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Theorem mideulem2 28743
Description: Lemma for opphllem 28744, which is itself used for mideu 28747. (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 7440 . . 3 (𝑦 = 𝐵 → (𝑅𝐿𝑦) = (𝑅𝐿𝐵))
21breq1d 5152 . 2 (𝑦 = 𝐵 → ((𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵) ↔ (𝑅𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵)))
3 oveq2 7440 . . 3 (𝑦 = 𝑀 → (𝑅𝐿𝑦) = (𝑅𝐿𝑀))
43breq1d 5152 . 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 28639 . . 3 (𝜑 → (𝐴𝐿𝐵) ∈ ran 𝐿)
14 opphllem.1 . . 3 (𝜑𝑅𝑃)
1512adantr 480 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝐴𝐵)
1615neneqd 2944 . . . . 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 28720 . . . . . . . . . 10 (𝜑 → (𝐴𝐿𝑂) ∈ ran 𝐿)
215, 7, 8, 9, 10, 17, 20tglnne 28637 . . . . . . . . 9 (𝜑𝐴𝑂)
225, 6, 7, 9, 10, 17, 11, 14, 18, 21tgcgrneq 28492 . . . . . . . 8 (𝜑𝐵𝑅)
2322adantr 480 . . . . . . 7 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝐵𝑅)
2423necomd 2995 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝑅𝐵)
2524neneqd 2944 . . . . 5 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → ¬ 𝑅 = 𝐵)
2616, 25jca 511 . . . 4 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → (¬ 𝐴 = 𝐵 ∧ ¬ 𝑅 = 𝐵))
27 mideu.s . . . . . 6 𝑆 = (pInvG‘𝐺)
289adantr 480 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝐺 ∈ TarskiG)
2910adantr 480 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝐴𝑃)
3011adantr 480 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝐵𝑃)
3114adantr 480 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝑅𝑃)
32 mideulem.2 . . . . . . . . 9 (𝜑𝑄𝑃)
33 mideulem.5 . . . . . . . . . . . . 13 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑄𝐿𝐵))
348, 9, 33perpln2 28720 . . . . . . . . . . . 12 (𝜑 → (𝑄𝐿𝐵) ∈ ran 𝐿)
355, 7, 8, 9, 32, 11, 34tglnne 28637 . . . . . . . . . . 11 (𝜑𝑄𝐵)
365, 7, 8, 9, 32, 11, 35tglinerflx2 28643 . . . . . . . . . 10 (𝜑𝐵 ∈ (𝑄𝐿𝐵))
375, 6, 7, 8, 9, 13, 34, 33perpcom 28722 . . . . . . . . . . 11 (𝜑 → (𝑄𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵))
385, 7, 8, 9, 10, 11, 12tglinecom 28644 . . . . . . . . . . 11 (𝜑 → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
3937, 38breqtrd 5168 . . . . . . . . . 10 (𝜑 → (𝑄𝐿𝐵)(⟂G‘𝐺)(𝐵𝐿𝐴))
405, 6, 7, 8, 9, 32, 11, 36, 10, 39perprag 28735 . . . . . . . . 9 (𝜑 → ⟨“𝑄𝐵𝐴”⟩ ∈ (∟G‘𝐺))
41 opphllem.2 . . . . . . . . . 10 (𝜑𝑅 ∈ (𝐵𝐼𝑄))
425, 8, 7, 9, 11, 14, 32, 41btwncolg3 28566 . . . . . . . . 9 (𝜑 → (𝑄 ∈ (𝐵𝐿𝑅) ∨ 𝐵 = 𝑅))
435, 6, 7, 8, 27, 9, 32, 11, 10, 14, 40, 35, 42ragcol 28708 . . . . . . . 8 (𝜑 → ⟨“𝑅𝐵𝐴”⟩ ∈ (∟G‘𝐺))
445, 6, 7, 8, 27, 9, 14, 11, 10, 43ragcom 28707 . . . . . . 7 (𝜑 → ⟨“𝐴𝐵𝑅”⟩ ∈ (∟G‘𝐺))
4544adantr 480 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → ⟨“𝐴𝐵𝑅”⟩ ∈ (∟G‘𝐺))
46 animorrl 982 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → (𝑅 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
475, 6, 7, 8, 27, 28, 29, 30, 31, 45, 46ragflat3 28715 . . . . 5 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → (𝐴 = 𝐵𝑅 = 𝐵))
48 oran 991 . . . . 5 ((𝐴 = 𝐵𝑅 = 𝐵) ↔ ¬ (¬ 𝐴 = 𝐵 ∧ ¬ 𝑅 = 𝐵))
4947, 48sylib 218 . . . 4 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → ¬ (¬ 𝐴 = 𝐵 ∧ ¬ 𝑅 = 𝐵))
