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Theorem opphllem 26513
Description: Lemma 8.24 of [Schwabhauser] p. 66. This is used later for mideulem 26514 and later for opphl 26532. (Contributed by Thierry Arnoux, 21-Dec-2019.)
Hypotheses
Ref Expression
colperpex.p 𝑃 = (Base‘𝐺)
colperpex.d = (dist‘𝐺)
colperpex.i 𝐼 = (Itv‘𝐺)
colperpex.l 𝐿 = (LineG‘𝐺)
colperpex.g (𝜑𝐺 ∈ TarskiG)
mideu.s 𝑆 = (pInvG‘𝐺)
mideu.1 (𝜑𝐴𝑃)
mideu.2 (𝜑𝐵𝑃)
mideulem.1 (𝜑𝐴𝐵)
mideulem.2 (𝜑𝑄𝑃)
mideulem.3 (𝜑𝑂𝑃)
mideulem.4 (𝜑𝑇𝑃)
mideulem.5 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))
mideulem.6 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
mideulem.7 (𝜑𝑇 ∈ (𝐴𝐿𝐵))
mideulem.8 (𝜑𝑇 ∈ (𝑄𝐼𝑂))
opphllem.1 (𝜑𝑅𝑃)
opphllem.2 (𝜑𝑅 ∈ (𝐵𝐼𝑄))
opphllem.3 (𝜑 → (𝐴 𝑂) = (𝐵 𝑅))
Assertion
Ref Expression
opphllem (𝜑 → ∃𝑥𝑃 (𝐵 = ((𝑆𝑥)‘𝐴) ∧ 𝑂 = ((𝑆𝑥)‘𝑅)))
Distinct variable groups:   𝑥,   𝑥,𝐴   𝑥,𝐵   𝑥,𝐼   𝑥,𝑂   𝑥,𝑃   𝑥,𝑄   𝑥,𝑅   𝑥,𝑇   𝜑,𝑥
Allowed substitution hints:   𝑆(𝑥)   𝐺(𝑥)   𝐿(𝑥)

Proof of Theorem opphllem
Dummy variables 𝑚 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 colperpex.p . . . 4 𝑃 = (Base‘𝐺)
2 colperpex.d . . . 4 = (dist‘𝐺)
3 colperpex.i . . . 4 𝐼 = (Itv‘𝐺)
4 colperpex.l . . . 4 𝐿 = (LineG‘𝐺)
5 mideu.s . . . 4 𝑆 = (pInvG‘𝐺)
6 colperpex.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
76adantr 483 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐺 ∈ TarskiG)
8 eqid 2819 . . . 4 (𝑆𝑥) = (𝑆𝑥)
9 mideu.2 . . . . 5 (𝜑𝐵𝑃)
109adantr 483 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐵𝑃)
11 mideulem.3 . . . . 5 (𝜑𝑂𝑃)
1211adantr 483 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑂𝑃)
13 mideu.1 . . . . 5 (𝜑𝐴𝑃)
1413adantr 483 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐴𝑃)
15 opphllem.1 . . . . 5 (𝜑𝑅𝑃)
1615adantr 483 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑅𝑃)
17 simprl 769 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥𝑃)
18 mideulem.1 . . . . . . . . . . . . . 14 (𝜑𝐴𝐵)
1918necomd 3069 . . . . . . . . . . . . 13 (𝜑𝐵𝐴)
2019neneqd 3019 . . . . . . . . . . . 12 (𝜑 → ¬ 𝐵 = 𝐴)
2120adantr 483 . . . . . . . . . . 11 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝐵 = 𝐴)
22 mideulem.6 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
234, 6, 22perpln2 26489 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴𝐿𝑂) ∈ ran 𝐿)
241, 3, 4, 6, 13, 11, 23tglnne 26406 . . . . . . . . . . . . . 14 (𝜑𝐴𝑂)
2524necomd 3069 . . . . . . . . . . . . 13 (𝜑𝑂𝐴)
2625neneqd 3019 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑂 = 𝐴)
2726adantr 483 . . . . . . . . . . 11 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝑂 = 𝐴)
2821, 27jca 514 . . . . . . . . . 10 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
296adantr 483 . . . . . . . . . . . 12 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐺 ∈ TarskiG)
309adantr 483 . . . . . . . . . . . 12 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐵𝑃)
3113adantr 483 . . . . . . . . . . . 12 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐴𝑃)
3211adantr 483 . . . . . . . . . . . 12 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝑂𝑃)
331, 3, 4, 6, 9, 13, 19tglinerflx2 26412 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ (𝐵𝐿𝐴))
341, 3, 4, 6, 13, 9, 18tglinecom 26413 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
3534, 22eqbrtrrd 5081 . . . . . . . . . . . . . 14 (𝜑 → (𝐵𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝑂))
361, 2, 3, 4, 6, 9, 13, 33, 11, 35perprag 26504 . . . . . . . . . . . . 13 (𝜑 → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
