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Theorem opphllem 26449
Description: Lemma 8.24 of [Schwabhauser] p. 66. This is used later for mideulem 26450 and later for opphl 26468. (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 2821 . . . 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 767 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥𝑃)
18 mideulem.1 . . . . . . . . . . . . . 14 (𝜑𝐴𝐵)
1918necomd 3071 . . . . . . . . . . . . 13 (𝜑𝐵𝐴)
2019neneqd 3021 . . . . . . . . . . . 12 (𝜑 → ¬ 𝐵 = 𝐴)
2120adantr 481 . . . . . . . . . . 11 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝐵 = 𝐴)
22 mideulem.6 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
234, 6, 22perpln2 26425 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴𝐿𝑂) ∈ ran 𝐿)
241, 3, 4, 6, 13, 11, 23tglnne 26342 . . . . . . . . . . . . . 14 (𝜑𝐴𝑂)
2524necomd 3071 . . . . . . . . . . . . 13 (𝜑𝑂𝐴)
2625neneqd 3021 . . . . . . . . . . . 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 26348 . . . . . . . . . . . . . 14 (𝜑𝐴 ∈ (𝐵𝐿𝐴))
341, 3, 4, 6, 13, 9, 18tglinecom 26349 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
3534, 22eqbrtrrd 5082 . . . . . . . . . . . . . 14 (𝜑 → (𝐵𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝑂))
361, 2, 3, 4, 6, 9, 13, 33, 11, 35perprag 26440 . . . . . . . . . . . . 13 (𝜑 → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
3736adantr 481 . . . . . . . . . . . 12 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
38 simpr 485 . . . . . . . . . . . . 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 26420 . . . . . . . . . . 11 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (𝐵 = 𝐴𝑂 = 𝐴))
41 oran 983 . . . . . . . . . . 11 ((𝐵 = 𝐴𝑂 = 𝐴) ↔ ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
4240, 41sylib 219 . . . . . . . . . 10 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
4328, 42pm2.65da 813 . . . . . . . . 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 2935 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
4718adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐴𝐵)
4847neneqd 3021 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝐴 = 𝐵)
4946, 48jca 512 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (¬ 𝑂 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵))
50 pm4.56 982 . . . . . 6 ((¬ 𝑂 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵) ↔ ¬ (𝑂 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
5149, 50sylib 219 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ (𝑂 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
521, 4, 3, 7, 14, 10, 12, 51ncolrot2 26277 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ (𝐵 ∈ (𝑂𝐿𝐴) ∨ 𝑂 = 𝐴))
53 simprrr 778 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝑅𝐼𝑂))
541, 2, 3, 7, 16, 17, 12, 53tgbtwncom 26202 . . . . 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 777 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝑇𝐼𝐵))
601, 3, 4, 7, 56, 14, 10, 17, 58, 59coltr3 26362 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐴𝐿𝐵))
6143, 34neleqtrrd 2935 . . . . . . 7 (𝜑 → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
6261adantr 481 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
63 nelne2 3115 . . . . . 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 26204 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑂𝑅)
661, 2, 3, 4, 5, 6, 9, 13, 11israg 26411 . . . . . . . 8 (𝜑 → (⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺) ↔ (𝐵 𝑂) = (𝐵 ((𝑆𝐴)‘𝑂))))
6736, 66mpbid 233 . . . . . . 7 (𝜑 → (𝐵 𝑂) = (𝐵 ((𝑆𝐴)‘𝑂)))
6867ad3antrrr 726 . . . . . 6 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝐵 𝑂) = (𝐵 ((𝑆𝐴)‘𝑂)))
696ad3antrrr 726 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝐺 ∈ TarskiG)
70 eqid 2821 . . . . . . . . 9 (𝑆𝐴) = (𝑆𝐴)
711, 2, 3, 4, 5, 7, 14, 70, 12mircl 26375 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ((𝑆𝐴)‘𝑂) ∈ 𝑃)
7271ad2antrr 722 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → ((𝑆𝐴)‘𝑂) ∈ 𝑃)
7313ad3antrrr 726 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝐴𝑃)
7411ad3antrrr 726 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝑂𝑃)
7515ad3antrrr 726 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝑅𝑃)
769ad3antrrr 726 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝐵𝑃)
77 simplr 765 . . . . . . 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 26372 . . . . . . 7 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝐴 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑂))
79 eqid 2821 . . . . . . . . 9 (𝑆𝐵) = (𝑆𝐵)
801, 2, 3, 4, 5, 69, 76, 79, 77mirbtwn 26372 . . . . . . . 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 722 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝐺 ∈ TarskiG)
8373ad2antrr 722 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝐴𝑃)
8476ad2antrr 722 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝐵𝑃)
8547ad4antr 728 . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑄𝑃)
8874ad2antrr 722 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑂𝑃)
8956ad4antr 728 . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑇 ∈ (𝑄𝐼𝑂))
9675ad2antrr 722 . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝑥𝑃)
102101ad2antrr 722 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑥𝑃)
103 simp-5r 782 . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . 15 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑠𝑃)
108 simpllr 772 . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑥 𝑠) = (𝑥 𝑅))
111110ad2antrr 722 . . . . . . . . . . . . . . 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 765 . . . . . . . . . . . . . . 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 26448 . . . . . . . . . . . . . 14 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝐵 = 𝑚)
