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Theorem opphllem6 25444
Description: First part of Lemma 9.4 of [Schwabhauser] p. 68. (Contributed by Thierry Arnoux, 3-Mar-2020.)
Hypotheses
Ref Expression
hpg.p 𝑃 = (Base‘𝐺)
hpg.d = (dist‘𝐺)
hpg.i 𝐼 = (Itv‘𝐺)
hpg.o 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
opphl.l 𝐿 = (LineG‘𝐺)
opphl.d (𝜑𝐷 ∈ ran 𝐿)
opphl.g (𝜑𝐺 ∈ TarskiG)
opphl.k 𝐾 = (hlG‘𝐺)
opphllem5.n 𝑁 = ((pInvG‘𝐺)‘𝑀)
opphllem5.a (𝜑𝐴𝑃)
opphllem5.c (𝜑𝐶𝑃)
opphllem5.r (𝜑𝑅𝐷)
opphllem5.s (𝜑𝑆𝐷)
opphllem5.m (𝜑𝑀𝑃)
opphllem5.o (𝜑𝐴𝑂𝐶)
opphllem5.p (𝜑𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
opphllem5.q (𝜑𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
opphllem5.u (𝜑𝑈𝑃)
opphllem6.v (𝜑 → (𝑁𝑅) = 𝑆)
Assertion
Ref Expression
opphllem6 (𝜑 → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
Distinct variable groups:   𝐷,𝑎,𝑏   𝐼,𝑎,𝑏   𝑃,𝑎,𝑏   𝑡,𝐴   𝑡,𝐷   𝑡,𝑅   𝑡,𝐶   𝑡,𝐺   𝑡,𝐿   𝑡,𝑈   𝑡,𝐼   𝑡,𝐾   𝑡,𝑀   𝑡,𝑂   𝑡,𝑁   𝑡,𝑃   𝑡,𝑆   𝜑,𝑡   𝑡,   𝑡,𝑎,𝑏
Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐴(𝑎,𝑏)   𝐶(𝑎,𝑏)   𝑅(𝑎,𝑏)   𝑆(𝑎,𝑏)   𝑈(𝑎,𝑏)   𝐺(𝑎,𝑏)   𝐾(𝑎,𝑏)   𝐿(𝑎,𝑏)   𝑀(𝑎,𝑏)   (𝑎,𝑏)   𝑁(𝑎,𝑏)   𝑂(𝑎,𝑏)

Proof of Theorem opphllem6
StepHypRef Expression
1 hpg.p . . . 4 𝑃 = (Base‘𝐺)
2 hpg.d . . . 4 = (dist‘𝐺)
3 hpg.i . . . 4 𝐼 = (Itv‘𝐺)
4 opphl.l . . . 4 𝐿 = (LineG‘𝐺)
5 eqid 2610 . . . 4 (pInvG‘𝐺) = (pInvG‘𝐺)
6 opphl.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
76adantr 480 . . . 4 ((𝜑𝑅 = 𝑆) → 𝐺 ∈ TarskiG)
8 opphllem5.n . . . 4 𝑁 = ((pInvG‘𝐺)‘𝑀)
9 opphl.k . . . 4 𝐾 = (hlG‘𝐺)
10 opphllem5.m . . . . 5 (𝜑𝑀𝑃)
1110adantr 480 . . . 4 ((𝜑𝑅 = 𝑆) → 𝑀𝑃)
12 opphllem5.a . . . . 5 (𝜑𝐴𝑃)
1312adantr 480 . . . 4 ((𝜑𝑅 = 𝑆) → 𝐴𝑃)
14 opphllem5.c . . . . 5 (𝜑𝐶𝑃)
1514adantr 480 . . . 4 ((𝜑𝑅 = 𝑆) → 𝐶𝑃)
16 opphllem5.u . . . . 5 (𝜑𝑈𝑃)
1716adantr 480 . . . 4 ((𝜑𝑅 = 𝑆) → 𝑈𝑃)
18 opphl.d . . . . . . . 8 (𝜑𝐷 ∈ ran 𝐿)
19 opphllem5.r . . . . . . . 8 (𝜑𝑅𝐷)
201, 4, 3, 6, 18, 19tglnpt 25244 . . . . . . 7 (𝜑𝑅𝑃)
21 opphllem5.p . . . . . . . 8 (𝜑𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
224, 6, 21perpln2 25406 . . . . . . 7 (𝜑 → (𝐴𝐿𝑅) ∈ ran 𝐿)
231, 3, 4, 6, 12, 20, 22tglnne 25323 . . . . . 6 (𝜑𝐴𝑅)
2423adantr 480 . . . . 5 ((𝜑𝑅 = 𝑆) → 𝐴𝑅)
25 opphllem6.v . . . . . . . 8 (𝜑 → (𝑁𝑅) = 𝑆)
2625adantr 480 . . . . . . 7 ((𝜑𝑅 = 𝑆) → (𝑁𝑅) = 𝑆)
27 simpr 476 . . . . . . 7 ((𝜑𝑅 = 𝑆) → 𝑅 = 𝑆)
2826, 27eqtr4d 2647 . . . . . 6 ((𝜑𝑅 = 𝑆) → (𝑁𝑅) = 𝑅)
291, 2, 3, 4, 5, 6, 10, 8, 20mirinv 25361 . . . . . . 7 (𝜑 → ((𝑁𝑅) = 𝑅𝑀 = 𝑅))
3029adantr 480 . . . . . 6 ((𝜑𝑅 = 𝑆) → ((𝑁𝑅) = 𝑅𝑀 = 𝑅))
3128, 30mpbid 221 . . . . 5 ((𝜑𝑅 = 𝑆) → 𝑀 = 𝑅)
3224, 31neeqtrrd 2856 . . . 4 ((𝜑𝑅 = 𝑆) → 𝐴𝑀)
33 opphllem5.s . . . . . . . 8 (𝜑𝑆𝐷)
341, 4, 3, 6, 18, 33tglnpt 25244 . . . . . . 7 (𝜑𝑆𝑃)
35 opphllem5.q . . . . . . . 8 (𝜑𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
364, 6, 35perpln2 25406 . . . . . . 7 (𝜑 → (𝐶𝐿𝑆) ∈ ran 𝐿)
371, 3, 4, 6, 14, 34, 36tglnne 25323 . . . . . 6 (𝜑𝐶𝑆)
3837adantr 480 . . . . 5 ((𝜑𝑅 = 𝑆) → 𝐶𝑆)
3931, 27eqtrd 2644 . . . . 5 ((𝜑𝑅 = 𝑆) → 𝑀 = 𝑆)
4038, 39neeqtrrd 2856 . . . 4 ((𝜑𝑅 = 𝑆) → 𝐶𝑀)
41 simpr 476 . . . . . . . 8 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 = 𝑡) → 𝑅 = 𝑡)
426ad3antrrr 762 . . . . . . . . . 10 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
4342adantr 480 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝐺 ∈ TarskiG)