3736adantr 483 . . . . . . . . . . . 12 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
38 simpr 487 . . . . . . . . . . . . 13 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝑂 ∈ (𝐵𝐿𝐴))
3938orcd 869 . . . . . . . . . . . 12 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (𝑂 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
401, 2, 3, 4, 5, 29, 30, 31, 32, 37, 39ragflat3 26484 . . . . . . . . . . 11 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (𝐵 = 𝐴𝑂 = 𝐴))
41 oran 986 . . . . . . . . . . 11 ((𝐵 = 𝐴𝑂 = 𝐴) ↔ ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
4240, 41sylib 220 . . . . . . . . . 10 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
4328, 42pm2.65da 815 . . . . . . . . 9 (𝜑 → ¬ 𝑂 ∈ (𝐵𝐿𝐴))
4443adantr 483 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝑂 ∈ (𝐵𝐿𝐴))
4534adantr 483 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
4644, 45neleqtrrd 2933 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
4718adantr 483 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐴𝐵)
4847neneqd 3019 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝐴 = 𝐵)
4946, 48jca 514 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (¬ 𝑂 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵))
50 pm4.56 985 . . . . . 6 ((¬ 𝑂 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵) ↔ ¬ (𝑂 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
5149, 50sylib 220 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ (𝑂 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
521, 4, 3, 7, 14, 10, 12, 51ncolrot2 26341 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ (𝐵 ∈ (𝑂𝐿𝐴) ∨ 𝑂 = 𝐴))
53 simprrr 780 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝑅𝐼𝑂))
541, 2, 3, 7, 16, 17, 12, 53tgbtwncom 26266 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝑂𝐼𝑅))
55 mideulem.4 . . . . . . . 8 (𝜑𝑇𝑃)
5655adantr 483 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑇𝑃)
57 mideulem.7 . . . . . . . 8 (𝜑𝑇 ∈ (𝐴𝐿𝐵))
5857adantr 483 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑇 ∈ (𝐴𝐿𝐵))
59 simprrl 779 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝑇𝐼𝐵))
601, 3, 4, 7, 56, 14, 10, 17, 58, 59coltr3 26426 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐴𝐿𝐵))
6143, 34neleqtrrd 2933 . . . . . . 7 (𝜑 → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
6261adantr 483 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
63 nelne2 3113 . . . . . 6 ((𝑥 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝑂 ∈ (𝐴𝐿𝐵)) → 𝑥𝑂)
6460, 62, 63syl2anc 586 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥𝑂)
651, 2, 3, 7, 12, 17, 16, 54, 64tgbtwnne 26268 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑂𝑅)
661, 2, 3, 4, 5, 6, 9, 13, 11israg 26475 . . . . . . . 8 (𝜑 → (⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 𝑂) = (𝐵 ((𝑆𝐴)‘𝑂))))
6736, 66mpbid 234 . . . . . . 7 (𝜑 → (𝐵 𝑂) = (𝐵 ((𝑆𝐴)‘𝑂)))
6867ad3antrrr 728 . . . . . 6 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝐵 𝑂) = (𝐵 ((𝑆𝐴)‘𝑂)))
696ad3antrrr 728 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝐺 ∈ TarskiG)
70 eqid 2819 . . . . . . . . 9 (𝑆𝐴) = (𝑆𝐴)
711, 2, 3, 4, 5, 7, 14, 70, 12mircl 26439 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ((𝑆𝐴)‘𝑂) ∈ 𝑃)
7271ad2antrr 724 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → ((𝑆𝐴)‘𝑂) ∈ 𝑃)
7313ad3antrrr 728 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝐴𝑃)
7411ad3antrrr 728 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝑂𝑃)
