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Theorem mideulem2 27676
Description: Lemma for opphllem 27677, which is itself used for mideu 27680. (Contributed by Thierry Arnoux, 19-Feb-2020.)
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))
mideulem2.1 (𝜑𝑋𝑃)
mideulem2.2 (𝜑𝑋 ∈ (𝑇𝐼𝐵))
mideulem2.3 (𝜑𝑋 ∈ (𝑅𝐼𝑂))
mideulem2.4 (𝜑𝑍𝑃)
mideulem2.5 (𝜑𝑋 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))
mideulem2.6 (𝜑 → (𝑋 𝑍) = (𝑋 𝑅))
mideulem2.7 (𝜑𝑀𝑃)
mideulem2.8 (𝜑𝑅 = ((𝑆𝑀)‘𝑍))
Assertion
Ref Expression
mideulem2 (𝜑𝐵 = 𝑀)

Proof of Theorem mideulem2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 oveq2 7365 . . 3 (𝑦 = 𝐵 → (𝑅𝐿𝑦) = (𝑅𝐿𝐵))
21breq1d 5115 . 2 (𝑦 = 𝐵 → ((𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵) ↔ (𝑅𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵)))
3 oveq2 7365 . . 3 (𝑦 = 𝑀 → (𝑅𝐿𝑦) = (𝑅𝐿𝑀))
43breq1d 5115 . 2 (𝑦 = 𝑀 → ((𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵) ↔ (𝑅𝐿𝑀)(⟂G‘𝐺)(𝐴𝐿𝐵)))
5 colperpex.p . . 3 𝑃 = (Base‘𝐺)
6 colperpex.d . . 3 = (dist‘𝐺)
7 colperpex.i . . 3 𝐼 = (Itv‘𝐺)
8 colperpex.l . . 3 𝐿 = (LineG‘𝐺)
9 colperpex.g . . 3 (𝜑𝐺 ∈ TarskiG)
10 mideu.1 . . . 4 (𝜑𝐴𝑃)
11 mideu.2 . . . 4 (𝜑𝐵𝑃)
12 mideulem.1 . . . 4 (𝜑𝐴𝐵)
135, 7, 8, 9, 10, 11, 12tgelrnln 27572 . . 3 (𝜑 → (𝐴𝐿𝐵) ∈ ran 𝐿)
14 opphllem.1 . . 3 (𝜑𝑅𝑃)
1512adantr 481 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝐴𝐵)
1615neneqd 2948 . . . . 5 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → ¬ 𝐴 = 𝐵)
17 mideulem.3 . . . . . . . . 9 (𝜑𝑂𝑃)
18 opphllem.3 . . . . . . . . 9 (𝜑 → (𝐴 𝑂) = (𝐵 𝑅))
19 mideulem.6 . . . . . . . . . . 11 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝑂))
208, 9, 19perpln2 27653 . . . . . . . . . 10 (𝜑 → (𝐴𝐿𝑂) ∈ ran 𝐿)
215, 7, 8, 9, 10, 17, 20tglnne 27570 . . . . . . . . 9 (𝜑𝐴𝑂)
225, 6, 7, 9, 10, 17, 11, 14, 18, 21tgcgrneq 27425 . . . . . . . 8 (𝜑𝐵𝑅)
2322adantr 481 . . . . . . 7 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝐵𝑅)
2423necomd 2999 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝑅𝐵)
2524neneqd 2948 . . . . 5 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → ¬ 𝑅 = 𝐵)
2616, 25jca 512 . . . 4 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → (¬ 𝐴 = 𝐵 ∧ ¬ 𝑅 = 𝐵))
27 mideu.s . . . . . 6 𝑆 = (pInvG‘𝐺)
289adantr 481 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝐺 ∈ TarskiG)
2910adantr 481 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝐴𝑃)
3011adantr 481 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝐵𝑃)
3114adantr 481 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝑅𝑃)
32 mideulem.2 . . . . . . . . 9 (𝜑𝑄𝑃)
33 mideulem.5 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))
348, 9, 33perpln2 27653 . . . . . . . . . . . 12 (𝜑 → (𝑄𝐿𝐵) ∈ ran 𝐿)
355, 7, 8, 9, 32, 11, 34tglnne 27570 . . . . . . . . . . 11 (𝜑𝑄𝐵)
365, 7, 8, 9, 32, 11, 35tglinerflx2 27576 . . . . . . . . . 10 (𝜑𝐵 ∈ (𝑄𝐿𝐵))
