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Theorem opphllem 27677
Description: Lemma 8.24 of [Schwabhauser] p. 66. This is used later for mideulem 27678 and later for opphl 27696. (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 481 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐺 ∈ TarskiG)
8 eqid 2736 . . . 4 (𝑆𝑥) = (𝑆𝑥)
9 mideu.2 . . . . 5 (𝜑𝐵𝑃)
109adantr 481 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐵𝑃)
11 mideulem.3 . . . . 5 (𝜑𝑂𝑃)
1211adantr 481 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑂𝑃)
13 mideu.1 . . . . 5 (𝜑𝐴𝑃)
1413adantr 481 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐴𝑃)
15 opphllem.1 . . . . 5 (𝜑𝑅𝑃)
1615adantr 481 . . . 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 2999 . . . . . . . . . . . . 13 (𝜑𝐵𝐴)
2019neneqd 2948 . . . . . . . . . . . 12 (𝜑 → ¬ 𝐵 = 𝐴)
2120adantr 481 . . . . . . . . . . 11 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝐵 = 𝐴)
22 mideulem.6 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
234, 6, 22perpln2 27653 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴𝐿𝑂) ∈ ran 𝐿)
241, 3, 4, 6, 13, 11, 23tglnne 27570 . . . . . . . . . . . . . 14 (𝜑𝐴𝑂)
2524necomd 2999 . . . . . . . . . . . . 13 (𝜑𝑂𝐴)
2625neneqd 2948 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑂 = 𝐴)
2726adantr 481 . . . . . . . . . . 11 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝑂 = 𝐴)
2821, 27jca 512 . . . . . . . . . 10 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
296adantr 481 . . . . . . . . . . . 12 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐺 ∈ TarskiG)
309adantr 481 . . . . . . . . . . . 12 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐵𝑃)
3113adantr 481 . . . . . . . . . . . 12 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐴𝑃)
3211adantr 481 . . . . . . . . . . . 12 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝑂𝑃)
331, 3, 4, 6, 9, 13, 19tglinerflx2 27576 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ (𝐵𝐿𝐴))
341, 3, 4, 6, 13, 9, 18tglinecom 27577 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
3534, 22eqbrtrrd 5129 . . . . . . . . . . . . . 14 (𝜑 → (𝐵𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝑂))
361, 2, 3, 4, 6, 9, 13, 33, 11, 35perprag 27668 . . . . . . . . . . . . 13 (𝜑 → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
3736adantr 481 . . . . . . . . . . . 12 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
38 simpr 485 . . . . . . . . . . . . 13 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝑂 ∈ (𝐵𝐿𝐴))
3938orcd 871 . . . . . . . . . . . 12 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (𝑂 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
401, 2, 3, 4, 5, 29, 30, 31, 32, 37, 39ragflat3 27648 . . . . . . . . . . 11 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (𝐵 = 𝐴𝑂 = 𝐴))
41 oran 988 . . . . . . . . . . 11 ((𝐵 = 𝐴𝑂 = 𝐴) ↔ ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
4240, 41sylib 217 . . . . . . . . . 10 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
4328, 42pm2.65da 815 . . . . . . . . 9 (𝜑 → ¬ 𝑂 ∈ (𝐵𝐿𝐴))
4443adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝑂 ∈ (𝐵𝐿𝐴))
4534adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
4644, 45neleqtrrd 2860 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
4718adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐴𝐵)
4847neneqd 2948 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝐴 = 𝐵)
4946, 48jca 512 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (¬ 𝑂 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵))
50 pm4.56 987 . . . . . 6 ((¬ 𝑂 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵) ↔ ¬ (𝑂 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
5149, 50sylib 217 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ (𝑂 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
521, 4, 3, 7, 14, 10, 12, 51ncolrot2 27505 . . . 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 27430 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝑂𝐼𝑅))
55 mideulem.4 . . . . . . . 8 (𝜑𝑇𝑃)
5655adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑇𝑃)
57 mideulem.7 . . . . . . . 8 (𝜑𝑇 ∈ (𝐴𝐿𝐵))
5857adantr 481 . . . . . . 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 27590 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐴𝐿𝐵))
6143, 34neleqtrrd 2860 . . . . . . 7 (𝜑 → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
6261adantr 481 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
63 nelne2 3042 . . . . . 6 ((𝑥 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝑂 ∈ (𝐴𝐿𝐵)) → 𝑥𝑂)
6460, 62, 63syl2anc 584 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥𝑂)
651, 2, 3, 7, 12, 17, 16, 54, 64tgbtwnne 27432 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑂𝑅)
661, 2, 3, 4, 5, 6, 9, 13, 11israg 27639 . . . . . . . 8 (𝜑 → (⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 𝑂) = (𝐵 ((𝑆𝐴)‘𝑂))))
6736, 66mpbid 231 . . . . . . 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 2736 . . . . . . . . 9 (𝑆𝐴) = (𝑆𝐴)
711, 2, 3, 4, 5, 7, 14, 70, 12mircl 27603 . . . . . . . 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 27600 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝐴 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑂))
79 eqid 2736 . . . . . . . . 9 (𝑆𝐵) = (𝑆𝐵)
801, 2, 3, 4, 5, 69, 76, 79, 77mirbtwn 27600 . . . . . . . 8 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝐵 ∈ (((𝑆𝐵)‘𝑠)𝐼𝑠))
81 simpr 485 . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑥 ∈ (𝑇𝐼𝐵))