7515ad3antrrr 728 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝑅𝑃)
769ad3antrrr 728 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝐵𝑃)
77 simplr 767 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝑠𝑃)
781, 2, 3, 4, 5, 69, 73, 70, 74mirbtwn 26436 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝐴 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑂))
79 eqid 2819 . . . . . . . . 9 (𝑆𝐵) = (𝑆𝐵)
801, 2, 3, 4, 5, 69, 76, 79, 77mirbtwn 26436 . . . . . . . 8 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝐵 ∈ (((𝑆𝐵)‘𝑠)𝐼𝑠))
81 simpr 487 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑅 = ((𝑆𝑚)‘𝑠))
8269ad2antrr 724 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝐺 ∈ TarskiG)
8373ad2antrr 724 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝐴𝑃)
8476ad2antrr 724 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝐵𝑃)
8547ad4antr 730 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝐴𝐵)
86 mideulem.2 . . . . . . . . . . . . . . . 16 (𝜑𝑄𝑃)
8786ad5antr 732 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑄𝑃)
8874ad2antrr 724 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑂𝑃)
8956ad4antr 730 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑇𝑃)
90 mideulem.5 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))
9190ad5antr 732 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))
9222ad5antr 732 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
9358ad4antr 730 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑇 ∈ (𝐴𝐿𝐵))
94 mideulem.8 . . . . . . . . . . . . . . . 16 (𝜑𝑇 ∈ (𝑄𝐼𝑂))
9594ad5antr 732 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑇 ∈ (𝑄𝐼𝑂))
9675ad2antrr 724 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑅𝑃)
97 opphllem.2 . . . . . . . . . . . . . . . 16 (𝜑𝑅 ∈ (𝐵𝐼𝑄))
9897ad5antr 732 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑅 ∈ (𝐵𝐼𝑄))
99 opphllem.3 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴 𝑂) = (𝐵 𝑅))
10099ad5antr 732 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → (𝐴 𝑂) = (𝐵 𝑅))
10117ad2antrr 724 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝑥𝑃)
102101ad2antrr 724 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑥𝑃)
103 simp-5r 784 . . . . . . . . . . . . . . . . 17 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂))))
104103simprd 498 . . . . . . . . . . . . . . . 16 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))
105104simpld 497 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑥 ∈ (𝑇𝐼𝐵))
106104simprd 498 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑥 ∈ (𝑅𝐼𝑂))
10777ad2antrr 724 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑠𝑃)
108 simpllr 774 . . . . . . . . . . . . . . . 16 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅)))
109108simpld 497 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠))
110 simprr 771 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑥 𝑠) = (𝑥 𝑅))
111110ad2antrr 724 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → (𝑥 𝑠) = (𝑥 𝑅))
112 simplr 767 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑚𝑃)
1131, 2, 3, 4, 82, 5, 83, 84, 85, 87, 88, 89, 91, 92, 93, 95, 96, 98, 100, 102, 105, 106, 107, 109, 111, 112, 81mideulem2 26512 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝐵 = 𝑚)
114113eqcomd 2825 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑚 = 𝐵)
115114fveq2d 6667 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → (𝑆𝑚) = (𝑆𝐵))
116115fveq1d 6665 . . . . . . . . . . 11 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → ((𝑆𝑚)‘𝑠) = ((𝑆𝐵)‘𝑠))
