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Theorem opphllem6 25362
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 2609 . . . 4 (pInvG‘𝐺) = (pInvG‘𝐺)
6 opphl.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
76adantr 479 . . . 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 479 . . . 4 ((𝜑𝑅 = 𝑆) → 𝑀𝑃)
12 opphllem5.a . . . . 5 (𝜑𝐴𝑃)
1312adantr 479 . . . 4 ((𝜑𝑅 = 𝑆) → 𝐴𝑃)
14 opphllem5.c . . . . 5 (𝜑𝐶𝑃)
1514adantr 479 . . . 4 ((𝜑𝑅 = 𝑆) → 𝐶𝑃)
16 opphllem5.u . . . . 5 (𝜑𝑈𝑃)
1716adantr 479 . . . 4 ((𝜑𝑅 = 𝑆) → 𝑈𝑃)
18 opphl.d . . . . . . . 8 (𝜑𝐷 ∈ ran 𝐿)
19 opphllem5.r . . . . . . . 8 (𝜑𝑅𝐷)
201, 4, 3, 6, 18, 19tglnpt 25162 . . . . . . 7 (𝜑𝑅𝑃)
21 opphllem5.p . . . . . . . 8 (𝜑𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
224, 6, 21perpln2 25324 . . . . . . 7 (𝜑 → (𝐴𝐿𝑅) ∈ ran 𝐿)
231, 3, 4, 6, 12, 20, 22tglnne 25241 . . . . . 6 (𝜑𝐴𝑅)
2423adantr 479 . . . . 5 ((𝜑𝑅 = 𝑆) → 𝐴𝑅)
25 opphllem6.v . . . . . . . 8 (𝜑 → (𝑁𝑅) = 𝑆)
2625adantr 479 . . . . . . 7 ((𝜑𝑅 = 𝑆) → (𝑁𝑅) = 𝑆)
27 simpr 475 . . . . . . 7 ((𝜑𝑅 = 𝑆) → 𝑅 = 𝑆)
2826, 27eqtr4d 2646 . . . . . 6 ((𝜑𝑅 = 𝑆) → (𝑁𝑅) = 𝑅)
291, 2, 3, 4, 5, 6, 10, 8, 20mirinv 25279 . . . . . . 7 (𝜑 → ((𝑁𝑅) = 𝑅𝑀 = 𝑅))
3029adantr 479 . . . . . 6 ((𝜑𝑅 = 𝑆) → ((𝑁𝑅) = 𝑅𝑀 = 𝑅))
3128, 30mpbid 220 . . . . 5 ((𝜑𝑅 = 𝑆) → 𝑀 = 𝑅)
3224, 31neeqtrrd 2855 . . . 4 ((𝜑𝑅 = 𝑆) → 𝐴𝑀)
33 opphllem5.s . . . . . . . 8 (𝜑𝑆𝐷)
341, 4, 3, 6, 18, 33tglnpt 25162 . . . . . . 7 (𝜑𝑆𝑃)
35 opphllem5.q . . . . . . . 8 (𝜑𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
364, 6, 35perpln2 25324 . . . . . . 7 (𝜑 → (𝐶𝐿𝑆) ∈ ran 𝐿)
371, 3, 4, 6, 14, 34, 36tglnne 25241 . . . . . 6 (𝜑𝐶𝑆)
3837adantr 479 . . . . 5 ((𝜑𝑅 = 𝑆) → 𝐶𝑆)
3931, 27eqtrd 2643 . . . . 5 ((𝜑𝑅 = 𝑆) → 𝑀 = 𝑆)
4038, 39neeqtrrd 2855 . . . 4 ((𝜑𝑅 = 𝑆) → 𝐶𝑀)
41 simpr 475 . . . . . . . 8 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 = 𝑡) → 𝑅 = 𝑡)
426ad3antrrr 761 . . . . . . . . . 10 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
4342adantr 479 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝐺 ∈ TarskiG)
4414ad3antrrr 761 . . . . . . . . . 10 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐶𝑃)
4544adantr 479 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝐶𝑃)
4620ad3antrrr 761 . . . . . . . . . 10 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅𝑃)
4746adantr 479 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅𝑃)
4818ad3antrrr 761 . . . . . . . . . . 11 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ ran 𝐿)
49 simplr 787 . . . . . . . . . . 11 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡𝐷)
501, 4, 3, 42, 48, 49tglnpt 25162 . . . . . . . . . 10 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡𝑃)
5150adantr 479 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑡𝑃)
5212ad3antrrr 761 . . . . . . . . . 10 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐴𝑃)
5352adantr 479 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝐴𝑃)
5434ad3antrrr 761 . . . . . . . . . . 11 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑆𝑃)
5554adantr 479 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑆𝑃)
56 simpllr 794 . . . . . . . . . . . 12 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑆)
571, 3, 4, 6, 14, 34, 37tglinerflx2 25247 . . . . . . . . . . . . 13 (𝜑𝑆 ∈ (𝐶𝐿𝑆))
5857ad3antrrr 761 . . . . . . . . . . . 12 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑆 ∈ (𝐶𝐿𝑆))
5956, 58eqeltrd 2687 . . . . . . . . . . 11 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐶𝐿𝑆))
6059adantr 479 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅 ∈ (𝐶𝐿𝑆))
611, 3, 4, 6, 14, 34, 37tgelrnln 25243 . . . . . . . . . . . . 13 (𝜑 → (𝐶𝐿𝑆) ∈ ran 𝐿)
621, 2, 3, 4, 6, 18, 61, 35perpcom 25326 . . . . . . . . . . . 12 (𝜑 → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
6362ad4antr 763 . . . . . . . . . . 11 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
64 simpr 475 . . . . . . . . . . . 12 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅𝑡)
