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Theorem outpasch 27986
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 = (Baseβ€˜πΊ)
outpasch.i 𝐼 = (Itvβ€˜πΊ)
outpasch.l 𝐿 = (LineGβ€˜πΊ)
outpasch.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
outpasch.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
outpasch.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
outpasch.c (πœ‘ β†’ 𝐢 ∈ 𝑃)
outpasch.r (πœ‘ β†’ 𝑅 ∈ 𝑃)
outpasch.q (πœ‘ β†’ 𝑄 ∈ 𝑃)
outpasch.1 (πœ‘ β†’ 𝐢 ∈ (𝐴𝐼𝑅))
outpasch.2 (πœ‘ β†’ 𝑄 ∈ (𝐡𝐼𝐢))
Assertion
Ref Expression
outpasch (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑃 (π‘₯ ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼π‘₯)))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡   π‘₯,𝐢   π‘₯,𝐺   π‘₯,𝐼   π‘₯,𝐿   π‘₯,𝑃   π‘₯,𝑄   π‘₯,𝑅   πœ‘,π‘₯
Proof of Theorem outpasch
Dummy variables 𝑑 π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 outpasch.a . . . . . 6 (πœ‘ β†’ 𝐴 ∈ 𝑃)
21adantr 482 . . . . 5 ((πœ‘ ∧ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ 𝐴 ∈ 𝑃)
3 simpr 486 . . . . . . 7 (((πœ‘ ∧ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ π‘₯ = 𝐴) β†’ π‘₯ = 𝐴)
43eleq1d 2819 . . . . . 6 (((πœ‘ ∧ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ π‘₯ = 𝐴) β†’ (π‘₯ ∈ (𝐴𝐼𝐡) ↔ 𝐴 ∈ (𝐴𝐼𝐡)))
53oveq2d 7420 . . . . . . 7 (((πœ‘ ∧ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ π‘₯ = 𝐴) β†’ (𝑅𝐼π‘₯) = (𝑅𝐼𝐴))
65eleq2d 2820 . . . . . 6 (((πœ‘ ∧ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ π‘₯ = 𝐴) β†’ (𝑄 ∈ (𝑅𝐼π‘₯) ↔ 𝑄 ∈ (𝑅𝐼𝐴)))
74, 6anbi12d 632 . . . . 5 (((πœ‘ ∧ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ π‘₯ = 𝐴) β†’ ((π‘₯ ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼π‘₯)) ↔ (𝐴 ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼𝐴))))
8 outpasch.p . . . . . . . 8 𝑃 = (Baseβ€˜πΊ)
9 eqid 2733 . . . . . . . 8 (distβ€˜πΊ) = (distβ€˜πΊ)
10 outpasch.i . . . . . . . 8 𝐼 = (Itvβ€˜πΊ)
11 outpasch.g . . . . . . . 8 (πœ‘ β†’ 𝐺 ∈ TarskiG)
12 outpasch.b . . . . . . . 8 (πœ‘ β†’ 𝐡 ∈ 𝑃)
138, 9, 10, 11, 1, 12tgbtwntriv1 27722 . . . . . . 7 (πœ‘ β†’ 𝐴 ∈ (𝐴𝐼𝐡))
1413adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ 𝐴 ∈ (𝐴𝐼𝐡))
1511adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ 𝐺 ∈ TarskiG)
16 outpasch.r . . . . . . . 8 (πœ‘ β†’ 𝑅 ∈ 𝑃)
1716adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ 𝑅 ∈ 𝑃)
18 outpasch.q . . . . . . . 8 (πœ‘ β†’ 𝑄 ∈ 𝑃)
1918adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ 𝑄 ∈ 𝑃)
20 outpasch.c . . . . . . . 8 (πœ‘ β†’ 𝐢 ∈ 𝑃)
2120adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ 𝐢 ∈ 𝑃)
22 simpr 486 . . . . . . 7 ((πœ‘ ∧ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ 𝑄 ∈ (𝑅𝐼𝐢))
23 outpasch.1 . . . . . . . . 9 (πœ‘ β†’ 𝐢 ∈ (𝐴𝐼𝑅))
248, 9, 10, 11, 1, 20, 16, 23tgbtwncom 27719 . . . . . . . 8 (πœ‘ β†’ 𝐢 ∈ (𝑅𝐼𝐴))
2524adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ 𝐢 ∈ (𝑅𝐼𝐴))
268, 9, 10, 15, 17, 19, 21, 2, 22, 25tgbtwnexch 27729 . . . . . 6 ((πœ‘ ∧ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ 𝑄 ∈ (𝑅𝐼𝐴))
