 . . . . . . 7 ((((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) ∧ 𝐶 ∈ (𝑄𝐼𝑅)) → 𝑄 ∈ (𝐵𝐼𝑅))
648, 9, 10, 49, 50, 51, 52, 63tgbtwncom 26202 . . . . . 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 26268 . . . . . . . . 9 (𝜑 → ((𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶) ↔ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅))))
6766biimpa 477 . . . . . . . 8 ((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)))
68 3orcoma 1085 . . . . . . . . 9 ((𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)) ↔ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)))
69 3orass 1082 . . . . . . . . 9 ((𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)) ↔ (𝑄 ∈ (𝑅𝐼𝐶) ∨ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅))))
7068, 69bitr3i 278 . . . . . . . 8 ((𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝑄 ∈ (𝑅𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)) ↔ (𝑄 ∈ (𝑅𝐼𝐶) ∨ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅))))
7167, 70sylib 219 . . . . . . 7 ((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → (𝑄 ∈ (𝑅𝐼𝐶) ∨ (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅))))
7271orcanai 996 . . . . . 6 (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → (𝑅 ∈ (𝑄𝐼𝐶) ∨ 𝐶 ∈ (𝑄𝐼𝑅)))
7348, 64, 72mpjaodan 952 . . . . 5 (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → 𝑄 ∈ (𝑅𝐼𝐵))
7437, 73jca 512 . . . 4 (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵)))
7530, 35, 74rspcedvd 3625 . . 3 (((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝑄 ∈ (𝑅𝐼𝐶)) → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
7629, 75pm2.61dan 809 . 2 ((𝜑 ∧ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
7712ad2antrr 722 . . . 4 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵𝑃)
7834adantl 482 . . . 4 ((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 = 𝐵) → ((𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)) ↔ (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵))))
7936ad2antrr 722 . . . . 5 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ (𝐴𝐼𝐵))
8011ad2antrr 722 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐺 ∈ TarskiG)
8116ad2antrr 722 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅𝑃)
8218ad2antrr 722 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄𝑃)
8320ad2antrr 722 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶𝑃)
84 simplr 765 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶))
85 simpr 485 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ (𝑅𝐿𝑄))
8611adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐺 ∈ TarskiG)
8716adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑅𝑃)
8818adantr 481 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄𝑃)
8920adantr 481 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐶𝑃)
90 simpr 485 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶))
918, 10, 65, 86, 87, 88, 89, 90ncolne1 26339 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑅𝑄)
928, 10, 65, 86, 87, 88, 91tglinerflx2 26348 . . . . . . . 8 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ∈ (𝑅𝐿𝑄))
9392adantr 481 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝑅𝐿𝑄))
948, 65, 10, 86, 88, 89, 87, 90ncolcom 26275 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ (𝑅 ∈ (𝐶𝐿𝑄) ∨ 𝐶 = 𝑄))
958, 65, 10, 86, 89, 88, 87, 94ncolrot1 26276 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ (𝐶 ∈ (𝑄𝐿𝑅) ∨ 𝑄 = 𝑅))
968, 10, 65, 86, 89, 88, 87, 95ncolne1 26339 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐶𝑄)
9796adantr 481 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶𝑄)
9846ad2antrr 722 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝐶𝐼𝐵))
998, 10, 65, 80, 83, 82, 77, 97, 98btwnlng3 26335 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ (𝐶𝐿𝑄))
1008, 10, 65, 80, 83, 82, 97tglinerflx2 26348 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝐶𝐿𝑄))
1018, 10, 65, 80, 81, 82, 83, 82, 84, 85, 93, 99, 100tglineinteq 26359 . . . . . 6 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 = 𝑄)
1028, 9, 10, 11, 16, 12tgbtwntriv2 26201 . . . . . . 7 (𝜑𝐵 ∈ (𝑅𝐼𝐵))
103102ad2antrr 722 . . . . . 6 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵 ∈ (𝑅𝐼𝐵))
104101, 103eqeltrrd 2914 . . . . 5 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝑅𝐼𝐵))
10579, 104jca 512 . . . 4 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → (𝐵 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝐵)))
10677, 78, 105rspcedvd 3625 . . 3 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ 𝐵 ∈ (𝑅𝐿𝑄)) → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