5026, 49pm2.65da 816 . . 3 (𝜑 → ¬ 𝑅 ∈ (𝐴𝐿𝐵))
515, 6, 7, 8, 9, 13, 14, 50foot 28731 . 2 (𝜑 → ∃!𝑦 ∈ (𝐴𝐿𝐵)(𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵))
525, 7, 8, 9, 10, 11, 12tglinerflx2 28643 . 2 (𝜑𝐵 ∈ (𝐴𝐿𝐵))
53 mideulem2.1 . . 3 (𝜑𝑋𝑃)
5412neneqd 2944 . . . . 5 (𝜑 → ¬ 𝐴 = 𝐵)
55 oveq2 7440 . . . . . . 7 (𝑦 = 𝐴 → (𝑅𝐿𝑦) = (𝑅𝐿𝐴))
5655breq1d 5152 . . . . . 6 (𝑦 = 𝐴 → ((𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵) ↔ (𝑅𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝐵)))
5751adantr 480 . . . . . 6 ((𝜑𝑋 = 𝐴) → ∃!𝑦 ∈ (𝐴𝐿𝐵)(𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵))
585, 7, 8, 9, 10, 11, 12tglinerflx1 28642 . . . . . . 7 (𝜑𝐴 ∈ (𝐴𝐿𝐵))
5958adantr 480 . . . . . 6 ((𝜑𝑋 = 𝐴) → 𝐴 ∈ (𝐴𝐿𝐵))
6052adantr 480 . . . . . 6 ((𝜑𝑋 = 𝐴) → 𝐵 ∈ (𝐴𝐿𝐵))
619adantr 480 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝐺 ∈ TarskiG)
6214adantr 480 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑅𝑃)
6310adantr 480 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝐴𝑃)
6450, 54jca 511 . . . . . . . . . . . 12 (𝜑 → (¬ 𝑅 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵))
65 pm4.56 990 . . . . . . . . . . . 12 ((¬ 𝑅 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵) ↔ ¬ (𝑅 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
6664, 65sylib 218 . . . . . . . . . . 11 (𝜑 → ¬ (𝑅 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
675, 7, 8, 9, 14, 10, 11, 66ncolne1 28634 . . . . . . . . . 10 (𝜑𝑅𝐴)
6867adantr 480 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑅𝐴)
695, 7, 8, 61, 62, 63, 68tglinecom 28644 . . . . . . . 8 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐴) = (𝐴𝐿𝑅))
7068necomd 2995 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝐴𝑅)
7117adantr 480 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑂𝑃)
7221necomd 2995 . . . . . . . . . 10 (𝜑𝑂𝐴)
7372adantr 480 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑂𝐴)
7453adantr 480 . . . . . . . . . . 11 ((𝜑𝑋 = 𝐴) → 𝑋𝑃)
75 simpr 484 . . . . . . . . . . . 12 ((𝜑𝑋 = 𝐴) → 𝑋 = 𝐴)
7675, 70eqnetrd 3007 . . . . . . . . . . 11 ((𝜑𝑋 = 𝐴) → 𝑋𝑅)
77 mideulem2.3 . . . . . . . . . . . . . . . 16 (𝜑𝑋 ∈ (𝑅𝐼𝑂))
785, 6, 7, 9, 14, 53, 17, 77tgbtwncom 28497 . . . . . . . . . . . . . . 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 28657 . . . . . . . . . . . . . . . 16 (𝜑𝑋 ∈ (𝐴𝐿𝐵))
8312necomd 2995 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐵𝐴)
8483neneqd 2944 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ¬ 𝐵 = 𝐴)
8584adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝐵 = 𝐴)
8672neneqd 2944 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ¬ 𝑂 = 𝐴)
8786adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝑂 = 𝐴)
8885, 87jca 511 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
899adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐺 ∈ TarskiG)
9011adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐵𝑃)
9110adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐴𝑃)
9217adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝑂𝑃)
935, 7, 8, 9, 11, 10, 83tglinerflx2 28643 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐴 ∈ (𝐵𝐿𝐴))