2714, 26jca 513 . . . . 5 ((πœ‘ ∧ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ (𝐴 ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼𝐴)))
282, 7, 27rspcedvd 3614 . . . 4 ((πœ‘ ∧ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ βˆƒπ‘₯ ∈ 𝑃 (π‘₯ ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼π‘₯)))
2928adantlr 714 . . 3 (((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ βˆƒπ‘₯ ∈ 𝑃 (π‘₯ ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼π‘₯)))
3012ad2antrr 725 . . . 4 (((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ 𝐡 ∈ 𝑃)
31 eleq1 2822 . . . . . 6 (π‘₯ = 𝐡 β†’ (π‘₯ ∈ (𝐴𝐼𝐡) ↔ 𝐡 ∈ (𝐴𝐼𝐡)))
32 oveq2 7412 . . . . . . 7 (π‘₯ = 𝐡 β†’ (𝑅𝐼π‘₯) = (𝑅𝐼𝐡))
3332eleq2d 2820 . . . . . 6 (π‘₯ = 𝐡 β†’ (𝑄 ∈ (𝑅𝐼π‘₯) ↔ 𝑄 ∈ (𝑅𝐼𝐡)))
3431, 33anbi12d 632 . . . . 5 (π‘₯ = 𝐡 β†’ ((π‘₯ ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼π‘₯)) ↔ (𝐡 ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼𝐡))))
3534adantl 483 . . . 4 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ π‘₯ = 𝐡) β†’ ((π‘₯ ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼π‘₯)) ↔ (𝐡 ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼𝐡))))
368, 9, 10, 11, 1, 12tgbtwntriv2 27718 . . . . . 6 (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐡))
3736ad2antrr 725 . . . . 5 (((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ 𝐡 ∈ (𝐴𝐼𝐡))
3811ad3antrrr 729 . . . . . . 7 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝑅 ∈ (𝑄𝐼𝐢)) β†’ 𝐺 ∈ TarskiG)
3920ad3antrrr 729 . . . . . . 7 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝑅 ∈ (𝑄𝐼𝐢)) β†’ 𝐢 ∈ 𝑃)
4016ad3antrrr 729 . . . . . . 7 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝑅 ∈ (𝑄𝐼𝐢)) β†’ 𝑅 ∈ 𝑃)
4118ad3antrrr 729 . . . . . . 7 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝑅 ∈ (𝑄𝐼𝐢)) β†’ 𝑄 ∈ 𝑃)
4212ad3antrrr 729 . . . . . . 7 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝑅 ∈ (𝑄𝐼𝐢)) β†’ 𝐡 ∈ 𝑃)
43 simpr 486 . . . . . . . 8 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝑅 ∈ (𝑄𝐼𝐢)) β†’ 𝑅 ∈ (𝑄𝐼𝐢))
448, 9, 10, 38, 41, 40, 39, 43tgbtwncom 27719 . . . . . . 7 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝑅 ∈ (𝑄𝐼𝐢)) β†’ 𝑅 ∈ (𝐢𝐼𝑄))
45 outpasch.2 . . . . . . . . 9 (πœ‘ β†’ 𝑄 ∈ (𝐡𝐼𝐢))
468, 9, 10, 11, 12, 18, 20, 45tgbtwncom 27719 . . . . . . . 8 (πœ‘ β†’ 𝑄 ∈ (𝐢𝐼𝐡))
4746ad3antrrr 729 . . . . . . 7 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝑅 ∈ (𝑄𝐼𝐢)) β†’ 𝑄 ∈ (𝐢𝐼𝐡))
488, 9, 10, 38, 39, 40, 41, 42, 44, 47tgbtwnexch3 27725 . . . . . 6 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝑅 ∈ (𝑄𝐼𝐢)) β†’ 𝑄 ∈ (𝑅𝐼𝐡))
4911ad3antrrr 729 . . . . . . 7 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝐢 ∈ (𝑄𝐼𝑅)) β†’ 𝐺 ∈ TarskiG)
5012ad3antrrr 729 . . . . . . 7 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝐢 ∈ (𝑄𝐼𝑅)) β†’ 𝐡 ∈ 𝑃)
5118ad3antrrr 729 . . . . . . 7 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝐢 ∈ (𝑄𝐼𝑅)) β†’ 𝑄 ∈ 𝑃)
