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Theorem mideulem2 28760
Description: Lemma for opphllem 28761, which is itself used for mideu 28764. (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 7456 . . 3 (𝑦 = 𝐵 → (𝑅𝐿𝑦) = (𝑅𝐿𝐵))
21breq1d 5176 . 2 (𝑦 = 𝐵 → ((𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵) ↔ (𝑅𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵)))
3 oveq2 7456 . . 3 (𝑦 = 𝑀 → (𝑅𝐿𝑦) = (𝑅𝐿𝑀))
43breq1d 5176 . 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 28656 . . 3 (𝜑 → (𝐴𝐿𝐵) ∈ ran 𝐿)
14 opphllem.1 . . 3 (𝜑𝑅𝑃)
1512adantr 480 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝐴𝐵)
1615neneqd 2951 . . . . 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 28737 . . . . . . . . . 10 (𝜑 → (𝐴𝐿𝑂) ∈ ran 𝐿)
215, 7, 8, 9, 10, 17, 20tglnne 28654 . . . . . . . . 9 (𝜑𝐴𝑂)
225, 6, 7, 9, 10, 17, 11, 14, 18, 21tgcgrneq 28509 . . . . . . . 8 (𝜑𝐵𝑅)
2322adantr 480 . . . . . . 7 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝐵𝑅)
2423necomd 3002 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝑅𝐵)
2524neneqd 2951 . . . . 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 28737 . . . . . . . . . . . 12 (𝜑 → (𝑄𝐿𝐵) ∈ ran 𝐿)
355, 7, 8, 9, 32, 11, 34tglnne 28654 . . . . . . . . . . 11 (𝜑𝑄𝐵)
365, 7, 8, 9, 32, 11, 35tglinerflx2 28660 . . . . . . . . . 10 (𝜑𝐵 ∈ (𝑄𝐿𝐵))
375, 6, 7, 8, 9, 13, 34, 33perpcom 28739 . . . . . . . . . . 11 (𝜑 → (𝑄𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵))
385, 7, 8, 9, 10, 11, 12tglinecom 28661 . . . . . . . . . . 11 (𝜑 → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
3937, 38breqtrd 5192 . . . . . . . . . 10 (𝜑 → (𝑄𝐿𝐵)(⟂G‘𝐺)(𝐵𝐿𝐴))
405, 6, 7, 8, 9, 32, 11, 36, 10, 39perprag 28752 . . . . . . . . 9 (𝜑 → ⟨“𝑄𝐵𝐴”⟩ ∈ (∟G‘𝐺))
41 opphllem.2 . . . . . . . . . 10 (𝜑𝑅 ∈ (𝐵𝐼𝑄))
425, 8, 7, 9, 11, 14, 32, 41btwncolg3 28583 . . . . . . . . 9 (𝜑 → (𝑄 ∈ (𝐵𝐿𝑅) ∨ 𝐵 = 𝑅))
435, 6, 7, 8, 27, 9, 32, 11, 10, 14, 40, 35, 42ragcol 28725 . . . . . . . 8 (𝜑 → ⟨“𝑅𝐵𝐴”⟩ ∈ (∟G‘𝐺))
445, 6, 7, 8, 27, 9, 14, 11, 10, 43ragcom 28724 . . . . . . 7 (𝜑 → ⟨“𝐴𝐵𝑅”⟩ ∈ (∟G‘𝐺))
4544adantr 480 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → ⟨“𝐴𝐵𝑅”⟩ ∈ (∟G‘𝐺))
46 animorrl 981 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → (𝑅 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
475, 6, 7, 8, 27, 28, 29, 30, 31, 45, 46ragflat3 28732 . . . . 5 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → (𝐴 = 𝐵𝑅 = 𝐵))
48 oran 990 . . . . 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 28748 . 2 (𝜑 → ∃!𝑦 ∈ (𝐴𝐿𝐵)(𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵))
525, 7, 8, 9, 10, 11, 12tglinerflx2 28660 . 2 (𝜑𝐵 ∈ (𝐴𝐿𝐵))
53 mideulem2.1 . . 3 (𝜑𝑋𝑃)
5412neneqd 2951 . . . . 5 (𝜑 → ¬ 𝐴 = 𝐵)
55 oveq2 7456 . . . . . . 7 (𝑦 = 𝐴 → (𝑅𝐿𝑦) = (𝑅𝐿𝐴))
5655breq1d 5176 . . . . . 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 28659 . . . . . . 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 pm4.56 989 . . . . . . . . . . . 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 28651 . . . . . . . . . 10 (𝜑𝑅𝐴)