9438, 19eqbrtrrd 5166 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐵𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝑂))
955, 6, 7, 8, 9, 11, 10, 93, 17, 94perprag 28735 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
9695adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
97 animorrl 982 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (𝑂 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
985, 6, 7, 8, 27, 89, 90, 91, 92, 96, 97ragflat3 28715 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (𝐵 = 𝐴𝑂 = 𝐴))
99 oran 991 . . . . . . . . . . . . . . . . . . 19 ((𝐵 = 𝐴𝑂 = 𝐴) ↔ ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
10098, 99sylib 218 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
10188, 100pm2.65da 816 . . . . . . . . . . . . . . . . 17 (𝜑 → ¬ 𝑂 ∈ (𝐵𝐿𝐴))
102101, 38neleqtrrd 2863 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
103 nelne2 3039 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝑂 ∈ (𝐴𝐿𝐵)) → 𝑋𝑂)
10482, 102, 103syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑𝑋𝑂)
1055, 6, 7, 9, 17, 53, 14, 78, 104tgbtwnne 28499 . . . . . . . . . . . . . 14 (𝜑𝑂𝑅)
106105adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑋 = 𝐴) → 𝑂𝑅)
107106necomd 2995 . . . . . . . . . . . 12 ((𝜑𝑋 = 𝐴) → 𝑅𝑂)
10877adantr 480 . . . . . . . . . . . 12 ((𝜑𝑋 = 𝐴) → 𝑋 ∈ (𝑅𝐼𝑂))
1095, 7, 8, 61, 62, 71, 74, 107, 108btwnlng1 28628 . . . . . . . . . . 11 ((𝜑𝑋 = 𝐴) → 𝑋 ∈ (𝑅𝐿𝑂))
1105, 7, 8, 61, 74, 62, 71, 76, 109, 107lnrot2 28633 . . . . . . . . . 10 ((𝜑𝑋 = 𝐴) → 𝑂 ∈ (𝑋𝐿𝑅))
11175oveq1d 7447 . . . . . . . . . 10 ((𝜑𝑋 = 𝐴) → (𝑋𝐿𝑅) = (𝐴𝐿𝑅))
112110, 111eleqtrd 2842 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑂 ∈ (𝐴𝐿𝑅))
1135, 7, 8, 61, 63, 62, 70, 71, 73, 112tglineelsb2 28641 . . . . . . . 8 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝑅) = (𝐴𝐿𝑂))
11469, 113eqtrd 2776 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐴) = (𝐴𝐿𝑂))
1155, 6, 7, 8, 9, 13, 20, 19perpcom 28722 . . . . . . . 8 (𝜑 → (𝐴𝐿𝑂)(⟂G‘𝐺)(𝐴𝐿𝐵))
116115adantr 480 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝑂)(⟂G‘𝐺)(𝐴𝐿𝐵))
117114, 116eqbrtrd 5164 . . . . . 6 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝐵))
11813adantr 480 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝐵) ∈ ran 𝐿)
11922necomd 2995 . . . . . . . . 9 (𝜑𝑅𝐵)
1205, 7, 8, 9, 14, 11, 119tgelrnln 28639 . . . . . . . 8 (𝜑 → (𝑅𝐿𝐵) ∈ ran 𝐿)
121120adantr 480 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐵) ∈ ran 𝐿)
1225, 7, 8, 9, 14, 11, 119tglinerflx2 28643 . . . . . . . . . 10 (𝜑𝐵 ∈ (𝑅𝐿𝐵))
12352, 122elind 4199 . . . . . . . . 9 (𝜑𝐵 ∈ ((𝐴𝐿𝐵) ∩ (𝑅𝐿𝐵)))
1245, 7, 8, 9, 14, 11, 119tglinerflx1 28642 . . . . . . . . 9 (𝜑𝑅 ∈ (𝑅𝐿𝐵))
1255, 6, 7, 8, 9, 13, 120, 123, 58, 124, 12, 119, 44ragperp 28726 . . . . . . . 8 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑅𝐿𝐵))
126125adantr 480 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑅𝐿𝐵))
1275, 6, 7, 8, 61, 118, 121, 126perpcom 28722 . . . . . 6 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵))
12856, 2, 57, 59, 60, 117, 127reu2eqd 3741 . . . . 5 ((𝜑𝑋 = 𝐴) → 𝐴 = 𝐵)
12954, 128mtand 815 . . . 4 (𝜑 → ¬ 𝑋 = 𝐴)
130129neqned 2946 . . 3 (𝜑𝑋𝐴)
131 mideulem2.7 . . 3 (𝜑𝑀𝑃)
132130necomd 2995 . . . 4 (𝜑𝐴𝑋)