5216ad3antrrr 729 . . . . . . 7 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝐢 ∈ (𝑄𝐼𝑅)) β†’ 𝑅 ∈ 𝑃)
5320ad3antrrr 729 . . . . . . . 8 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝐢 ∈ (𝑄𝐼𝑅)) β†’ 𝐢 ∈ 𝑃)
54 simpr 486 . . . . . . . . . . 11 (((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝐢 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐢) β†’ 𝑄 = 𝐢)
558, 9, 10, 11, 16, 20tgbtwntriv2 27718 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐢 ∈ (𝑅𝐼𝐢))
5655ad4antr 731 . . . . . . . . . . 11 (((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝐢 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐢) β†’ 𝐢 ∈ (𝑅𝐼𝐢))
5754, 56eqeltrd 2834 . . . . . . . . . 10 (((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝐢 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐢) β†’ 𝑄 ∈ (𝑅𝐼𝐢))
58 simpllr 775 . . . . . . . . . 10 (((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝐢 ∈ (𝑄𝐼𝑅)) ∧ 𝑄 = 𝐢) β†’ Β¬ 𝑄 ∈ (𝑅𝐼𝐢))
5957, 58pm2.65da 816 . . . . . . . . 9 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝐢 ∈ (𝑄𝐼𝑅)) β†’ Β¬ 𝑄 = 𝐢)
6059neqned 2948 . . . . . . . 8 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝐢 ∈ (𝑄𝐼𝑅)) β†’ 𝑄 β‰  𝐢)
6145ad3antrrr 729 . . . . . . . 8 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝐢 ∈ (𝑄𝐼𝑅)) β†’ 𝑄 ∈ (𝐡𝐼𝐢))
62 simpr 486 . . . . . . . 8 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝐢 ∈ (𝑄𝐼𝑅)) β†’ 𝐢 ∈ (𝑄𝐼𝑅))
638, 9, 10, 49, 50, 51, 53, 52, 60, 61, 62tgbtwnouttr 27728 . . . . . . 7 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝐢 ∈ (𝑄𝐼𝑅)) β†’ 𝑄 ∈ (𝐡𝐼𝑅))
648, 9, 10, 49, 50, 51, 52, 63tgbtwncom 27719 . . . . . 6 ((((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) ∧ 𝐢 ∈ (𝑄𝐼𝑅)) β†’ 𝑄 ∈ (𝑅𝐼𝐡))
65 outpasch.l . . . . . . . . . 10 𝐿 = (LineGβ€˜πΊ)
668, 65, 10, 11, 18, 20, 16tgcolg 27785 . . . . . . . . 9 (πœ‘ β†’ ((𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢) ↔ (𝑅 ∈ (𝑄𝐼𝐢) ∨ 𝑄 ∈ (𝑅𝐼𝐢) ∨ 𝐢 ∈ (𝑄𝐼𝑅))))
6766biimpa 478 . . . . . . . 8 ((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ (𝑅 ∈ (𝑄𝐼𝐢) ∨ 𝑄 ∈ (𝑅𝐼𝐢) ∨ 𝐢 ∈ (𝑄𝐼𝑅)))
68 3orcoma 1094 . . . . . . . . 9 ((𝑄 ∈ (𝑅𝐼𝐢) ∨ 𝑅 ∈ (𝑄𝐼𝐢) ∨ 𝐢 ∈ (𝑄𝐼𝑅)) ↔ (𝑅 ∈ (𝑄𝐼𝐢) ∨ 𝑄 ∈ (𝑅𝐼𝐢) ∨ 𝐢 ∈ (𝑄𝐼𝑅)))
69 3orass 1091 . . . . . . . . 9 ((𝑄 ∈ (𝑅𝐼𝐢) ∨ 𝑅 ∈ (𝑄𝐼𝐢) ∨ 𝐢 ∈ (𝑄𝐼𝑅)) ↔ (𝑄 ∈ (𝑅𝐼𝐢) ∨ (𝑅 ∈ (𝑄𝐼𝐢) ∨ 𝐢 ∈ (𝑄𝐼𝑅))))
7068, 69bitr3i 277 . . . . . . . 8 ((𝑅 ∈ (𝑄𝐼𝐢) ∨ 𝑄 ∈ (𝑅𝐼𝐢) ∨ 𝐢 ∈ (𝑄𝐼𝑅)) ↔ (𝑄 ∈ (𝑅𝐼𝐢) ∨ (𝑅 ∈ (𝑄𝐼𝐢) ∨ 𝐢 ∈ (𝑄𝐼𝑅))))
7167, 70sylib 217 . . . . . . 7 ((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ (𝑄 ∈ (𝑅𝐼𝐢) ∨ (𝑅 ∈ (𝑄𝐼𝐢) ∨ 𝐢 ∈ (𝑄𝐼𝑅))))
7271orcanai 1002 . . . . . 6 (((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ (𝑅 ∈ (𝑄𝐼𝐢) ∨ 𝐢 ∈ (𝑄𝐼𝑅)))
7348, 64, 72mpjaodan 958 . . . . 5 (((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ 𝑄 ∈ (𝑅𝐼𝐡))
7437, 73jca 513 . . . 4 (((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ (𝐡 ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼𝐡)))