6867adantr 480 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑅𝐴)
695, 7, 8, 61, 62, 63, 68tglinecom 28661 . . . . . . . 8 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐴) = (𝐴𝐿𝑅))
7068necomd 3002 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝐴𝑅)
7117adantr 480 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑂𝑃)
7221necomd 3002 . . . . . . . . . 10 (𝜑𝑂𝐴)
7372adantr 480 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑂𝐴)
7453adantr 480 . . . . . . . . . . 11 ((𝜑𝑋 = 𝐴) → 𝑋𝑃)
75 simpr 484 . . . . . . . . . . . 12 ((𝜑𝑋 = 𝐴) → 𝑋 = 𝐴)
7675, 70eqnetrd 3014 . . . . . . . . . . 11 ((𝜑𝑋 = 𝐴) → 𝑋𝑅)
77 mideulem2.3 . . . . . . . . . . . . . . . 16 (𝜑𝑋 ∈ (𝑅𝐼𝑂))
785, 6, 7, 9, 14, 53, 17, 77tgbtwncom 28514 . . . . . . . . . . . . . . 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 28674 . . . . . . . . . . . . . . . 16 (𝜑𝑋 ∈ (𝐴𝐿𝐵))
8312necomd 3002 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐵𝐴)
8483neneqd 2951 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ¬ 𝐵 = 𝐴)
8584adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝐵 = 𝐴)
8672neneqd 2951 . . . . . . . . . . . . . . . . . . . 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 28660 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐴 ∈ (𝐵𝐿𝐴))
9438, 19eqbrtrrd 5190 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐵𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝑂))
955, 6, 7, 8, 9, 11, 10, 93, 17, 94perprag 28752 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
9695adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
97 animorrl 981 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (𝑂 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
985, 6, 7, 8, 27, 89, 90, 91, 92, 96, 97ragflat3 28732 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (𝐵 = 𝐴𝑂 = 𝐴))
99 oran 990 . . . . . . . . . . . . . . . . . . 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 2867 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
103 nelne2 3046 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝑂 ∈ (𝐴𝐿𝐵)) → 𝑋𝑂)
10482, 102, 103syl2anc 583 . . . . . . . . . . . . . . 15 (𝜑𝑋𝑂)
1055, 6, 7, 9, 17, 53, 14, 78, 104tgbtwnne 28516 . . . . . . . . . . . . . 14 (𝜑𝑂𝑅)
106105adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑋 = 𝐴) → 𝑂𝑅)
107106necomd 3002 . . . . . . . . . . . 12 ((𝜑𝑋 = 𝐴) → 𝑅𝑂)
10877adantr 480 . . . . . . . . . . . 12 ((𝜑𝑋 = 𝐴) → 𝑋 ∈ (𝑅𝐼𝑂))
1095, 7, 8, 61, 62, 71, 74, 107, 108btwnlng1 28645 . . . . . . . . . . 11 ((𝜑𝑋 = 𝐴) → 𝑋 ∈ (𝑅𝐿𝑂))
1105, 7, 8, 61, 74, 62, 71, 76, 109, 107lnrot2 28650 . . . . . . . . . 10 ((𝜑𝑋 = 𝐴) → 𝑂 ∈ (𝑋𝐿𝑅))
11175oveq1d 7463 . . . . . . . . . 10 ((𝜑𝑋 = 𝐴) → (𝑋𝐿𝑅) = (𝐴𝐿𝑅))
112110, 111eleqtrd 2846 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑂 ∈ (𝐴𝐿𝑅))