4414ad3antrrr 762 . . . . . . . . . 10 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐶𝑃)
4544adantr 480 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝐶𝑃)
4620ad3antrrr 762 . . . . . . . . . 10 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅𝑃)
4746adantr 480 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅𝑃)
4818ad3antrrr 762 . . . . . . . . . . 11 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ ran 𝐿)
49 simplr 788 . . . . . . . . . . 11 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡𝐷)
501, 4, 3, 42, 48, 49tglnpt 25244 . . . . . . . . . 10 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡𝑃)
5150adantr 480 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑡𝑃)
5212ad3antrrr 762 . . . . . . . . . 10 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐴𝑃)
5352adantr 480 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝐴𝑃)
5434ad3antrrr 762 . . . . . . . . . . 11 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑆𝑃)
5554adantr 480 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑆𝑃)
56 simpllr 795 . . . . . . . . . . . 12 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑆)
571, 3, 4, 6, 14, 34, 37tglinerflx2 25329 . . . . . . . . . . . . 13 (𝜑𝑆 ∈ (𝐶𝐿𝑆))
5857ad3antrrr 762 . . . . . . . . . . . 12 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑆 ∈ (𝐶𝐿𝑆))
5956, 58eqeltrd 2688 . . . . . . . . . . 11 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐶𝐿𝑆))
6059adantr 480 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅 ∈ (𝐶𝐿𝑆))
611, 3, 4, 6, 14, 34, 37tgelrnln 25325 . . . . . . . . . . . . 13 (𝜑 → (𝐶𝐿𝑆) ∈ ran 𝐿)
621, 2, 3, 4, 6, 18, 61, 35perpcom 25408 . . . . . . . . . . . 12 (𝜑 → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
6362ad4antr 764 . . . . . . . . . . 11 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
64 simpr 476 . . . . . . . . . . . 12 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅𝑡)
6548adantr 480 . . . . . . . . . . . 12 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝐷 ∈ ran 𝐿)
6619ad3antrrr 762 . . . . . . . . . . . . 13 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅𝐷)
6766adantr 480 . . . . . . . . . . . 12 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅𝐷)
6849adantr 480 . . . . . . . . . . . 12 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑡𝐷)
691, 3, 4, 43, 47, 51, 64, 64, 65, 67, 68tglinethru 25331 . . . . . . . . . . 11 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝐷 = (𝑅𝐿𝑡))
7063, 69breqtrd 4609 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)(𝑅𝐿𝑡))
711, 2, 3, 4, 43, 45, 55, 60, 51, 70perprag 25418 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → ⟨“𝐶𝑅𝑡”⟩ ∈ (∟G‘𝐺))
721, 3, 4, 6, 12, 20, 23tglinerflx2 25329 . . . . . . . . . . . 12 (𝜑𝑅 ∈ (𝐴𝐿𝑅))
7372ad3antrrr 762 . . . . . . . . . . 11 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐴𝐿𝑅))
7473adantr 480 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅 ∈ (𝐴𝐿𝑅))
751, 3, 4, 6, 12, 20, 23tgelrnln 25325 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝐿𝑅) ∈ ran 𝐿)
761, 2, 3, 4, 6, 18, 75, 21perpcom 25408 . . . . . . . . . . . 12 (𝜑 → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
7776ad4antr 764 . . . . . . . . . . 11 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
7877, 69breqtrd 4609 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)(𝑅𝐿𝑡))