375, 6, 7, 8, 9, 13, 34, 33perpcom 27655 . . . . . . . . . . 11 (𝜑 → (𝑄𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵))
385, 7, 8, 9, 10, 11, 12tglinecom 27577 . . . . . . . . . . 11 (𝜑 → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
3937, 38breqtrd 5131 . . . . . . . . . 10 (𝜑 → (𝑄𝐿𝐵)(⟂G‘𝐺)(𝐵𝐿𝐴))
405, 6, 7, 8, 9, 32, 11, 36, 10, 39perprag 27668 . . . . . . . . 9 (𝜑 → ⟨“𝑄𝐵𝐴”⟩ ∈ (∟G‘𝐺))
41 opphllem.2 . . . . . . . . . 10 (𝜑𝑅 ∈ (𝐵𝐼𝑄))
425, 8, 7, 9, 11, 14, 32, 41btwncolg3 27499 . . . . . . . . 9 (𝜑 → (𝑄 ∈ (𝐵𝐿𝑅) ∨ 𝐵 = 𝑅))
435, 6, 7, 8, 27, 9, 32, 11, 10, 14, 40, 35, 42ragcol 27641 . . . . . . . 8 (𝜑 → ⟨“𝑅𝐵𝐴”⟩ ∈ (∟G‘𝐺))
445, 6, 7, 8, 27, 9, 14, 11, 10, 43ragcom 27640 . . . . . . 7 (𝜑 → ⟨“𝐴𝐵𝑅”⟩ ∈ (∟G‘𝐺))
4544adantr 481 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → ⟨“𝐴𝐵𝑅”⟩ ∈ (∟G‘𝐺))
46 animorrl 979 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → (𝑅 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
475, 6, 7, 8, 27, 28, 29, 30, 31, 45, 46ragflat3 27648 . . . . 5 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → (𝐴 = 𝐵𝑅 = 𝐵))
48 oran 988 . . . . 5 ((𝐴 = 𝐵𝑅 = 𝐵) ↔ ¬ (¬ 𝐴 = 𝐵 ∧ ¬ 𝑅 = 𝐵))
4947, 48sylib 217 . . . 4 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → ¬ (¬ 𝐴 = 𝐵 ∧ ¬ 𝑅 = 𝐵))
5026, 49pm2.65da 815 . . 3 (𝜑 → ¬ 𝑅 ∈ (𝐴𝐿𝐵))
515, 6, 7, 8, 9, 13, 14, 50foot 27664 . 2 (𝜑 → ∃!𝑦 ∈ (𝐴𝐿𝐵)(𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵))
525, 7, 8, 9, 10, 11, 12tglinerflx2 27576 . 2 (𝜑𝐵 ∈ (𝐴𝐿𝐵))
53 mideulem2.1 . . 3 (𝜑𝑋𝑃)
5412neneqd 2948 . . . . 5 (𝜑 → ¬ 𝐴 = 𝐵)
55 oveq2 7365 . . . . . . 7 (𝑦 = 𝐴 → (𝑅𝐿𝑦) = (𝑅𝐿𝐴))
5655breq1d 5115 . . . . . 6 (𝑦 = 𝐴 → ((𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵) ↔ (𝑅𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝐵)))
5751adantr 481 . . . . . 6 ((𝜑𝑋 = 𝐴) → ∃!𝑦 ∈ (𝐴𝐿𝐵)(𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵))
585, 7, 8, 9, 10, 11, 12tglinerflx1 27575 . . . . . . 7 (𝜑𝐴 ∈ (𝐴𝐿𝐵))
5958adantr 481 . . . . . 6 ((𝜑𝑋 = 𝐴) → 𝐴 ∈ (𝐴𝐿𝐵))
6052adantr 481 . . . . . 6 ((𝜑𝑋 = 𝐴) → 𝐵 ∈ (𝐴𝐿𝐵))
619adantr 481 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝐺 ∈ TarskiG)
6214adantr 481 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑅𝑃)
6310adantr 481 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝐴𝑃)
6450, 54jca 512 . . . . . . . . . . . 12 (𝜑 → (¬ 𝑅 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵))
65 pm4.56 987 . . . . . . . . . . . 12 ((¬ 𝑅 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵) ↔ ¬ (𝑅 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
6664, 65sylib 217 . . . . . . . . . . 11 (𝜑 → ¬ (𝑅 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
675, 7, 8, 9, 14, 10, 11, 66ncolne1 27567 . . . . . . . . . 10 (𝜑𝑅𝐴)
6867adantr 481 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑅𝐴)