1135, 7, 8, 61, 63, 62, 70, 71, 73, 112tglineelsb2 28658 . . . . . . . 8 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝑅) = (𝐴𝐿𝑂))
11469, 113eqtrd 2780 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐴) = (𝐴𝐿𝑂))
1155, 6, 7, 8, 9, 13, 20, 19perpcom 28739 . . . . . . . 8 (𝜑 → (𝐴𝐿𝑂)(⟂G‘𝐺)(𝐴𝐿𝐵))
116115adantr 480 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝑂)(⟂G‘𝐺)(𝐴𝐿𝐵))
117114, 116eqbrtrd 5188 . . . . . 6 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝐵))
11813adantr 480 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝐵) ∈ ran 𝐿)
11922necomd 3002 . . . . . . . . 9 (𝜑𝑅𝐵)
1205, 7, 8, 9, 14, 11, 119tgelrnln 28656 . . . . . . . 8 (𝜑 → (𝑅𝐿𝐵) ∈ ran 𝐿)
121120adantr 480 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐵) ∈ ran 𝐿)
1225, 7, 8, 9, 14, 11, 119tglinerflx2 28660 . . . . . . . . . 10 (𝜑𝐵 ∈ (𝑅𝐿𝐵))
12352, 122elind 4223 . . . . . . . . 9 (𝜑𝐵 ∈ ((𝐴𝐿𝐵) ∩ (𝑅𝐿𝐵)))
1245, 7, 8, 9, 14, 11, 119tglinerflx1 28659 . . . . . . . . 9 (𝜑𝑅 ∈ (𝑅𝐿𝐵))
1255, 6, 7, 8, 9, 13, 120, 123, 58, 124, 12, 119, 44ragperp 28743 . . . . . . . 8 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑅𝐿𝐵))
126125adantr 480 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑅𝐿𝐵))
1275, 6, 7, 8, 61, 118, 121, 126perpcom 28739 . . . . . 6 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵))
12856, 2, 57, 59, 60, 117, 127reu2eqd 3758 . . . . 5 ((𝜑𝑋 = 𝐴) → 𝐴 = 𝐵)
12954, 128mtand 815 . . . 4 (𝜑 → ¬ 𝑋 = 𝐴)
130129neqned 2953 . . 3 (𝜑𝑋𝐴)
131 mideulem2.7 . . 3 (𝜑𝑀𝑃)
132130necomd 3002 . . . 4 (𝜑𝐴𝑋)
133 eqid 2740 . . . . 5 (𝑆𝐴) = (𝑆𝐴)
134 eqid 2740 . . . . 5 (𝑆𝑀) = (𝑆𝑀)
1355, 6, 7, 8, 27, 9, 10, 133, 17mircl 28687 . . . . 5 (𝜑 → ((𝑆𝐴)‘𝑂) ∈ 𝑃)
136 mideulem2.4 . . . . 5 (𝜑𝑍𝑃)
137 mideulem2.5 . . . . 5 (𝜑𝑋 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))
13882orcd 872 . . . . . . . . 9 (𝜑 → (𝑋 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
1395, 8, 7, 9, 10, 11, 53, 138colcom 28584 . . . . . . . 8 (𝜑 → (𝑋 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
1405, 8, 7, 9, 11, 10, 53, 139colrot1 28585 . . . . . . 7 (𝜑 → (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
1415, 6, 7, 8, 27, 9, 11, 10, 17, 53, 95, 83, 140ragcol 28725 . . . . . 6 (𝜑 → ⟨“𝑋𝐴𝑂”⟩ ∈ (∟G‘𝐺))
1425, 6, 7, 8, 27, 9, 53, 10, 17israg 28723 . . . . . 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 2746 . . . . 5 (𝜑 → (𝑋 𝑅) = (𝑋 𝑍))
146 eqidd 2741 . . . . 5 (𝜑 → ((𝑆𝐴)‘𝑂) = ((𝑆𝐴)‘𝑂))
147 mideulem2.8 . . . . . . . 8 (𝜑𝑅 = ((𝑆𝑀)‘𝑍))
148147eqcomd 2746 . . . . . . 7 (𝜑 → ((𝑆𝑀)‘𝑍) = 𝑅)
1495, 6, 7, 8, 27, 9, 131, 134, 136, 148mircom 28689 . . . . . 6 (𝜑 → ((𝑆𝑀)‘𝑅) = 𝑍)
150149eqcomd 2746 . . . . 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 28717 . . . 4 (𝜑𝑋 ∈ (𝐴𝐼𝑀))
1525, 7, 8, 9, 10, 53, 131, 132, 151btwnlng3 28647 . . 3 (𝜑𝑀 ∈ (𝐴𝐿𝑋))
1535, 7, 8, 9, 10, 11, 12, 53, 130, 82, 131, 152tglineeltr 28657 . 2 (𝜑𝑀 ∈ (𝐴𝐿𝐵))
1545, 6, 7, 8, 9, 13, 120, 125perpcom 28739 . 2 (𝜑 → (𝑅𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵))