7530, 35, 74rspcedvd 3614 . . 3 (((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝑄 ∈ (𝑅𝐼𝐢)) β†’ βˆƒπ‘₯ ∈ 𝑃 (π‘₯ ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼π‘₯)))
7629, 75pm2.61dan 812 . 2 ((πœ‘ ∧ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ βˆƒπ‘₯ ∈ 𝑃 (π‘₯ ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼π‘₯)))
7712ad2antrr 725 . . . 4 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐡 ∈ 𝑃)
7834adantl 483 . . . 4 ((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ = 𝐡) β†’ ((π‘₯ ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼π‘₯)) ↔ (𝐡 ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼𝐡))))
7936ad2antrr 725 . . . . 5 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐡 ∈ (𝐴𝐼𝐡))
8011ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐺 ∈ TarskiG)
8116ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝑅 ∈ 𝑃)
8218ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝑄 ∈ 𝑃)
8320ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐢 ∈ 𝑃)
84 simplr 768 . . . . . . 7 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢))
85 simpr 486 . . . . . . 7 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐡 ∈ (𝑅𝐿𝑄))
8611adantr 482 . . . . . . . . 9 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ 𝐺 ∈ TarskiG)
8716adantr 482 . . . . . . . . 9 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ 𝑅 ∈ 𝑃)
8818adantr 482 . . . . . . . . 9 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ 𝑄 ∈ 𝑃)
8920adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ 𝐢 ∈ 𝑃)
90 simpr 486 . . . . . . . . . 10 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢))
918, 10, 65, 86, 87, 88, 89, 90ncolne1 27856 . . . . . . . . 9 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ 𝑅 β‰  𝑄)
928, 10, 65, 86, 87, 88, 91tglinerflx2 27865 . . . . . . . 8 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ 𝑄 ∈ (𝑅𝐿𝑄))
9392adantr 482 . . . . . . 7 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝑄 ∈ (𝑅𝐿𝑄))
948, 65, 10, 86, 88, 89, 87, 90ncolcom 27792 . . . . . . . . . . 11 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ Β¬ (𝑅 ∈ (𝐢𝐿𝑄) ∨ 𝐢 = 𝑄))
958, 65, 10, 86, 89, 88, 87, 94ncolrot1 27793 . . . . . . . . . 10 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ Β¬ (𝐢 ∈ (𝑄𝐿𝑅) ∨ 𝑄 = 𝑅))
968, 10, 65, 86, 89, 88, 87, 95ncolne1 27856 . . . . . . . . 9 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ 𝐢 β‰  𝑄)
9796adantr 482 . . . . . . . 8 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐢 β‰  𝑄)
9846ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝑄 ∈ (𝐢𝐼𝐡))
998, 10, 65, 80, 83, 82, 77, 97, 98btwnlng3 27852 . . . . . . 7 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐡 ∈ (𝐢𝐿𝑄))
1008, 10, 65, 80, 83, 82, 97tglinerflx2 27865 . . . . . . 7 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝑄 ∈ (𝐢𝐿𝑄))
1018, 10, 65, 80, 81, 82, 83, 82, 84, 85, 93, 99, 100tglineinteq 27876 . . . . . 6 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐡 = 𝑄)
1028, 9, 10, 11, 16, 12tgbtwntriv2 27718 . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ (𝑅𝐼𝐡))