11781, 116eqtrd 2854 . . . . . . . . . 10 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑅 = ((𝑆𝐵)‘𝑠))
118 eqid 2819 . . . . . . . . . . 11 (𝑆𝑚) = (𝑆𝑚)
1191, 2, 3, 4, 5, 69, 118, 77, 75, 101, 110midexlem 26470 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → ∃𝑚𝑃 𝑅 = ((𝑆𝑚)‘𝑠))
120117, 119r19.29a 3287 . . . . . . . . 9 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝑅 = ((𝑆𝐵)‘𝑠))
121120oveq1d 7163 . . . . . . . 8 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑅𝐼𝑠) = (((𝑆𝐵)‘𝑠)𝐼𝑠))
12280, 121eleqtrrd 2914 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝐵 ∈ (𝑅𝐼𝑠))
1231, 2, 3, 4, 5, 69, 73, 70, 74mircgr 26435 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝐴 ((𝑆𝐴)‘𝑂)) = (𝐴 𝑂))
12499ad3antrrr 728 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝐴 𝑂) = (𝐵 𝑅))
125123, 124eqtrd 2854 . . . . . . . . 9 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝐴 ((𝑆𝐴)‘𝑂)) = (𝐵 𝑅))
1261, 2, 3, 69, 73, 72, 76, 75, 125tgcgrcomlr 26258 . . . . . . . 8 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (((𝑆𝐴)‘𝑂) 𝐴) = (𝑅 𝐵))
127120oveq2d 7164 . . . . . . . . 9 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝐵 𝑅) = (𝐵 ((𝑆𝐵)‘𝑠)))
1281, 2, 3, 4, 5, 69, 76, 79, 77mircgr 26435 . . . . . . . . 9 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝐵 ((𝑆𝐵)‘𝑠)) = (𝐵 𝑠))
129124, 127, 1283eqtrd 2858 . . . . . . . 8 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝐴 𝑂) = (𝐵 𝑠))
1301, 2, 3, 69, 72, 73, 74, 75, 76, 77, 78, 122, 126, 129tgcgrextend 26263 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (((𝑆𝐴)‘𝑂) 𝑂) = (𝑅 𝑠))
1311, 2, 3, 69, 72, 75axtgcgrrflx 26240 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (((𝑆𝐴)‘𝑂) 𝑅) = (𝑅 ((𝑆𝐴)‘𝑂)))
1321, 2, 3, 69, 74, 75axtgcgrrflx 26240 . . . . . . . 8 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑂 𝑅) = (𝑅 𝑂))
13353ad2antrr 724 . . . . . . . . 9 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝑥 ∈ (𝑅𝐼𝑂))
134 simprl 769 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠))
1351, 2, 3, 69, 72, 101, 77, 134tgbtwncom 26266 . . . . . . . . 9 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝑥 ∈ (𝑠𝐼((𝑆𝐴)‘𝑂)))
1361, 2, 3, 69, 101, 77, 101, 75, 110tgcgrcomlr 26258 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑠 𝑥) = (𝑅 𝑥))
137136eqcomd 2825 . . . . . . . . 9 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑅 𝑥) = (𝑠 𝑥))
13836ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
13947necomd 3069 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐵𝐴)
140139ad2antrr 724 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝐵𝐴)
14160ad2antrr 724 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝑥 ∈ (𝐴𝐿𝐵))
142141orcd 869 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑥 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
1431, 4, 3, 69, 73, 76, 101, 142colcom 26336 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑥 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
1441, 4, 3, 69, 76, 73, 101, 143colrot1 26337 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝐵 ∈ (𝐴𝐿𝑥) ∨ 𝐴 = 𝑥))
1451, 2, 3, 4, 5, 69, 76, 73, 74, 101, 138, 140, 144ragcol 26477 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → ⟨“𝑥𝐴𝑂”⟩ ∈ (∟G‘𝐺))
1461, 2, 3, 4, 5, 69, 101, 73, 74israg 26475 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (⟨“𝑥𝐴𝑂”⟩ ∈ (∟G‘𝐺) ↔ (𝑥 𝑂) = (𝑥 ((𝑆𝐴)‘𝑂))))
147145, 146mpbid 234 . . . . . . . . 9 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑥 𝑂) = (𝑥 ((𝑆𝐴)‘𝑂)))