107 eleq1 2900 . . . . . . . . . 10 (𝑡 = 𝑥 → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑥 ∈ (𝑎𝐼𝑏)))
108107cbvrexvw 3451 . . . . . . . . 9 (∃𝑡 ∈ (𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝑎𝐼𝑏))
109108anbi2i 622 . . . . . . . 8 (((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑡 ∈ (𝑅𝐿𝑄)𝑡 ∈ (𝑎𝐼𝑏)) ↔ ((𝑎 ∈ (𝑃 ∖ (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑅𝐿𝑄))) ∧ ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝑎𝐼𝑏)))
110109opabbii 5125 . . . . . . 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 722 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐺 ∈ TarskiG)
11216ad2antrr 722 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅𝑃)
11318ad2antrr 722 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄𝑃)
11491adantr 481 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅𝑄)
1158, 10, 65, 111, 112, 113, 114tgelrnln 26344 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → (𝑅𝐿𝑄) ∈ ran 𝐿)
116 eqid 2821 . . . . . . 7 (hlG‘𝐺) = (hlG‘𝐺)
11720ad2antrr 722 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶𝑃)
1181ad2antrr 722 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐴𝑃)
11912ad2antrr 722 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐵𝑃)
12092adantr 481 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝑅𝐿𝑄))
1218, 65, 10, 86, 88, 89, 87, 90ncolrot2 26277 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ (𝐶 ∈ (𝑅𝐿𝑄) ∨ 𝑅 = 𝑄))
122 pm2.45 875 . . . . . . . . . 10 (¬ (𝐶 ∈ (𝑅𝐿𝑄) ∨ 𝑅 = 𝑄) → ¬ 𝐶 ∈ (𝑅𝐿𝑄))
123121, 122syl 17 . . . . . . . . 9 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ¬ 𝐶 ∈ (𝑅𝐿𝑄))
124123adantr 481 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ¬ 𝐶 ∈ (𝑅𝐿𝑄))
125 simpr 485 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ¬ 𝐵 ∈ (𝑅𝐿𝑄))
12646ad2antrr 722 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑄 ∈ (𝐶𝐼𝐵))
1278, 9, 10, 110, 117, 119, 120, 124, 125, 126islnoppd 26454 . . . . . . 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 26347 . . . . . . . 8 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑅 ∈ (𝑅𝐿𝑄))
129128adantr 481 . . . . . . 7 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅 ∈ (𝑅𝐿𝑄))
13024ad2antrr 722 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶 ∈ (𝑅𝐼𝐴))
1318, 10, 65, 86, 89, 87, 88, 121ncolne1 26339 . . . . . . . . . 10 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐶𝑅)
132131adantr 481 . . . . . . . . 9 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝐶𝑅)
1338, 9, 10, 111, 112, 117, 118, 130, 132tgbtwnne 26204 . . . . . . . 8 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → 𝑅𝐴)
1348, 10, 116, 112, 118, 117, 111, 118, 130, 133, 132btwnhl1 26326 . . . . . . 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 26468 . . . . . 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 26453 . . . . . 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 233 . . . . 5 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ((¬ 𝐴 ∈ (𝑅𝐿𝑄) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝐴𝐼𝐵)))
138137simprd 496 . . . 4 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ∃𝑥 ∈ (𝑅𝐿𝑄)𝑥 ∈ (𝐴𝐼𝐵))
139111ad2antrr 722 . . . . . . . 8 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
140115ad2antrr 722 . . . . . . . 8 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → (𝑅𝐿𝑄) ∈ ran 𝐿)
141 simplr 765 . . . . . . . 8 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑥 ∈ (𝑅𝐿𝑄))
1428, 65, 10, 139, 140, 141tglnpt 26263 . . . . . . 7 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑥𝑃)
143 simpr 485 . . . . . . 7 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑥 ∈ (𝐴𝐼𝐵))
144139ad2antrr 722 . . . . . . . . . 10 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐺 ∈ TarskiG)
14587ad3antrrr 726 . . . . . . . . . . 11 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑅𝑃)
146145ad2antrr 722 . . . . . . . . . 10 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑅𝑃)
14788ad5antr 730 . . . . . . . . . 10 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄𝑃)
148117ad2antrr 722 . . . . . . . . . . 11 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐶𝑃)
149148ad2antrr 722 . . . . . . . . . 10 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐶𝑃)
15090ad5antr 730 . . . . . . . . . 10 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶))
151 simplr 765 . . . . . . . . . . 11 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡𝑃)
152114ad4antr 728 . . . . . . . . . . 11 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑅𝑄)