103102ad2antrr 725 . . . . . 6 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐡 ∈ (𝑅𝐼𝐡))
104101, 103eqeltrrd 2835 . . . . 5 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝑄 ∈ (𝑅𝐼𝐡))
10579, 104jca 513 . . . 4 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ (𝐡 ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼𝐡)))
10677, 78, 105rspcedvd 3614 . . 3 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ βˆƒπ‘₯ ∈ 𝑃 (π‘₯ ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼π‘₯)))
107 eleq1 2822 . . . . . . . . . 10 (𝑑 = π‘₯ β†’ (𝑑 ∈ (π‘ŽπΌπ‘) ↔ π‘₯ ∈ (π‘ŽπΌπ‘)))
108107cbvrexvw 3236 . . . . . . . . 9 (βˆƒπ‘‘ ∈ (𝑅𝐿𝑄)𝑑 ∈ (π‘ŽπΌπ‘) ↔ βˆƒπ‘₯ ∈ (𝑅𝐿𝑄)π‘₯ ∈ (π‘ŽπΌπ‘))
109108anbi2i 624 . . . . . . . 8 (((π‘Ž ∈ (𝑃 βˆ– (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 βˆ– (𝑅𝐿𝑄))) ∧ βˆƒπ‘‘ ∈ (𝑅𝐿𝑄)𝑑 ∈ (π‘ŽπΌπ‘)) ↔ ((π‘Ž ∈ (𝑃 βˆ– (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 βˆ– (𝑅𝐿𝑄))) ∧ βˆƒπ‘₯ ∈ (𝑅𝐿𝑄)π‘₯ ∈ (π‘ŽπΌπ‘)))
110109opabbii 5214 . . . . . . 7 {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 βˆ– (𝑅𝐿𝑄))) ∧ βˆƒπ‘‘ ∈ (𝑅𝐿𝑄)𝑑 ∈ (π‘ŽπΌπ‘))} = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 βˆ– (𝑅𝐿𝑄))) ∧ βˆƒπ‘₯ ∈ (𝑅𝐿𝑄)π‘₯ ∈ (π‘ŽπΌπ‘))}
11111ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐺 ∈ TarskiG)
11216ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝑅 ∈ 𝑃)
11318ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝑄 ∈ 𝑃)
11491adantr 482 . . . . . . . 8 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝑅 β‰  𝑄)
1158, 10, 65, 111, 112, 113, 114tgelrnln 27861 . . . . . . 7 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ (𝑅𝐿𝑄) ∈ ran 𝐿)
116 eqid 2733 . . . . . . 7 (hlGβ€˜πΊ) = (hlGβ€˜πΊ)
11720ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐢 ∈ 𝑃)
1181ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐴 ∈ 𝑃)
11912ad2antrr 725 . . . . . . 7 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐡 ∈ 𝑃)
12092adantr 482 . . . . . . . 8 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝑄 ∈ (𝑅𝐿𝑄))
1218, 65, 10, 86, 88, 89, 87, 90ncolrot2 27794 . . . . . . . . . 10 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ Β¬ (𝐢 ∈ (𝑅𝐿𝑄) ∨ 𝑅 = 𝑄))
122 pm2.45 881 . . . . . . . . . 10 (Β¬ (𝐢 ∈ (𝑅𝐿𝑄) ∨ 𝑅 = 𝑄) β†’ Β¬ 𝐢 ∈ (𝑅𝐿𝑄))
123121, 122syl 17 . . . . . . . . 9 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ Β¬ 𝐢 ∈ (𝑅𝐿𝑄))
124123adantr 482 . . . . . . . 8 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ Β¬ 𝐢 ∈ (𝑅𝐿𝑄))
125 simpr 486 . . . . . . . 8 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ Β¬ 𝐡 ∈ (𝑅𝐿𝑄))
12646ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝑄 ∈ (𝐢𝐼𝐡))
1278, 9, 10, 110, 117, 119, 120, 124, 125, 126islnoppd 27971 . . . . . . 7 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐢{βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 βˆ– (𝑅𝐿𝑄))) ∧ βˆƒπ‘‘ ∈ (𝑅𝐿𝑄)𝑑 ∈ (π‘ŽπΌπ‘))}𝐡)
1288, 10, 65, 86, 87, 88, 91tglinerflx1 27864 . . . . . . . 8 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ 𝑅 ∈ (𝑅𝐿𝑄))
