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Theorem opphllem6 28760
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 2737 . . . 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 28557 . . . . . . 7 (𝜑𝑅𝑃)
21 opphllem5.p . . . . . . . 8 (𝜑𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
224, 6, 21perpln2 28719 . . . . . . 7 (𝜑 → (𝐴𝐿𝑅) ∈ ran 𝐿)
231, 3, 4, 6, 12, 20, 22tglnne 28636 . . . . . 6 (𝜑𝐴𝑅)
2423adantr 480 . . . . 5 ((𝜑𝑅 = 𝑆) → 𝐴𝑅)
25 opphllem6.v . . . . . . . 8 (𝜑 → (𝑁𝑅) = 𝑆)
2625adantr 480 . . . . . . 7 ((𝜑𝑅 = 𝑆) → (𝑁𝑅) = 𝑆)
27 simpr 484 . . . . . . 7 ((𝜑𝑅 = 𝑆) → 𝑅 = 𝑆)
2826, 27eqtr4d 2780 . . . . . 6 ((𝜑𝑅 = 𝑆) → (𝑁𝑅) = 𝑅)
291, 2, 3, 4, 5, 6, 10, 8, 20mirinv 28674 . . . . . . 7 (𝜑 → ((𝑁𝑅) = 𝑅𝑀 = 𝑅))
3029adantr 480 . . . . . 6 ((𝜑𝑅 = 𝑆) → ((𝑁𝑅) = 𝑅𝑀 = 𝑅))
3128, 30mpbid 232 . . . . 5 ((𝜑𝑅 = 𝑆) → 𝑀 = 𝑅)
3224, 31neeqtrrd 3015 . . . 4 ((𝜑𝑅 = 𝑆) → 𝐴𝑀)
33 opphllem5.s . . . . . . . 8 (𝜑𝑆𝐷)
341, 4, 3, 6, 18, 33tglnpt 28557 . . . . . . 7 (𝜑𝑆𝑃)
35 opphllem5.q . . . . . . . 8 (𝜑𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
364, 6, 35perpln2 28719 . . . . . . 7 (𝜑 → (𝐶𝐿𝑆) ∈ ran 𝐿)
371, 3, 4, 6, 14, 34, 36tglnne 28636 . . . . . 6 (𝜑𝐶𝑆)
3837adantr 480 . . . . 5 ((𝜑𝑅 = 𝑆) → 𝐶𝑆)
3931, 27eqtrd 2777 . . . . 5 ((𝜑𝑅 = 𝑆) → 𝑀 = 𝑆)
4038, 39neeqtrrd 3015 . . . 4 ((𝜑𝑅 = 𝑆) → 𝐶𝑀)
41 simpr 484 . . . . . . . 8 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅 = 𝑡) → 𝑅 = 𝑡)
426ad4antr 732 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝐺 ∈ TarskiG)
4314ad4antr 732 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝐶𝑃)
4420ad4antr 732 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅𝑃)
456ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
4618ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ ran 𝐿)
47 simplr 769 . . . . . . . . . . 11 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡𝐷)
481, 4, 3, 45, 46, 47tglnpt 28557 . . . . . . . . . 10 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡𝑃)
4948adantr 480 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑡𝑃)
5012ad4antr 732 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝐴𝑃)
5134ad4antr 732 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑆𝑃)
52 simpllr 776 . . . . . . . . . . . 12 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑆)
531, 3, 4, 6, 14, 34, 37tglinerflx2 28642 . . . . . . . . . . . . 13 (𝜑𝑆 ∈ (𝐶𝐿𝑆))
5453ad3antrrr 730 . . . . . . . . . . . 12 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑆 ∈ (𝐶𝐿𝑆))
5552, 54eqeltrd 2841 . . . . . . . . . . 11 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐶𝐿𝑆))
5655adantr 480 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅 ∈ (𝐶𝐿𝑆))
571, 2, 3, 4, 6, 18, 36, 35perpcom 28721 . . . . . . . . . . . 12 (𝜑 → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
5857ad4antr 732 . . . . . . . . . . 11 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)𝐷)
59 simpr 484 . . . . . . . . . . . 12 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅𝑡)
6018ad4antr 732 . . . . . . . . . . . 12 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝐷 ∈ ran 𝐿)
6119ad4antr 732 . . . . . . . . . . . 12 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅𝐷)
62 simpllr 776 . . . . . . . . . . . 12 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑡𝐷)
631, 3, 4, 42, 44, 49, 59, 59, 60, 61, 62tglinethru 28644 . . . . . . . . . . 11 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝐷 = (𝑅𝐿𝑡))
6458, 63breqtrd 5169 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → (𝐶𝐿𝑆)(⟂G‘𝐺)(𝑅𝐿𝑡))