6548adantr 479 . . . . . . . . . . . 12 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝐷 ∈ ran 𝐿)
6619ad3antrrr 761 . . . . . . . . . . . . 13 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅𝐷)
6766adantr 479 . . . . . . . . . . . 12 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅𝐷)
6849adantr 479 . . . . . . . . . . . 12 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑡𝐷)
691, 3, 4, 43, 47, 51, 64, 64, 65, 67, 68tglinethru 25249 . . . . . . . . . . 11 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝐷 = (𝑅𝐿𝑡))
7063, 69breqtrd 4603 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)(𝑅𝐿𝑡))
711, 2, 3, 4, 43, 45, 55, 60, 51, 70perprag 25336 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → ⟨“𝐶𝑅𝑡”⟩ ∈ (∟G‘𝐺))
721, 3, 4, 6, 12, 20, 23tglinerflx2 25247 . . . . . . . . . . . 12 (𝜑𝑅 ∈ (𝐴𝐿𝑅))
7372ad3antrrr 761 . . . . . . . . . . 11 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐴𝐿𝑅))
7473adantr 479 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅 ∈ (𝐴𝐿𝑅))
751, 3, 4, 6, 12, 20, 23tgelrnln 25243 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝐿𝑅) ∈ ran 𝐿)
761, 2, 3, 4, 6, 18, 75, 21perpcom 25326 . . . . . . . . . . . 12 (𝜑 → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
7776ad4antr 763 . . . . . . . . . . 11 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
7877, 69breqtrd 4603 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)(𝑅𝐿𝑡))
791, 2, 3, 4, 43, 53, 47, 74, 51, 78perprag 25336 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → ⟨“𝐴𝑅𝑡”⟩ ∈ (∟G‘𝐺))
80 simplr 787 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑡 ∈ (𝐴𝐼𝐶))
811, 2, 3, 43, 53, 51, 45, 80tgbtwncom 25100 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑡 ∈ (𝐶𝐼𝐴))
821, 2, 3, 4, 5, 43, 45, 47, 51, 53, 71, 79, 81ragflat2 25316 . . . . . . . 8 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅 = 𝑡)
8341, 82pm2.61dane 2868 . . . . . . 7 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑡)
84 simpr 475 . . . . . . 7 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ (𝐴𝐼𝐶))
8583, 84eqeltrd 2687 . . . . . 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 25349 . . . . . . . . 9 (𝜑 → (𝐴𝑂𝐶 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐶𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐶))))
8986, 88mpbid 220 . . . . . . . 8 (𝜑 → ((¬ 𝐴𝐷 ∧ ¬ 𝐶𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐶)))
9089simprd 477 . . . . . . 7 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐶))
9190adantr 479 . . . . . 6 ((𝜑𝑅 = 𝑆) → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐶))
9285, 91r19.29a 3059 . . . . 5 ((𝜑𝑅 = 𝑆) → 𝑅 ∈ (𝐴𝐼𝐶))
9331, 92eqeltrd 2687 . . . 4 ((𝜑𝑅 = 𝑆) → 𝑀 ∈ (𝐴𝐼𝐶))
941, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 17, 32, 40, 93mirbtwnhl 25293 . . 3 ((𝜑𝑅 = 𝑆) → (𝑈(𝐾𝑀)𝐴 ↔ (𝑁𝑈)(𝐾𝑀)𝐶))
9531fveq2d 6092 . . . 4 ((𝜑𝑅 = 𝑆) → (𝐾𝑀) = (𝐾𝑅))
9695breqd 4588 . . 3 ((𝜑𝑅 = 𝑆) → (𝑈(𝐾𝑀)𝐴𝑈(𝐾𝑅)𝐴))
9739fveq2d 6092 . . . 4 ((𝜑𝑅 = 𝑆) → (𝐾𝑀) = (𝐾𝑆))
9897breqd 4588 . . 3 ((𝜑𝑅 = 𝑆) → ((𝑁𝑈)(𝐾𝑀)𝐶 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
9994, 96, 983bitr3d 296 . 2 ((𝜑𝑅 = 𝑆) → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
10018ad2antrr 757 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐷 ∈ ran 𝐿)
1016ad2antrr 757 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐺 ∈ TarskiG)
10212ad2antrr 757 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐴𝑃)
10314ad2antrr 757 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐶𝑃)
10419ad2antrr 757 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝑅𝐷)
10533ad2antrr 757 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝑆𝐷)
10610ad2antrr 757 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝑀𝑃)
10786ad2antrr 757 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐴𝑂𝐶)
10821ad2antrr 757 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