129128adantr 482 . . . . . . 7 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝑅 ∈ (𝑅𝐿𝑄))
13024ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐢 ∈ (𝑅𝐼𝐴))
1318, 10, 65, 86, 89, 87, 88, 121ncolne1 27856 . . . . . . . . . 10 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ 𝐢 β‰  𝑅)
132131adantr 482 . . . . . . . . 9 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐢 β‰  𝑅)
1338, 9, 10, 111, 112, 117, 118, 130, 132tgbtwnne 27721 . . . . . . . 8 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝑅 β‰  𝐴)
1348, 10, 116, 112, 118, 117, 111, 118, 130, 133, 132btwnhl1 27843 . . . . . . 7 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐢((hlGβ€˜πΊ)β€˜π‘…)𝐴)
1358, 9, 10, 110, 65, 115, 111, 116, 117, 118, 119, 127, 129, 134opphl 27985 . . . . . 6 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ 𝐴{βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 βˆ– (𝑅𝐿𝑄))) ∧ βˆƒπ‘‘ ∈ (𝑅𝐿𝑄)𝑑 ∈ (π‘ŽπΌπ‘))}𝐡)
1368, 9, 10, 110, 118, 119islnopp 27970 . . . . . 6 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ (𝐴{βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– (𝑅𝐿𝑄)) ∧ 𝑏 ∈ (𝑃 βˆ– (𝑅𝐿𝑄))) ∧ βˆƒπ‘‘ ∈ (𝑅𝐿𝑄)𝑑 ∈ (π‘ŽπΌπ‘))}𝐡 ↔ ((Β¬ 𝐴 ∈ (𝑅𝐿𝑄) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ βˆƒπ‘₯ ∈ (𝑅𝐿𝑄)π‘₯ ∈ (𝐴𝐼𝐡))))
137135, 136mpbid 231 . . . . 5 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ ((Β¬ 𝐴 ∈ (𝑅𝐿𝑄) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ βˆƒπ‘₯ ∈ (𝑅𝐿𝑄)π‘₯ ∈ (𝐴𝐼𝐡)))
138137simprd 497 . . . 4 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ βˆƒπ‘₯ ∈ (𝑅𝐿𝑄)π‘₯ ∈ (𝐴𝐼𝐡))
139111ad2antrr 725 . . . . . . . 8 (((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) β†’ 𝐺 ∈ TarskiG)
140115ad2antrr 725 . . . . . . . 8 (((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) β†’ (𝑅𝐿𝑄) ∈ ran 𝐿)
141 simplr 768 . . . . . . . 8 (((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) β†’ π‘₯ ∈ (𝑅𝐿𝑄))
1428, 65, 10, 139, 140, 141tglnpt 27780 . . . . . . 7 (((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) β†’ π‘₯ ∈ 𝑃)
143 simpr 486 . . . . . . 7 (((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) β†’ π‘₯ ∈ (𝐴𝐼𝐡))
144139ad2antrr 725 . . . . . . . . . 10 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝐺 ∈ TarskiG)
14587ad3antrrr 729 . . . . . . . . . . 11 (((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) β†’ 𝑅 ∈ 𝑃)
146145ad2antrr 725 . . . . . . . . . 10 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝑅 ∈ 𝑃)
14788ad5antr 733 . . . . . . . . . 10 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝑄 ∈ 𝑃)
148117ad2antrr 725 . . . . . . . . . . 11 (((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) β†’ 𝐢 ∈ 𝑃)
149148ad2antrr 725 . . . . . . . . . 10 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝐢 ∈ 𝑃)
15090ad5antr 733 . . . . . . . . . 10 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢))
151 simplr 768 . . . . . . . . . . 11 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝑑 ∈ 𝑃)