695, 7, 8, 61, 62, 63, 68tglinecom 27577 . . . . . . . 8 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐴) = (𝐴𝐿𝑅))
7068necomd 2999 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝐴𝑅)
7117adantr 481 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑂𝑃)
7221necomd 2999 . . . . . . . . . 10 (𝜑𝑂𝐴)
7372adantr 481 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑂𝐴)
7453adantr 481 . . . . . . . . . . 11 ((𝜑𝑋 = 𝐴) → 𝑋𝑃)
75 simpr 485 . . . . . . . . . . . 12 ((𝜑𝑋 = 𝐴) → 𝑋 = 𝐴)
7675, 70eqnetrd 3011 . . . . . . . . . . 11 ((𝜑𝑋 = 𝐴) → 𝑋𝑅)
77 mideulem2.3 . . . . . . . . . . . . . . . 16 (𝜑𝑋 ∈ (𝑅𝐼𝑂))
785, 6, 7, 9, 14, 53, 17, 77tgbtwncom 27430 . . . . . . . . . . . . . . 15 (𝜑𝑋 ∈ (𝑂𝐼𝑅))
79 mideulem.4 . . . . . . . . . . . . . . . . 17 (𝜑𝑇𝑃)
80 mideulem.7 . . . . . . . . . . . . . . . . 17 (𝜑𝑇 ∈ (𝐴𝐿𝐵))
81 mideulem2.2 . . . . . . . . . . . . . . . . 17 (𝜑𝑋 ∈ (𝑇𝐼𝐵))
825, 7, 8, 9, 79, 10, 11, 53, 80, 81coltr3 27590 . . . . . . . . . . . . . . . 16 (𝜑𝑋 ∈ (𝐴𝐿𝐵))
8312necomd 2999 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐵𝐴)
8483neneqd 2948 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ¬ 𝐵 = 𝐴)
8584adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝐵 = 𝐴)
8672neneqd 2948 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ¬ 𝑂 = 𝐴)
8786adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝑂 = 𝐴)
8885, 87jca 512 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
899adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐺 ∈ TarskiG)
9011adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐵𝑃)
9110adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐴𝑃)
9217adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝑂𝑃)
935, 7, 8, 9, 11, 10, 83tglinerflx2 27576 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐴 ∈ (𝐵𝐿𝐴))
9438, 19eqbrtrrd 5129 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐵𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝑂))
955, 6, 7, 8, 9, 11, 10, 93, 17, 94perprag 27668 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
9695adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
97 animorrl 979 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (𝑂 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
985, 6, 7, 8, 27, 89, 90, 91, 92, 96, 97ragflat3 27648 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (𝐵 = 𝐴𝑂 = 𝐴))
99 oran 988 . . . . . . . . . . . . . . . . . . 19 ((𝐵 = 𝐴𝑂 = 𝐴) ↔ ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
10098, 99sylib 217 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
10188, 100pm2.65da 815 . . . . . . . . . . . . . . . . 17 (𝜑 → ¬ 𝑂 ∈ (𝐵𝐿𝐴))
102101, 38neleqtrrd 2860 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
103 nelne2 3042 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝑂 ∈ (𝐴𝐿𝐵)) → 𝑋𝑂)
10482, 102, 103syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑𝑋𝑂)
1055, 6, 7, 9, 17, 53, 14, 78, 104tgbtwnne 27432 . . . . . . . . . . . . . 14 (𝜑𝑂𝑅)