153142ad2antrr 722 . . . . . . . . . . . 12 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑥𝑃)
15491necomd 3071 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄𝑅)
155154ad5antr 730 . . . . . . . . . . . . 13 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄𝑅)
156141ad2antrr 722 . . . . . . . . . . . . 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 26336 . . . . . . . . . . . 12 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑥 ∈ (𝑄𝐿𝑅))
158 simprl 767 . . . . . . . . . . . 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 26362 . . . . . . . . . . 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 26336 . . . . . . . . . 10 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝑅𝐿𝑄))
16192ad5antr 730 . . . . . . . . . 10 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄 ∈ (𝑅𝐿𝑄))
16296ad5antr 730 . . . . . . . . . . 11 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐶𝑄)
163119ad2antrr 722 . . . . . . . . . . . . 13 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐵𝑃)
164163ad2antrr 722 . . . . . . . . . . . 12 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐵𝑃)
16512adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐵𝑃)
16696necomd 3071 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄𝐶)
16745adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ∈ (𝐵𝐼𝐶))
1688, 10, 65, 86, 88, 89, 165, 166, 167btwnlng2 26334 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝐵 ∈ (𝑄𝐿𝐶))
169168ad5antr 730 . . . . . . . . . . . 12 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝐵 ∈ (𝑄𝐿𝐶))
170 simprr 769 . . . . . . . . . . . . 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 26202 . . . . . . . . . . . 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 26362 . . . . . . . . . . 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 26336 . . . . . . . . . 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 26348 . . . . . . . . . . 11 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → 𝑄 ∈ (𝐶𝐿𝑄))
175174ad5antr 730 . . . . . . . . . 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 26359 . . . . . . . . 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 26202 . . . . . . . . 9 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑡 ∈ (𝑅𝐼𝑥))
178176, 177eqeltrrd 2914 . . . . . . . 8 (((((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) ∧ 𝑡𝑃) ∧ (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵))) → 𝑄 ∈ (𝑅𝐼𝑥))
179118ad2antrr 722 . . . . . . . . 9 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝐴𝑃)
1808, 9, 10, 139, 179, 142, 163, 143tgbtwncom 26202 . . . . . . . . 9 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑥 ∈ (𝐵𝐼𝐴))
18124ad4antr 728 . . . . . . . . 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 26181 . . . . . . . 8 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → ∃𝑡𝑃 (𝑡 ∈ (𝑥𝐼𝑅) ∧ 𝑡 ∈ (𝐶𝐼𝐵)))
183178, 182r19.29a 3289 . . . . . . 7 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → 𝑄 ∈ (𝑅𝐼𝑥))
184142, 143, 183jca32 516 . . . . . 6 (((((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝑅𝐿𝑄)) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → (𝑥𝑃 ∧ (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥))))
185184expl 458 . . . . 5 (((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) ∧ ¬ 𝐵 ∈ (𝑅𝐿𝑄)) → ((𝑥 ∈ (𝑅𝐿𝑄) ∧ 𝑥 ∈ (𝐴𝐼𝐵)) → (𝑥𝑃 ∧ (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))))
186185reximdv2 3271 . . . 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 809 . 2 ((𝜑 ∧ ¬ (𝑅 ∈ (𝑄𝐿𝐶) ∨ 𝑄 = 𝐶)) → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
18976, 188pm2.61dan 809 1 (𝜑 → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐵) ∧ 𝑄 ∈ (𝑅𝐼𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3o 1078   = wceq 1528  wcel 2105  wne 3016  wrex 3139  cdif 3932   class class class wbr 5058  {copab 5120  ran crn 5550  cfv 6349  (class class class)co 7145  Basecbs 16473  distcds 16564  TarskiGcstrkg 26144  Itvcitv 26150  LineGclng 26151  hlGchlg 26314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-pm 8399  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-dju 9319  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-n0 11887  df-xnn0 11957  df-z 11971  df-uz 12233  df-fz 12883  df-fzo 13024  df-hash 13681  df-word 13852  df-concat 13913  df-s1 13940  df-s2 14200  df-s3 14201  df-trkgc 26162  df-trkgb 26163  df-trkgcb 26164  df-trkgld 26166  df-trkg 26167  df-cgrg 26225  df-leg 26297  df-hlg 26315  df-mir 26367  df-rag 26408  df-perpg 26410
This theorem is referenced by:  hlpasch  26470
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