133 eqid 2736 . . . . 5 (𝑆𝐴) = (𝑆𝐴)
134 eqid 2736 . . . . 5 (𝑆𝑀) = (𝑆𝑀)
1355, 6, 7, 8, 27, 9, 10, 133, 17mircl 28670 . . . . 5 (𝜑 → ((𝑆𝐴)‘𝑂) ∈ 𝑃)
136 mideulem2.4 . . . . 5 (𝜑𝑍𝑃)
137 mideulem2.5 . . . . 5 (𝜑𝑋 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))
13882orcd 873 . . . . . . . . 9 (𝜑 → (𝑋 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
1395, 8, 7, 9, 10, 11, 53, 138colcom 28567 . . . . . . . 8 (𝜑 → (𝑋 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
1405, 8, 7, 9, 11, 10, 53, 139colrot1 28568 . . . . . . 7 (𝜑 → (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
1415, 6, 7, 8, 27, 9, 11, 10, 17, 53, 95, 83, 140ragcol 28708 . . . . . 6 (𝜑 → ⟨“𝑋𝐴𝑂”⟩ ∈ (∟G‘𝐺))
1425, 6, 7, 8, 27, 9, 53, 10, 17israg 28706 . . . . . 6 (𝜑 → (⟨“𝑋𝐴𝑂”⟩ ∈ (∟G‘𝐺) ↔ (𝑋 𝑂) = (𝑋 ((𝑆𝐴)‘𝑂))))
143141, 142mpbid 232 . . . . 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 28672 . . . . . 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 28700 . . . 4 (𝜑𝑋 ∈ (𝐴𝐼𝑀))
1525, 7, 8, 9, 10, 53, 131, 132, 151btwnlng3 28630 . . 3 (𝜑𝑀 ∈ (𝐴𝐿𝑋))
1535, 7, 8, 9, 10, 11, 12, 53, 130, 82, 131, 152tglineeltr 28640 . 2 (𝜑𝑀 ∈ (𝐴𝐿𝐵))
1545, 6, 7, 8, 9, 13, 120, 125perpcom 28722 . 2 (𝜑 → (𝑅𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵))
155 nelne2 3039 . . . . . 6 ((𝑀 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝑅 ∈ (𝐴𝐿𝐵)) → 𝑀𝑅)
156153, 50, 155syl2anc 584 . . . . 5 (𝜑𝑀𝑅)
157156necomd 2995 . . . 4 (𝜑𝑅𝑀)
1585, 7, 8, 9, 14, 131, 157tgelrnln 28639 . . 3 (𝜑 → (𝑅𝐿𝑀) ∈ ran 𝐿)
1595, 7, 8, 9, 14, 131, 157tglinerflx2 28643 . . . . 5 (𝜑𝑀 ∈ (𝑅𝐿𝑀))
160153, 159elind 4199 . . . 4 (𝜑𝑀 ∈ ((𝐴𝐿𝐵) ∩ (𝑅𝐿𝑀)))
1615, 7, 8, 9, 14, 131, 157tglinerflx1 28642 . . . 4 (𝜑𝑅 ∈ (𝑅𝐿𝑀))
162 simpr 484 . . . . . . . 8 ((𝜑𝑀 = 𝑋) → 𝑀 = 𝑋)
1639adantr 480 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → 𝐺 ∈ TarskiG)
164131adantr 480 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → 𝑀𝑃)
16510adantr 480 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → 𝐴𝑃)
16617adantr 480 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → 𝑂𝑃)
167135adantr 480 . . . . . . . . . . 11 ((𝜑𝑀 = 𝑋) → ((𝑆𝐴)‘𝑂) ∈ 𝑃)
168143adantr 480 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → (𝑋 𝑂) = (𝑋 ((𝑆𝐴)‘𝑂)))
169162oveq1d 7447 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → (𝑀 𝑂) = (𝑋 𝑂))
170162oveq1d 7447 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → (𝑀 ((𝑆𝐴)‘𝑂)) = (𝑋 ((𝑆𝐴)‘𝑂)))
171168, 169, 1703eqtr4rd 2787 . . . . . . . . . . 11 ((𝜑𝑀 = 𝑋) → (𝑀 ((𝑆𝐴)‘𝑂)) = (𝑀 𝑂))
172136adantr 480 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑍𝑃)
17314adantr 480 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑅𝑃)
174147adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑀 = 𝑋) → 𝑅 = ((𝑆𝑀)‘𝑍))
175174oveq2d 7448 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → (𝑀 𝑅) = (𝑀 ((𝑆𝑀)‘𝑍)))
1765, 6, 7, 8, 27, 163, 164, 134, 172mircgr 28666 . . . . . . . . . . . . . . 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 28489 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → (𝑅 𝑀) = (𝑍 𝑀))
17982adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑀 = 𝑋) → 𝑋 ∈ (𝐴𝐿𝐵))
180162, 179eqeltrd 2840 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝐴𝐿𝐵))