106105adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑋 = 𝐴) → 𝑂𝑅)
107106necomd 2999 . . . . . . . . . . . 12 ((𝜑𝑋 = 𝐴) → 𝑅𝑂)
10877adantr 481 . . . . . . . . . . . 12 ((𝜑𝑋 = 𝐴) → 𝑋 ∈ (𝑅𝐼𝑂))
1095, 7, 8, 61, 62, 71, 74, 107, 108btwnlng1 27561 . . . . . . . . . . 11 ((𝜑𝑋 = 𝐴) → 𝑋 ∈ (𝑅𝐿𝑂))
1105, 7, 8, 61, 74, 62, 71, 76, 109, 107lnrot2 27566 . . . . . . . . . 10 ((𝜑𝑋 = 𝐴) → 𝑂 ∈ (𝑋𝐿𝑅))
11175oveq1d 7372 . . . . . . . . . 10 ((𝜑𝑋 = 𝐴) → (𝑋𝐿𝑅) = (𝐴𝐿𝑅))
112110, 111eleqtrd 2840 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑂 ∈ (𝐴𝐿𝑅))
1135, 7, 8, 61, 63, 62, 70, 71, 73, 112tglineelsb2 27574 . . . . . . . 8 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝑅) = (𝐴𝐿𝑂))
11469, 113eqtrd 2776 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐴) = (𝐴𝐿𝑂))
1155, 6, 7, 8, 9, 13, 20, 19perpcom 27655 . . . . . . . 8 (𝜑 → (𝐴𝐿𝑂)(⟂G‘𝐺)(𝐴𝐿𝐵))
116115adantr 481 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝑂)(⟂G‘𝐺)(𝐴𝐿𝐵))
117114, 116eqbrtrd 5127 . . . . . 6 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝐵))
11813adantr 481 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝐵) ∈ ran 𝐿)
11922necomd 2999 . . . . . . . . 9 (𝜑𝑅𝐵)
1205, 7, 8, 9, 14, 11, 119tgelrnln 27572 . . . . . . . 8 (𝜑 → (𝑅𝐿𝐵) ∈ ran 𝐿)
121120adantr 481 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐵) ∈ ran 𝐿)
1225, 7, 8, 9, 14, 11, 119tglinerflx2 27576 . . . . . . . . . 10 (𝜑𝐵 ∈ (𝑅𝐿𝐵))
12352, 122elind 4154 . . . . . . . . 9 (𝜑𝐵 ∈ ((𝐴𝐿𝐵) ∩ (𝑅𝐿𝐵)))
1245, 7, 8, 9, 14, 11, 119tglinerflx1 27575 . . . . . . . . 9 (𝜑𝑅 ∈ (𝑅𝐿𝐵))
1255, 6, 7, 8, 9, 13, 120, 123, 58, 124, 12, 119, 44ragperp 27659 . . . . . . . 8 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑅𝐿𝐵))
126125adantr 481 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑅𝐿𝐵))
1275, 6, 7, 8, 61, 118, 121, 126perpcom 27655 . . . . . 6 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵))
12856, 2, 57, 59, 60, 117, 127reu2eqd 3694 . . . . 5 ((𝜑𝑋 = 𝐴) → 𝐴 = 𝐵)
12954, 128mtand 814 . . . 4 (𝜑 → ¬ 𝑋 = 𝐴)
130129neqned 2950 . . 3 (𝜑𝑋𝐴)
131 mideulem2.7 . . 3 (𝜑𝑀𝑃)
132130necomd 2999 . . . 4 (𝜑𝐴𝑋)
133 eqid 2736 . . . . 5 (𝑆𝐴) = (𝑆𝐴)
134 eqid 2736 . . . . 5 (𝑆𝑀) = (𝑆𝑀)
1355, 6, 7, 8, 27, 9, 10, 133, 17mircl 27603 . . . . 5 (𝜑 → ((𝑆𝐴)‘𝑂) ∈ 𝑃)
136 mideulem2.4 . . . . 5 (𝜑𝑍𝑃)
137 mideulem2.5 . . . . 5 (𝜑𝑋 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))
13882orcd 871 . . . . . . . . 9 (𝜑 → (𝑋 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
1395, 8, 7, 9, 10, 11, 53, 138colcom 27500 . . . . . . . 8 (𝜑 → (𝑋 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
1405, 8, 7, 9, 11, 10, 53, 139colrot1 27501 . . . . . . 7 (𝜑 → (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
1415, 6, 7, 8, 27, 9, 11, 10, 17, 53, 95, 83, 140ragcol 27641 . . . . . 6 (𝜑 → ⟨“𝑋𝐴𝑂”⟩ ∈ (∟G‘𝐺))