10935ad2antrr 757 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
110 simpr 475 . . . . 5 ((𝜑𝑅𝑆) → 𝑅𝑆)
111110adantr 479 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝑅𝑆)
112 simpr 475 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴))
11316ad2antrr 757 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝑈𝑃)
11425ad2antrr 757 . . . 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 25359 . . 3 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
11618ad2antrr 757 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐷 ∈ ran 𝐿)
1176adantr 479 . . . . . 6 ((𝜑𝑅𝑆) → 𝐺 ∈ TarskiG)
118117adantr 479 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐺 ∈ TarskiG)
11914ad2antrr 757 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐶𝑃)
12012adantr 479 . . . . . 6 ((𝜑𝑅𝑆) → 𝐴𝑃)
121120adantr 479 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐴𝑃)
12233adantr 479 . . . . . 6 ((𝜑𝑅𝑆) → 𝑆𝐷)
123122adantr 479 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑆𝐷)
12419adantr 479 . . . . . 6 ((𝜑𝑅𝑆) → 𝑅𝐷)
125124adantr 479 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑅𝐷)
12610ad2antrr 757 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑀𝑃)
12786ad2antrr 757 . . . . . 6 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐴𝑂𝐶)
1281, 2, 3, 87, 4, 116, 118, 121, 119, 127oppcom 25354 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐶𝑂𝐴)
12935ad2antrr 757 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
13021adantr 479 . . . . . 6 ((𝜑𝑅𝑆) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
131130adantr 479 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
132110necomd 2836 . . . . . 6 ((𝜑𝑅𝑆) → 𝑆𝑅)
133132adantr 479 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑆𝑅)
134 simpr 475 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶))
13516ad2antrr 757 . . . . . 6 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑈𝑃)
1361, 2, 3, 4, 5, 118, 126, 8, 135mircl 25274 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → (𝑁𝑈) ∈ 𝑃)
13720adantr 479 . . . . . . 7 ((𝜑𝑅𝑆) → 𝑅𝑃)
138137adantr 479 . . . . . 6 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑅𝑃)
13925ad2antrr 757 . . . . . 6 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → (𝑁𝑅) = 𝑆)
1401, 2, 3, 4, 5, 118, 126, 8, 138, 139mircom 25276 . . . . 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 25359 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → ((𝑁𝑈)(𝐾𝑆)𝐶 ↔ (𝑁‘(𝑁𝑈))(𝐾𝑅)𝐴))
1421, 2, 3, 4, 5, 118, 126, 8, 135mirmir 25275 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → (𝑁‘(𝑁𝑈)) = 𝑈)
143142breq1d 4587 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → ((𝑁‘(𝑁𝑈))(𝐾𝑅)𝐴𝑈(𝐾𝑅)𝐴))
144141, 143bitr2d 267 . . 3 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
145 eqid 2609 . . . . 5 (≤G‘𝐺) = (≤G‘𝐺)
1461, 2, 3, 145, 6, 34, 14, 20, 12legtrid 25204 . . . 4 (𝜑 → ((𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴) ∨ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)))
147146adantr 479 . . 3 ((𝜑𝑅𝑆) → ((𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴) ∨ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)))
148115, 144, 147mpjaodan 822 . 2 ((𝜑𝑅𝑆) → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
14999, 148pm2.61dane 2868 1 (𝜑 → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1976  wne 2779  wrex 2896  cdif 3536   class class class wbr 4577  {copab 4636  ran crn 5029  cfv 5790  (class class class)co 6527  Basecbs 15641  distcds 15723  TarskiGcstrkg 25046  Itvcitv 25052  LineGclng 25053  ≤Gcleg 25195  hlGchlg 25213  pInvGcmir 25265  ⟂Gcperpg 25308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-hash 12935  df-word 13100  df-concat 13102  df-s1 13103  df-s2 13390  df-s3 13391  df-trkgc 25064  df-trkgb 25065  df-trkgcb 25066  df-trkg 25069  df-cgrg 25124  df-leg 25196  df-hlg 25214  df-mir 25266  df-rag 25307  df-perpg 25309
This theorem is referenced by:  opphl  25364
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