18150adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → ¬ 𝑅 ∈ (𝐴𝐿𝐵))
182180, 181, 155syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑀 = 𝑋) → 𝑀𝑅)
183182necomd 2995 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑅𝑀)
1845, 6, 7, 163, 173, 164, 172, 164, 178, 183tgcgrneq 28492 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑍𝑀)
1855, 6, 7, 8, 27, 9, 131, 134, 136mirbtwn 28667 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ (((𝑆𝑀)‘𝑍)𝐼𝑍))
186147oveq1d 7447 . . . . . . . . . . . . . . 15 (𝜑 → (𝑅𝐼𝑍) = (((𝑆𝑀)‘𝑍)𝐼𝑍))
187185, 186eleqtrrd 2843 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (𝑅𝐼𝑍))
188187adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑅𝐼𝑍))
1895, 6, 7, 163, 173, 164, 172, 188tgbtwncom 28497 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑍𝐼𝑅))
190137adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑀 = 𝑋) → 𝑋 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))
191162, 190eqeltrd 2840 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))
1925, 6, 7, 163, 167, 164, 172, 191tgbtwncom 28497 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑍𝐼((𝑆𝐴)‘𝑂)))
19377adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑋 ∈ (𝑅𝐼𝑂))
194162, 193eqeltrd 2840 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑅𝐼𝑂))
1955, 7, 163, 172, 164, 173, 167, 166, 184, 183, 189, 192, 194tgbtwnconn22 28588 . . . . . . . . . . 11 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑂))
1965, 6, 7, 8, 27, 163, 164, 134, 166, 167, 171, 195ismir 28668 . . . . . . . . . 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 28695 . . . . . . . 8 ((𝜑𝑀 = 𝑋) → 𝑀 = 𝐴)
199162, 198eqtr3d 2778 . . . . . . 7 ((𝜑𝑀 = 𝑋) → 𝑋 = 𝐴)
200129, 199mtand 815 . . . . . 6 (𝜑 → ¬ 𝑀 = 𝑋)
201200neqned 2946 . . . . 5 (𝜑𝑀𝑋)
202201necomd 2995 . . . 4 (𝜑𝑋𝑀)
203149oveq2d 7448 . . . . . 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 28706 . . . . 5 (𝜑 → (⟨“𝑋𝑀𝑅”⟩ ∈ (∟G‘𝐺) ↔ (𝑋 𝑅) = (𝑋 ((𝑆𝑀)‘𝑅))))
206204, 205mpbird 257 . . . 4 (𝜑 → ⟨“𝑋𝑀𝑅”⟩ ∈ (∟G‘𝐺))
2075, 6, 7, 8, 9, 13, 158, 160, 82, 161, 202, 157, 206ragperp 28726 . . 3 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑅𝐿𝑀))
2085, 6, 7, 8, 9, 13, 158, 207perpcom 28722 . 2 (𝜑 → (𝑅𝐿𝑀)(⟂G‘𝐺)(𝐴𝐿𝐵))
2092, 4, 51, 52, 153, 154, 208reu2eqd 3741 1 (𝜑𝐵 = 𝑀)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847   = wceq 1539  wcel 2107  wne 2939  ∃!wreu 3377   class class class wbr 5142  ran crn 5685  cfv 6560  (class class class)co 7432  ⟨“cs3 14882  Basecbs 17248  distcds 17307  TarskiGcstrkg 28436  Itvcitv 28442  LineGclng 28443  pInvGcmir 28661  ∟Gcrag 28702  ⟂Gcperpg 28704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-er 8746  df-map 8869  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-dju 9942  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-xnn0 12602  df-z 12616  df-uz 12880  df-fz 13549  df-fzo 13696  df-hash 14371  df-word 14554  df-concat 14610  df-s1 14635  df-s2 14888  df-s3 14889  df-trkgc 28457  df-trkgb 28458  df-trkgcb 28459  df-trkg 28462  df-cgrg 28520  df-leg 28592  df-mir 28662  df-rag 28703  df-perpg 28705
This theorem is referenced by:  opphllem  28744
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