791, 2, 3, 4, 43, 53, 47, 74, 51, 78perprag 25418 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → ⟨“𝐴𝑅𝑡”⟩ ∈ (∟G‘𝐺))
80 simplr 788 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑡 ∈ (𝐴𝐼𝐶))
811, 2, 3, 43, 53, 51, 45, 80tgbtwncom 25183 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑡 ∈ (𝐶𝐼𝐴))
821, 2, 3, 4, 5, 43, 45, 47, 51, 53, 71, 79, 81ragflat2 25398 . . . . . . . 8 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅 = 𝑡)
8341, 82pm2.61dane 2869 . . . . . . 7 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑡)
84 simpr 476 . . . . . . 7 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ (𝐴𝐼𝐶))
8583, 84eqeltrd 2688 . . . . . 6 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐴𝐼𝐶))
86 opphllem5.o . . . . . . . . 9 (𝜑𝐴𝑂𝐶)
87 hpg.o . . . . . . . . . 10 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
881, 2, 3, 87, 12, 14islnopp 25431 . . . . . . . . 9 (𝜑 → (𝐴𝑂𝐶 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐶𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐶))))
8986, 88mpbid 221 . . . . . . . 8 (𝜑 → ((¬ 𝐴𝐷 ∧ ¬ 𝐶𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐶)))
9089simprd 478 . . . . . . 7 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐶))
9190adantr 480 . . . . . 6 ((𝜑𝑅 = 𝑆) → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐶))
9285, 91r19.29a 3060 . . . . 5 ((𝜑𝑅 = 𝑆) → 𝑅 ∈ (𝐴𝐼𝐶))
9331, 92eqeltrd 2688 . . . 4 ((𝜑𝑅 = 𝑆) → 𝑀 ∈ (𝐴𝐼𝐶))
941, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 17, 32, 40, 93mirbtwnhl 25375 . . 3 ((𝜑𝑅 = 𝑆) → (𝑈(𝐾𝑀)𝐴 ↔ (𝑁𝑈)(𝐾𝑀)𝐶))
9531fveq2d 6107 . . . 4 ((𝜑𝑅 = 𝑆) → (𝐾𝑀) = (𝐾𝑅))
9695breqd 4594 . . 3 ((𝜑𝑅 = 𝑆) → (𝑈(𝐾𝑀)𝐴𝑈(𝐾𝑅)𝐴))
9739fveq2d 6107 . . . 4 ((𝜑𝑅 = 𝑆) → (𝐾𝑀) = (𝐾𝑆))
9897breqd 4594 . . 3 ((𝜑𝑅 = 𝑆) → ((𝑁𝑈)(𝐾𝑀)𝐶 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
9994, 96, 983bitr3d 297 . 2 ((𝜑𝑅 = 𝑆) → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
10018ad2antrr 758 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐷 ∈ ran 𝐿)
1016ad2antrr 758 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐺 ∈ TarskiG)
10212ad2antrr 758 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐴𝑃)
10314ad2antrr 758 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐶𝑃)
10419ad2antrr 758 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝑅𝐷)
10533ad2antrr 758 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝑆𝐷)
10610ad2antrr 758 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝑀𝑃)
10786ad2antrr 758 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐴𝑂𝐶)
10821ad2antrr 758 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
10935ad2antrr 758 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
110 simpr 476 . . . . 5 ((𝜑𝑅𝑆) → 𝑅𝑆)
111110adantr 480 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝑅𝑆)
112 simpr 476 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴))
11316ad2antrr 758 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝑈𝑃)
11425ad2antrr 758 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → (𝑁𝑅) = 𝑆)
1151, 2, 3, 87, 4, 100, 101, 9, 8, 102, 103, 104, 105, 106, 107, 108, 109, 111, 112, 113, 114opphllem3 25441 . . 3 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
11618ad2antrr 758 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐷 ∈ ran 𝐿)
1176adantr 480 . . . . . 6 ((𝜑𝑅𝑆) → 𝐺 ∈ TarskiG)