1425, 6, 7, 8, 27, 9, 53, 10, 17israg 27639 . . . . . 6 (𝜑 → (⟨“𝑋𝐴𝑂”⟩ ∈ (∟G‘𝐺) ↔ (𝑋 𝑂) = (𝑋 ((𝑆𝐴)‘𝑂))))
143141, 142mpbid 231 . . . . 5 (𝜑 → (𝑋 𝑂) = (𝑋 ((𝑆𝐴)‘𝑂)))
144 mideulem2.6 . . . . . 6 (𝜑 → (𝑋 𝑍) = (𝑋 𝑅))
145144eqcomd 2742 . . . . 5 (𝜑 → (𝑋 𝑅) = (𝑋 𝑍))
146 eqidd 2737 . . . . 5 (𝜑 → ((𝑆𝐴)‘𝑂) = ((𝑆𝐴)‘𝑂))
147 mideulem2.8 . . . . . . . 8 (𝜑𝑅 = ((𝑆𝑀)‘𝑍))
148147eqcomd 2742 . . . . . . 7 (𝜑 → ((𝑆𝑀)‘𝑍) = 𝑅)
1495, 6, 7, 8, 27, 9, 131, 134, 136, 148mircom 27605 . . . . . 6 (𝜑 → ((𝑆𝑀)‘𝑅) = 𝑍)
150149eqcomd 2742 . . . . 5 (𝜑𝑍 = ((𝑆𝑀)‘𝑅))
1515, 6, 7, 8, 27, 9, 133, 134, 17, 135, 53, 14, 136, 10, 131, 78, 137, 143, 145, 146, 150krippen 27633 . . . 4 (𝜑𝑋 ∈ (𝐴𝐼𝑀))
1525, 7, 8, 9, 10, 53, 131, 132, 151btwnlng3 27563 . . 3 (𝜑𝑀 ∈ (𝐴𝐿𝑋))
1535, 7, 8, 9, 10, 11, 12, 53, 130, 82, 131, 152tglineeltr 27573 . 2 (𝜑𝑀 ∈ (𝐴𝐿𝐵))
1545, 6, 7, 8, 9, 13, 120, 125perpcom 27655 . 2 (𝜑 → (𝑅𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵))
155 nelne2 3042 . . . . . 6 ((𝑀 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝑅 ∈ (𝐴𝐿𝐵)) → 𝑀𝑅)
156153, 50, 155syl2anc 584 . . . . 5 (𝜑𝑀𝑅)
157156necomd 2999 . . . 4 (𝜑𝑅𝑀)
1585, 7, 8, 9, 14, 131, 157tgelrnln 27572 . . 3 (𝜑 → (𝑅𝐿𝑀) ∈ ran 𝐿)
1595, 7, 8, 9, 14, 131, 157tglinerflx2 27576 . . . . 5 (𝜑𝑀 ∈ (𝑅𝐿𝑀))
160153, 159elind 4154 . . . 4 (𝜑𝑀 ∈ ((𝐴𝐿𝐵) ∩ (𝑅𝐿𝑀)))
1615, 7, 8, 9, 14, 131, 157tglinerflx1 27575 . . . 4 (𝜑𝑅 ∈ (𝑅𝐿𝑀))
162 simpr 485 . . . . . . . 8 ((𝜑𝑀 = 𝑋) → 𝑀 = 𝑋)
1639adantr 481 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → 𝐺 ∈ TarskiG)
164131adantr 481 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → 𝑀𝑃)
16510adantr 481 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → 𝐴𝑃)
16617adantr 481 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → 𝑂𝑃)
167135adantr 481 . . . . . . . . . . 11 ((𝜑𝑀 = 𝑋) → ((𝑆𝐴)‘𝑂) ∈ 𝑃)
168143adantr 481 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → (𝑋 𝑂) = (𝑋 ((𝑆𝐴)‘𝑂)))
169162oveq1d 7372 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → (𝑀 𝑂) = (𝑋 𝑂))
170162oveq1d 7372 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → (𝑀 ((𝑆𝐴)‘𝑂)) = (𝑋 ((𝑆𝐴)‘𝑂)))
171168, 169, 1703eqtr4rd 2787 . . . . . . . . . . 11 ((𝜑𝑀 = 𝑋) → (𝑀 ((𝑆𝐴)‘𝑂)) = (𝑀 𝑂))
172136adantr 481 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑍𝑃)
17314adantr 481 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑅𝑃)
174147adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑀 = 𝑋) → 𝑅 = ((𝑆𝑀)‘𝑍))
175174oveq2d 7373 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → (𝑀 𝑅) = (𝑀 ((𝑆𝑀)‘𝑍)))
1765, 6, 7, 8, 27, 163, 164, 134, 172mircgr 27599 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → (𝑀 ((𝑆𝑀)‘𝑍)) = (𝑀 𝑍))
177175, 176eqtrd 2776 . . . . . . . . . . . . . 14 ((𝜑𝑀 = 𝑋) → (𝑀 𝑅) = (𝑀 𝑍))
1785, 6, 7, 163, 164, 173, 164, 172, 177tgcgrcomlr 27422 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → (𝑅 𝑀) = (𝑍 𝑀))
17982adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑀 = 𝑋) → 𝑋 ∈ (𝐴𝐿𝐵))