152114ad4antr 731 . . . . . . . . . . 11 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝑅 β‰  𝑄)
153142ad2antrr 725 . . . . . . . . . . . 12 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ π‘₯ ∈ 𝑃)
15491necomd 2997 . . . . . . . . . . . . . 14 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ 𝑄 β‰  𝑅)
155154ad5antr 733 . . . . . . . . . . . . 13 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝑄 β‰  𝑅)
156141ad2antrr 725 . . . . . . . . . . . . 13 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ π‘₯ ∈ (𝑅𝐿𝑄))
1578, 10, 65, 144, 147, 146, 153, 155, 156lncom 27853 . . . . . . . . . . . 12 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ π‘₯ ∈ (𝑄𝐿𝑅))
158 simprl 770 . . . . . . . . . . . 12 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝑑 ∈ (π‘₯𝐼𝑅))
1598, 10, 65, 144, 153, 147, 146, 151, 157, 158coltr3 27879 . . . . . . . . . . 11 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝑑 ∈ (𝑄𝐿𝑅))
1608, 10, 65, 144, 146, 147, 151, 152, 159lncom 27853 . . . . . . . . . 10 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝑑 ∈ (𝑅𝐿𝑄))
16192ad5antr 733 . . . . . . . . . 10 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝑄 ∈ (𝑅𝐿𝑄))
16296ad5antr 733 . . . . . . . . . . 11 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝐢 β‰  𝑄)
163119ad2antrr 725 . . . . . . . . . . . . 13 (((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) β†’ 𝐡 ∈ 𝑃)
164163ad2antrr 725 . . . . . . . . . . . 12 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝐡 ∈ 𝑃)
16512adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ 𝐡 ∈ 𝑃)
16696necomd 2997 . . . . . . . . . . . . . 14 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ 𝑄 β‰  𝐢)
16745adantr 482 . . . . . . . . . . . . . 14 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ 𝑄 ∈ (𝐡𝐼𝐢))
1688, 10, 65, 86, 88, 89, 165, 166, 167btwnlng2 27851 . . . . . . . . . . . . 13 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ 𝐡 ∈ (𝑄𝐿𝐢))
169168ad5antr 733 . . . . . . . . . . . 12 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝐡 ∈ (𝑄𝐿𝐢))
170 simprr 772 . . . . . . . . . . . . 13 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝑑 ∈ (𝐢𝐼𝐡))
1718, 9, 10, 144, 149, 151, 164, 170tgbtwncom 27719 . . . . . . . . . . . 12 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝑑 ∈ (𝐡𝐼𝐢))
1728, 10, 65, 144, 164, 147, 149, 151, 169, 171coltr3 27879 . . . . . . . . . . 11 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝑑 ∈ (𝑄𝐿𝐢))
1738, 10, 65, 144, 149, 147, 151, 162, 172lncom 27853 . . . . . . . . . 10 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝑑 ∈ (𝐢𝐿𝑄))
1748, 10, 65, 86, 89, 88, 96tglinerflx2 27865 . . . . . . . . . . 11 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ 𝑄 ∈ (𝐢𝐿𝑄))
175174ad5antr 733 . . . . . . . . . 10 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝑄 ∈ (𝐢𝐿𝑄))