118117adantr 480 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐺 ∈ TarskiG)
11914ad2antrr 758 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐶𝑃)
12012adantr 480 . . . . . 6 ((𝜑𝑅𝑆) → 𝐴𝑃)
121120adantr 480 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐴𝑃)
12233adantr 480 . . . . . 6 ((𝜑𝑅𝑆) → 𝑆𝐷)
123122adantr 480 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑆𝐷)
12419adantr 480 . . . . . 6 ((𝜑𝑅𝑆) → 𝑅𝐷)
125124adantr 480 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑅𝐷)
12610ad2antrr 758 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑀𝑃)
12786ad2antrr 758 . . . . . 6 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐴𝑂𝐶)
1281, 2, 3, 87, 4, 116, 118, 121, 119, 127oppcom 25436 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐶𝑂𝐴)
12935ad2antrr 758 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
13021adantr 480 . . . . . 6 ((𝜑𝑅𝑆) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
131130adantr 480 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
132110necomd 2837 . . . . . 6 ((𝜑𝑅𝑆) → 𝑆𝑅)
133132adantr 480 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑆𝑅)
134 simpr 476 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶))
13516ad2antrr 758 . . . . . 6 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑈𝑃)
1361, 2, 3, 4, 5, 118, 126, 8, 135mircl 25356 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → (𝑁𝑈) ∈ 𝑃)
13720adantr 480 . . . . . . 7 ((𝜑𝑅𝑆) → 𝑅𝑃)
138137adantr 480 . . . . . 6 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑅𝑃)
13925ad2antrr 758 . . . . . 6 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → (𝑁𝑅) = 𝑆)
1401, 2, 3, 4, 5, 118, 126, 8, 138, 139mircom 25358 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → (𝑁𝑆) = 𝑅)
1411, 2, 3, 87, 4, 116, 118, 9, 8, 119, 121, 123, 125, 126, 128, 129, 131, 133, 134, 136, 140opphllem3 25441 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → ((𝑁𝑈)(𝐾𝑆)𝐶 ↔ (𝑁‘(𝑁𝑈))(𝐾𝑅)𝐴))
1421, 2, 3, 4, 5, 118, 126, 8, 135mirmir 25357 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → (𝑁‘(𝑁𝑈)) = 𝑈)
143142breq1d 4593 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → ((𝑁‘(𝑁𝑈))(𝐾𝑅)𝐴𝑈(𝐾𝑅)𝐴))
144141, 143bitr2d 268 . . 3 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
145 eqid 2610 . . . . 5 (≤G‘𝐺) = (≤G‘𝐺)
1461, 2, 3, 145, 6, 34, 14, 20, 12legtrid 25286 . . . 4 (𝜑 → ((𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴) ∨ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)))
147146adantr 480 . . 3 ((𝜑𝑅𝑆) → ((𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴) ∨ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)))
148115, 144, 147mpjaodan 823 . 2 ((𝜑𝑅𝑆) → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
14999, 148pm2.61dane 2869 1 (𝜑 → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383   = wceq 1475  wcel 1977  wne 2780  wrex 2897  cdif 3537   class class class wbr 4583  {copab 4642  ran crn 5039  cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135  LineGclng 25136  ≤Gcleg 25277  hlGchlg 25295  pInvGcmir 25347  ⟂Gcperpg 25390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-trkgc 25147  df-trkgb 25148  df-trkgcb 25149  df-trkg 25152  df-cgrg 25206  df-leg 25278  df-hlg 25296  df-mir 25348  df-rag 25389  df-perpg 25391
This theorem is referenced by:  opphl  25446
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