180162, 179eqeltrd 2838 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝐴𝐿𝐵))
18150adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → ¬ 𝑅 ∈ (𝐴𝐿𝐵))
182180, 181, 155syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑀 = 𝑋) → 𝑀𝑅)
183182necomd 2999 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑅𝑀)
1845, 6, 7, 163, 173, 164, 172, 164, 178, 183tgcgrneq 27425 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑍𝑀)
1855, 6, 7, 8, 27, 9, 131, 134, 136mirbtwn 27600 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ (((𝑆𝑀)‘𝑍)𝐼𝑍))
186147oveq1d 7372 . . . . . . . . . . . . . . 15 (𝜑 → (𝑅𝐼𝑍) = (((𝑆𝑀)‘𝑍)𝐼𝑍))
187185, 186eleqtrrd 2841 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (𝑅𝐼𝑍))
188187adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑅𝐼𝑍))
1895, 6, 7, 163, 173, 164, 172, 188tgbtwncom 27430 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑍𝐼𝑅))
190137adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑀 = 𝑋) → 𝑋 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))
191162, 190eqeltrd 2838 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))
1925, 6, 7, 163, 167, 164, 172, 191tgbtwncom 27430 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑍𝐼((𝑆𝐴)‘𝑂)))
19377adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑋 ∈ (𝑅𝐼𝑂))
194162, 193eqeltrd 2838 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑅𝐼𝑂))
1955, 7, 163, 172, 164, 173, 167, 166, 184, 183, 189, 192, 194tgbtwnconn22 27521 . . . . . . . . . . 11 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑂))
1965, 6, 7, 8, 27, 163, 164, 134, 166, 167, 171, 195ismir 27601 . . . . . . . . . 10 ((𝜑𝑀 = 𝑋) → ((𝑆𝐴)‘𝑂) = ((𝑆𝑀)‘𝑂))
197196eqcomd 2742 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → ((𝑆𝑀)‘𝑂) = ((𝑆𝐴)‘𝑂))
1985, 6, 7, 8, 27, 163, 164, 165, 166, 197miduniq1 27628 . . . . . . . 8 ((𝜑𝑀 = 𝑋) → 𝑀 = 𝐴)
199162, 198eqtr3d 2778 . . . . . . 7 ((𝜑𝑀 = 𝑋) → 𝑋 = 𝐴)
200129, 199mtand 814 . . . . . 6 (𝜑 → ¬ 𝑀 = 𝑋)
201200neqned 2950 . . . . 5 (𝜑𝑀𝑋)
202201necomd 2999 . . . 4 (𝜑𝑋𝑀)
203149oveq2d 7373 . . . . . 6 (𝜑 → (𝑋 ((𝑆𝑀)‘𝑅)) = (𝑋 𝑍))
204203, 144eqtr2d 2777 . . . . 5 (𝜑 → (𝑋 𝑅) = (𝑋 ((𝑆𝑀)‘𝑅)))
2055, 6, 7, 8, 27, 9, 53, 131, 14israg 27639 . . . . 5 (𝜑 → (⟨“𝑋𝑀𝑅”⟩ ∈ (∟G‘𝐺) ↔ (𝑋 𝑅) = (𝑋 ((𝑆𝑀)‘𝑅))))
206204, 205mpbird 256 . . . 4 (𝜑 → ⟨“𝑋𝑀𝑅”⟩ ∈ (∟G‘𝐺))
2075, 6, 7, 8, 9, 13, 158, 160, 82, 161, 202, 157, 206ragperp 27659 . . 3 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑅𝐿𝑀))
2085, 6, 7, 8, 9, 13, 158, 207perpcom 27655 . 2 (𝜑 → (𝑅𝐿𝑀)(⟂G‘𝐺)(𝐴𝐿𝐵))
2092, 4, 51, 52, 153, 154, 208reu2eqd 3694 1 (𝜑𝐵 = 𝑀)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 845   = wceq 1541  wcel 2106  wne 2943  ∃!wreu 3351   class class class wbr 5105  ran crn 5634  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:  opphllem  27677
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