651, 2, 3, 4, 42, 43, 51, 56, 49, 64perprag 28734 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → ⟨“𝐶𝑅𝑡”⟩ ∈ (∟G‘𝐺))
661, 3, 4, 6, 12, 20, 23tglinerflx2 28642 . . . . . . . . . . 11 (𝜑𝑅 ∈ (𝐴𝐿𝑅))
6766ad4antr 732 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅 ∈ (𝐴𝐿𝑅))
681, 2, 3, 4, 6, 18, 22, 21perpcom 28721 . . . . . . . . . . . 12 (𝜑 → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
6968ad4antr 732 . . . . . . . . . . 11 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)𝐷)
7069, 63breqtrd 5169 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → (𝐴𝐿𝑅)(⟂G‘𝐺)(𝑅𝐿𝑡))
711, 2, 3, 4, 42, 50, 44, 67, 49, 70perprag 28734 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → ⟨“𝐴𝑅𝑡”⟩ ∈ (∟G‘𝐺))
72 simplr 769 . . . . . . . . . 10 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑡 ∈ (𝐴𝐼𝐶))
731, 2, 3, 42, 50, 49, 43, 72tgbtwncom 28496 . . . . . . . . 9 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑡 ∈ (𝐶𝐼𝐴))
741, 2, 3, 4, 5, 42, 43, 44, 49, 50, 65, 71, 73ragflat2 28711 . . . . . . . 8 (((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) ∧ 𝑅𝑡) → 𝑅 = 𝑡)
7541, 74pm2.61dane 3029 . . . . . . 7 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 = 𝑡)
76 simpr 484 . . . . . . 7 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑡 ∈ (𝐴𝐼𝐶))
7775, 76eqeltrd 2841 . . . . . 6 ((((𝜑𝑅 = 𝑆) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐶)) → 𝑅 ∈ (𝐴𝐼𝐶))
78 opphllem5.o . . . . . . . . 9 (𝜑𝐴𝑂𝐶)
79 hpg.o . . . . . . . . . 10 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
801, 2, 3, 79, 12, 14islnopp 28747 . . . . . . . . 9 (𝜑 → (𝐴𝑂𝐶 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐶𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐶))))
8178, 80mpbid 232 . . . . . . . 8 (𝜑 → ((¬ 𝐴𝐷 ∧ ¬ 𝐶𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐶)))
8281simprd 495 . . . . . . 7 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐶))
8382adantr 480 . . . . . 6 ((𝜑𝑅 = 𝑆) → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐶))
8477, 83r19.29a 3162 . . . . 5 ((𝜑𝑅 = 𝑆) → 𝑅 ∈ (𝐴𝐼𝐶))
8531, 84eqeltrd 2841 . . . 4 ((𝜑𝑅 = 𝑆) → 𝑀 ∈ (𝐴𝐼𝐶))
861, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 17, 32, 40, 85mirbtwnhl 28688 . . 3 ((𝜑𝑅 = 𝑆) → (𝑈(𝐾𝑀)𝐴 ↔ (𝑁𝑈)(𝐾𝑀)𝐶))
8731fveq2d 6910 . . . 4 ((𝜑𝑅 = 𝑆) → (𝐾𝑀) = (𝐾𝑅))
8887breqd 5154 . . 3 ((𝜑𝑅 = 𝑆) → (𝑈(𝐾𝑀)𝐴𝑈(𝐾𝑅)𝐴))
8939fveq2d 6910 . . . 4 ((𝜑𝑅 = 𝑆) → (𝐾𝑀) = (𝐾𝑆))
9089breqd 5154 . . 3 ((𝜑𝑅 = 𝑆) → ((𝑁𝑈)(𝐾𝑀)𝐶 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
9186, 88, 903bitr3d 309 . 2 ((𝜑𝑅 = 𝑆) → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
9218ad2antrr 726 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐷 ∈ ran 𝐿)
936ad2antrr 726 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐺 ∈ TarskiG)
9412ad2antrr 726 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐴𝑃)
9514ad2antrr 726 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐶𝑃)
9619ad2antrr 726 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝑅𝐷)
9733ad2antrr 726 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝑆𝐷)
9810ad2antrr 726 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝑀𝑃)
9978ad2antrr 726 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐴𝑂𝐶)
10021ad2antrr 726 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
10135ad2antrr 726 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
102 simplr 769 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝑅𝑆)
103 simpr 484 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴))
10416ad2antrr 726 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → 𝑈𝑃)
10525ad2antrr 726 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → (𝑁𝑅) = 𝑆)