114113eqcomd 2827 . . . . . . . . . . . . 13 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → 𝑚 = 𝐵)
115114fveq2d 6668 . . . . . . . . . . . 12 ((((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) ∧ 𝑚𝑃) ∧ 𝑅 = ((𝑆𝑚)‘𝑠)) → (𝑆𝑚) = (𝑆𝐵))
116115fveq1d 6666 . . . . . . . . . . 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 2856 . . . . . . . . . 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 2821 . . . . . . . . . . 11 (𝑆𝑚) = (𝑆𝑚)
1191, 2, 3, 4, 5, 69, 118, 77, 75, 101, 110midexlem 26406 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → ∃𝑚𝑃 𝑅 = ((𝑆𝑚)‘𝑠))
120117, 119r19.29a 3289 . . . . . . . . 9 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝑅 = ((𝑆𝐵)‘𝑠))
121120oveq1d 7160 . . . . . . . 8 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑅𝐼𝑠) = (((𝑆𝐵)‘𝑠)𝐼𝑠))
12280, 121eleqtrrd 2916 . . . . . . 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 26371 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝐴 ((𝑆𝐴)‘𝑂)) = (𝐴 𝑂))
12499ad3antrrr 726 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝐴 𝑂) = (𝐵 𝑅))
125123, 124eqtrd 2856 . . . . . . . . 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 26194 . . . . . . . 8 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (((𝑆𝐴)‘𝑂) 𝐴) = (𝑅 𝐵))
127120oveq2d 7161 . . . . . . . . 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 26371 . . . . . . . . 9 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝐵 ((𝑆𝐵)‘𝑠)) = (𝐵 𝑠))
129124, 127, 1283eqtrd 2860 . . . . . . . 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 26199 . . . . . . 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 26176 . . . . . . 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 26176 . . . . . . . 8 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑂 𝑅) = (𝑅 𝑂))
13353ad2antrr 722 . . . . . . . . 9 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝑥 ∈ (𝑅𝐼𝑂))
134 simprl 767 . . . . . . . . . 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 26202 . . . . . . . . 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 26194 . . . . . . . . . 10 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑠 𝑥) = (𝑅 𝑥))
137136eqcomd 2827 . . . . . . . . 9 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑅 𝑥) = (𝑠 𝑥))
13836ad3antrrr 726 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
13947necomd 3071 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐵𝐴)
140139ad2antrr 722 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → 𝐵𝐴)
14160ad2antrr 722 . . . . . . . . . . . . . 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 26272 . . . . . . . . . . . 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 26273 . . . . . . . . . . 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 26413 . . . . . . . . . 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 26411 . . . . . . . . . 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 233 . . . . . . . . 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 26199 . . . . . . . 8 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝑅 𝑂) = (𝑠 ((𝑆𝐴)‘𝑂)))
149132, 148eqtrd 2856 . . . . . . 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 26222 . . . . . 6 ((((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) ∧ 𝑠𝑃) ∧ (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅))) → (𝐴 𝑅) = (𝐵 ((𝑆𝐴)‘𝑂)))
15168, 150eqtr4d 2859 . . . . 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 26178 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → ∃𝑠𝑃 (𝑥 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑠) ∧ (𝑥 𝑠) = (𝑥 𝑅)))
153151, 152r19.29a 3289 . . . 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 26194 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑂 𝐴) = (𝑅 𝐵))
156143, 152r19.29a 3289 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
1571, 4, 3, 7, 12, 16, 17, 54btwncolg1 26269 . . . 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 26403 . . 3 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝐵 = ((𝑆𝑥)‘𝐴))
1591, 2, 3, 4, 5, 7, 17, 8, 14mirbtwn 26372 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (((𝑆𝑥)‘𝐴)𝐼𝐴))
160158oveq1d 7160 . . . . . . 7 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐵𝐼𝐴) = (((𝑆𝑥)‘𝐴)𝐼𝐴))
161159, 160eleqtrrd 2916 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐵𝐼𝐴))
1621, 2, 3, 7, 10, 17, 14, 161tgbtwncom 26202 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → 𝑥 ∈ (𝐴𝐼𝐵))
1631, 2, 3, 7, 14, 10axtgcgrrflx 26176 . . . . 5 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝐴 𝐵) = (𝐵 𝐴))
164158oveq2d 7161 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 𝐵) = (𝑥 ((𝑆𝑥)‘𝐴)))
1651, 2, 3, 4, 5, 7, 17, 8, 14mircgr 26371 . . . . . 6 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 ((𝑆𝑥)‘𝐴)) = (𝑥 𝐴))
166164, 165eqtrd 2856 . . . . 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 26222 . . . 4 ((𝜑 ∧ (𝑥𝑃 ∧ (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))) → (𝑥 𝑂) = (𝑥 𝑅))
1681, 2, 3, 4, 5, 7, 17, 8, 16, 12, 167, 54ismir 26373 . . 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 26202 . . 3 (𝜑𝑇 ∈ (𝑂𝐼𝑄))
1711, 2, 3, 6, 11, 9, 86, 55, 15, 170, 97axtgpasch 26181 . 2 (𝜑 → ∃𝑥𝑃 (𝑥 ∈ (𝑇𝐼𝐵) ∧ 𝑥 ∈ (𝑅𝐼𝑂)))
172169, 171reximddv 3275 1 (𝜑 → ∃𝑥𝑃 (𝐵 = ((𝑆𝑥)‘𝐴) ∧ 𝑂 = ((𝑆𝑥)‘𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  wne 3016  wrex 3139   class class class wbr 5058  cfv 6349  (class class class)co 7145  ⟨“cs3 14194  Basecbs 16473  distcds 16564  TarskiGcstrkg 26144  Itvcitv 26150  LineGclng 26151  pInvGcmir 26366  ∟Gcrag 26407  ⟂Gcperpg 26409
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-trkg 26167  df-cgrg 26225  df-leg 26297  df-mir 26367  df-rag 26408  df-perpg 26410
This theorem is referenced by:  mideulem  26450  opphllem3  26463
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