106104simprd 496 . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . 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 27676 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝐵 = 𝑚)
114113eqcomd 2742 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑚 = 𝐵)
115114fveq2d 6846 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → (𝑆𝑚) = (𝑆𝐵))
116115fveq1d 6844 . . . . . . . . . . 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 2776 . . . . . . . . . 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 2736 . . . . . . . . . . 11 (𝑆𝑚) = (𝑆𝑚)
1191, 2, 3, 4, 5, 69, 118, 77, 75, 101, 110midexlem 27634 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → ∃𝑚𝑃 𝑅 = ((𝑆𝑚)‘𝑠))
120117, 119r19.29a 3159 . . . . . . . . 9 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝑅 = ((𝑆𝐵)‘𝑠))
121120oveq1d 7372 . . . . . . . 8 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑅𝐼𝑠) = (((𝑆𝐵)‘𝑠)𝐼𝑠))
12280, 121eleqtrrd 2841 . . . . . . 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 27599 . . . . . . . . . 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 2776 . . . . . . . . 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 27422 . . . . . . . 8 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (((𝑆𝐴)‘𝑂) 𝐴) = (𝑅 𝐵))
127120oveq2d 7373 . . . . . . . . 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 27599 . . . . . . . . 9 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝐵 ((𝑆𝐵)‘𝑠)) = (𝐵 𝑠))
129124, 127, 1283eqtrd 2780 . . . . . . . 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 27427 . . . . . . 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 27404 . . . . . . 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 27404 . . . . . . . 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 27430 . . . . . . . . 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 27422 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑠 𝑥) = (𝑅 𝑥))
137136eqcomd 2742 . . . . . . . . 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 2999 . . . . . . . . . . . 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 871 . . . . . . . . . . . . 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 27500 . . . . . . . . . . . 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 27501 . . . . . . . . . . 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 27641 . . . . . . . . . 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 27639 . . . . . . . . . 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 231 . . . . . . . . 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 27427 . . . . . . . 8 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑅 𝑂) = (𝑠 ((𝑆𝐴)‘𝑂)))
149132, 148eqtrd 2776 . . . . . . 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 27450 . . . . . 6 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝐴 𝑅) = (𝐵 ((𝑆𝐴)‘𝑂)))
15168, 150eqtr4d 2779 . . . . 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 27406 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ∃𝑠𝑃 (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅)))
153151, 152r19.29a 3159 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐵 𝑂) = (𝐴 𝑅))
15499adantr 481 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐴 𝑂) = (𝐵 𝑅))
1551, 2, 3, 7, 14, 12, 10, 16, 154tgcgrcomlr 27422 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑂 𝐴) = (𝑅 𝐵))
156143, 152r19.29a 3159 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
1571, 4, 3, 7, 12, 16, 17, 54btwncolg1 27497 . . . 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 27631 . . 3 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐵 = ((𝑆𝑥)‘𝐴))
1591, 2, 3, 4, 5, 7, 17, 8, 14mirbtwn 27600 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (((𝑆𝑥)‘𝐴)𝐼𝐴))
160158oveq1d 7372 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐵𝐼𝐴) = (((𝑆𝑥)‘𝐴)𝐼𝐴))
161159, 160eleqtrrd 2841 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐵𝐼𝐴))
1621, 2, 3, 7, 10, 17, 14, 161tgbtwncom 27430 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐴𝐼𝐵))
1631, 2, 3, 7, 14, 10axtgcgrrflx 27404 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐴 𝐵) = (𝐵 𝐴))
164158oveq2d 7373 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 𝐵) = (𝑥 ((𝑆𝑥)‘𝐴)))
1651, 2, 3, 4, 5, 7, 17, 8, 14mircgr 27599 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 ((𝑆𝑥)‘𝐴)) = (𝑥 𝐴))
166164, 165eqtrd 2776 . . . . 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 27450 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 𝑂) = (𝑥 𝑅))
1681, 2, 3, 4, 5, 7, 17, 8, 16, 12, 167, 54ismir 27601 . . 3 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑂 = ((𝑆𝑥)‘𝑅))
169158, 168jca 512 . 2 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐵 = ((𝑆𝑥)‘𝐴) ∧ 𝑂 = ((𝑆𝑥)‘𝑅)))
1701, 2, 3, 6, 86, 55, 11, 94tgbtwncom 27430 . . 3 (𝜑𝑇 ∈ (𝑂𝐼𝑄))
1711, 2, 3, 6, 11, 9, 86, 55, 15, 170, 97axtgpasch 27409 . 2 (𝜑 → ∃𝑥𝑃 (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))
172169, 171reximddv 3168 1 (𝜑 → ∃𝑥𝑃 (𝐵 = ((𝑆𝑥)‘𝐴) ∧ 𝑂 = ((𝑆𝑥)‘𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 845   = wceq 1541  wcel 2106  wne 2943  wrex 3073   class class class wbr 5105  cfv 6496  (class class class)co 7357  ⟨“cs3 14731  Basecbs 17083  distcds 17142  TarskiGcstrkg 27369  Itvcitv 27375  LineGclng 27376  pInvGcmir 27594  ∟Gcrag 27635  ⟂Gcperpg 27637
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-concat 14459  df-s1 14484  df-s2 14737  df-s3 14738  df-trkgc 27390  df-trkgb 27391  df-trkgcb 27392  df-trkg 27395  df-cgrg 27453  df-leg 27525  df-mir 27595  df-rag 27636  df-perpg 27638
This theorem is referenced by:  mideulem  27678  opphllem3  27691
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