1768, 10, 65, 144, 146, 147, 149, 147, 150, 160, 161, 173, 175tglineinteq 27876 . . . . . . . . 9 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝑑 = 𝑄)
1778, 9, 10, 144, 153, 151, 146, 158tgbtwncom 27719 . . . . . . . . 9 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝑑 ∈ (𝑅𝐼π‘₯))
178176, 177eqeltrrd 2835 . . . . . . . 8 (((((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) ∧ 𝑑 ∈ 𝑃) ∧ (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡))) β†’ 𝑄 ∈ (𝑅𝐼π‘₯))
179118ad2antrr 725 . . . . . . . . 9 (((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) β†’ 𝐴 ∈ 𝑃)
1808, 9, 10, 139, 179, 142, 163, 143tgbtwncom 27719 . . . . . . . . 9 (((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) β†’ π‘₯ ∈ (𝐡𝐼𝐴))
18124ad4antr 731 . . . . . . . . 9 (((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) β†’ 𝐢 ∈ (𝑅𝐼𝐴))
1828, 9, 10, 139, 163, 145, 179, 142, 148, 180, 181axtgpasch 27698 . . . . . . . 8 (((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) β†’ βˆƒπ‘‘ ∈ 𝑃 (𝑑 ∈ (π‘₯𝐼𝑅) ∧ 𝑑 ∈ (𝐢𝐼𝐡)))
183178, 182r19.29a 3163 . . . . . . 7 (((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) β†’ 𝑄 ∈ (𝑅𝐼π‘₯))
184142, 143, 183jca32 517 . . . . . 6 (((((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝑅𝐿𝑄)) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) β†’ (π‘₯ ∈ 𝑃 ∧ (π‘₯ ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼π‘₯))))
185184expl 459 . . . . 5 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ ((π‘₯ ∈ (𝑅𝐿𝑄) ∧ π‘₯ ∈ (𝐴𝐼𝐡)) β†’ (π‘₯ ∈ 𝑃 ∧ (π‘₯ ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼π‘₯)))))
186185reximdv2 3165 . . . 4 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ (βˆƒπ‘₯ ∈ (𝑅𝐿𝑄)π‘₯ ∈ (𝐴𝐼𝐡) β†’ βˆƒπ‘₯ ∈ 𝑃 (π‘₯ ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼π‘₯))))
187138, 186mpd 15 . . 3 (((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) ∧ Β¬ 𝐡 ∈ (𝑅𝐿𝑄)) β†’ βˆƒπ‘₯ ∈ 𝑃 (π‘₯ ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼π‘₯)))
188106, 187pm2.61dan 812 . 2 ((πœ‘ ∧ Β¬ (𝑅 ∈ (𝑄𝐿𝐢) ∨ 𝑄 = 𝐢)) β†’ βˆƒπ‘₯ ∈ 𝑃 (π‘₯ ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼π‘₯)))
18976, 188pm2.61dan 812 1 (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑃 (π‘₯ ∈ (𝐴𝐼𝐡) ∧ 𝑄 ∈ (𝑅𝐼π‘₯)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∨ w3o 1087   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆƒwrex 3071   βˆ– cdif 3944   class class class wbr 5147  {copab 5209  ran crn 5676  β€˜cfv 6540  (class class class)co 7404  Basecbs 17140  distcds 17202  TarskiGcstrkg 27658  Itvcitv 27664  LineGclng 27665  hlGchlg 27831
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-xnn0 12541  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-hash 14287  df-word 14461  df-concat 14517  df-s1 14542  df-s2 14795  df-s3 14796  df-trkgc 27679  df-trkgb 27680  df-trkgcb 27681  df-trkgld 27683  df-trkg 27684  df-cgrg 27742  df-leg 27814  df-hlg 27832  df-mir 27884  df-rag 27925  df-perpg 27927
This theorem is referenced by:  hlpasch  27987
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