1061, 2, 3, 79, 4, 92, 93, 9, 8, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105opphllem3 28757 . . 3 (((𝜑𝑅𝑆) ∧ (𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴)) → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
10718ad2antrr 726 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐷 ∈ ran 𝐿)
1086ad2antrr 726 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐺 ∈ TarskiG)
10914ad2antrr 726 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐶𝑃)
11012ad2antrr 726 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐴𝑃)
11133ad2antrr 726 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑆𝐷)
11219ad2antrr 726 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑅𝐷)
11310ad2antrr 726 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑀𝑃)
11478ad2antrr 726 . . . . . 6 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐴𝑂𝐶)
1151, 2, 3, 79, 4, 107, 108, 110, 109, 114oppcom 28752 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐶𝑂𝐴)
11635ad2antrr 726 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐷(⟂G‘𝐺)(𝐶𝐿𝑆))
11721ad2antrr 726 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝐷(⟂G‘𝐺)(𝐴𝐿𝑅))
118 simpr 484 . . . . . . 7 ((𝜑𝑅𝑆) → 𝑅𝑆)
119118necomd 2996 . . . . . 6 ((𝜑𝑅𝑆) → 𝑆𝑅)
120119adantr 480 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑆𝑅)
121 simpr 484 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶))
12216ad2antrr 726 . . . . . 6 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑈𝑃)
1231, 2, 3, 4, 5, 108, 113, 8, 122mircl 28669 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → (𝑁𝑈) ∈ 𝑃)
12420ad2antrr 726 . . . . . 6 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → 𝑅𝑃)
12525ad2antrr 726 . . . . . 6 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → (𝑁𝑅) = 𝑆)
1261, 2, 3, 4, 5, 108, 113, 8, 124, 125mircom 28671 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → (𝑁𝑆) = 𝑅)
1271, 2, 3, 79, 4, 107, 108, 9, 8, 109, 110, 111, 112, 113, 115, 116, 117, 120, 121, 123, 126opphllem3 28757 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → ((𝑁𝑈)(𝐾𝑆)𝐶 ↔ (𝑁‘(𝑁𝑈))(𝐾𝑅)𝐴))
1281, 2, 3, 4, 5, 108, 113, 8, 122mirmir 28670 . . . . 5 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → (𝑁‘(𝑁𝑈)) = 𝑈)
129128breq1d 5153 . . . 4 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → ((𝑁‘(𝑁𝑈))(𝐾𝑅)𝐴𝑈(𝐾𝑅)𝐴))
130127, 129bitr2d 280 . . 3 (((𝜑𝑅𝑆) ∧ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)) → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
131 eqid 2737 . . . . 5 (≤G‘𝐺) = (≤G‘𝐺)
1321, 2, 3, 131, 6, 34, 14, 20, 12legtrid 28599 . . . 4 (𝜑 → ((𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴) ∨ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)))
133132adantr 480 . . 3 ((𝜑𝑅𝑆) → ((𝑆 𝐶)(≤G‘𝐺)(𝑅 𝐴) ∨ (𝑅 𝐴)(≤G‘𝐺)(𝑆 𝐶)))
134106, 130, 133mpjaodan 961 . 2 ((𝜑𝑅𝑆) → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
13591, 134pm2.61dane 3029 1 (𝜑 → (𝑈(𝐾𝑅)𝐴 ↔ (𝑁𝑈)(𝐾𝑆)𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940  wrex 3070  cdif 3948   class class class wbr 5143  {copab 5205  ran crn 5686  cfv 6561  (class class class)co 7431  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  Itvcitv 28441  LineGclng 28442  ≤Gcleg 28590  hlGchlg 28608  pInvGcmir 28660  ⟂Gcperpg 28703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-s2 14887  df-s3 14888  df-trkgc 28456  df-trkgb 28457  df-trkgcb 28458  df-trkg 28461  df-cgrg 28519  df-leg 28591  df-hlg 28609  df-mir 28661  df-rag 28702  df-perpg 28704
This theorem is referenced by:  opphl  28762
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