1481, 2, 3, 69, 75, 101, 74, 77, 101, 72, 133, 135, 137, 147tgcgrextend 26263 . . . . . . . 8 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑅 𝑂) = (𝑠 ((𝑆𝐴)‘𝑂)))
149132, 148eqtrd 2854 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑂 𝑅) = (𝑠 ((𝑆𝐴)‘𝑂)))
1501, 2, 3, 69, 72, 73, 74, 75, 75, 76, 77, 72, 78, 122, 130, 129, 131, 149tgifscgr 26286 . . . . . 6 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝐴 𝑅) = (𝐵 ((𝑆𝐴)‘𝑂)))
15168, 150eqtr4d 2857 . . . . 5 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝐵 𝑂) = (𝐴 𝑅))
1521, 2, 3, 7, 71, 17, 17, 16axtgsegcon 26242 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ∃𝑠𝑃 (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅)))
153151, 152r19.29a 3287 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐵 𝑂) = (𝐴 𝑅))
15499adantr 483 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐴 𝑂) = (𝐵 𝑅))
1551, 2, 3, 7, 14, 12, 10, 16, 154tgcgrcomlr 26258 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑂 𝐴) = (𝑅 𝐵))
156143, 152r19.29a 3287 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
1571, 4, 3, 7, 12, 16, 17, 54btwncolg1 26333 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 ∈ (𝑂𝐿𝑅) ∨ 𝑂 = 𝑅))
1581, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 17, 52, 65, 153, 155, 156, 157symquadlem 26467 . . 3 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐵 = ((𝑆𝑥)‘𝐴))
1591, 2, 3, 4, 5, 7, 17, 8, 14mirbtwn 26436 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (((𝑆𝑥)‘𝐴)𝐼𝐴))
160158oveq1d 7163 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐵𝐼𝐴) = (((𝑆𝑥)‘𝐴)𝐼𝐴))
161159, 160eleqtrrd 2914 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐵𝐼𝐴))
1621, 2, 3, 7, 10, 17, 14, 161tgbtwncom 26266 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐴𝐼𝐵))
1631, 2, 3, 7, 14, 10axtgcgrrflx 26240 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐴 𝐵) = (𝐵 𝐴))
164158oveq2d 7164 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 𝐵) = (𝑥 ((𝑆𝑥)‘𝐴)))
1651, 2, 3, 4, 5, 7, 17, 8, 14mircgr 26435 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 ((𝑆𝑥)‘𝐴)) = (𝑥 𝐴))
166164, 165eqtrd 2854 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 𝐵) = (𝑥 𝐴))
1671, 2, 3, 7, 14, 17, 10, 12, 10, 17, 14, 16, 162, 161, 163, 166, 154, 153tgifscgr 26286 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 𝑂) = (𝑥 𝑅))
1681, 2, 3, 4, 5, 7, 17, 8, 16, 12, 167, 54ismir 26437 . . 3 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑂 = ((𝑆𝑥)‘𝑅))
169158, 168jca 514 . 2 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐵 = ((𝑆𝑥)‘𝐴) ∧ 𝑂 = ((𝑆𝑥)‘𝑅)))
1701, 2, 3, 6, 86, 55, 11, 94tgbtwncom 26266 . . 3 (𝜑𝑇 ∈ (𝑂𝐼𝑄))
1711, 2, 3, 6, 11, 9, 86, 55, 15, 170, 97axtgpasch 26245 . 2 (𝜑 → ∃𝑥𝑃 (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))
172169, 171reximddv 3273 1 (𝜑 → ∃𝑥𝑃 (𝐵 = ((𝑆𝑥)‘𝐴) ∧ 𝑂 = ((𝑆𝑥)‘𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wo 843   = wceq 1531  wcel 2108  wne 3014  wrex 3137   class class class wbr 5057  cfv 6348  (class class class)co 7148  ⟨“cs3 14196  Basecbs 16475  distcds 16566  TarskiGcstrkg 26208  Itvcitv 26214  LineGclng 26215  pInvGcmir 26430  ∟Gcrag 26471  ⟂Gcperpg 26473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  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 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  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-trkg 26231  df-cgrg 26289  df-leg 26361  df-mir 26431  df-rag 26472  df-perpg 26474
This theorem is referenced by:  mideulem  26514  opphllem3  26527
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