155 nelne2 3046 . . . . . 6 ((𝑀 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝑅 ∈ (𝐴𝐿𝐵)) → 𝑀𝑅)
156153, 50, 155syl2anc 583 . . . . 5 (𝜑𝑀𝑅)
157156necomd 3002 . . . 4 (𝜑𝑅𝑀)
1585, 7, 8, 9, 14, 131, 157tgelrnln 28656 . . 3 (𝜑 → (𝑅𝐿𝑀) ∈ ran 𝐿)
1595, 7, 8, 9, 14, 131, 157tglinerflx2 28660 . . . . 5 (𝜑𝑀 ∈ (𝑅𝐿𝑀))
160153, 159elind 4223 . . . 4 (𝜑𝑀 ∈ ((𝐴𝐿𝐵) ∩ (𝑅𝐿𝑀)))
1615, 7, 8, 9, 14, 131, 157tglinerflx1 28659 . . . 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 7463 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → (𝑀 𝑂) = (𝑋 𝑂))
170162oveq1d 7463 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → (𝑀 ((𝑆𝐴)‘𝑂)) = (𝑋 ((𝑆𝐴)‘𝑂)))
171168, 169, 1703eqtr4rd 2791 . . . . . . . . . . 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 7464 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → (𝑀 𝑅) = (𝑀 ((𝑆𝑀)‘𝑍)))
1765, 6, 7, 8, 27, 163, 164, 134, 172mircgr 28683 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → (𝑀 ((𝑆𝑀)‘𝑍)) = (𝑀 𝑍))
177175, 176eqtrd 2780 . . . . . . . . . . . . . 14 ((𝜑𝑀 = 𝑋) → (𝑀 𝑅) = (𝑀 𝑍))
1785, 6, 7, 163, 164, 173, 164, 172, 177tgcgrcomlr 28506 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → (𝑅 𝑀) = (𝑍 𝑀))
17982adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑀 = 𝑋) → 𝑋 ∈ (𝐴𝐿𝐵))
180162, 179eqeltrd 2844 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝐴𝐿𝐵))
18150adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → ¬ 𝑅 ∈ (𝐴𝐿𝐵))
182180, 181, 155syl2anc 583 . . . . . . . . . . . . . 14 ((𝜑𝑀 = 𝑋) → 𝑀𝑅)
183182necomd 3002 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑅𝑀)
1845, 6, 7, 163, 173, 164, 172, 164, 178, 183tgcgrneq 28509 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑍𝑀)
1855, 6, 7, 8, 27, 9, 131, 134, 136mirbtwn 28684 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ (((𝑆𝑀)‘𝑍)𝐼𝑍))
186147oveq1d 7463 . . . . . . . . . . . . . . 15 (𝜑 → (𝑅𝐼𝑍) = (((𝑆𝑀)‘𝑍)𝐼𝑍))
187185, 186eleqtrrd 2847 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (𝑅𝐼𝑍))
188187adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑅𝐼𝑍))
1895, 6, 7, 163, 173, 164, 172, 188tgbtwncom 28514 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑍𝐼𝑅))
190137adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑀 = 𝑋) → 𝑋 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))
191162, 190eqeltrd 2844 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))
1925, 6, 7, 163, 167, 164, 172, 191tgbtwncom 28514 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑍𝐼((𝑆𝐴)‘𝑂)))
19377adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑋 ∈ (𝑅𝐼𝑂))
194162, 193eqeltrd 2844 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑅𝐼𝑂))
1955, 7, 163, 172, 164, 173, 167, 166, 184, 183, 189, 192, 194tgbtwnconn22 28605 . . . . . . . . . . 11 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑂))
1965, 6, 7, 8, 27, 163, 164, 134, 166, 167, 171, 195ismir 28685 . . . . . . . . . 10 ((𝜑𝑀 = 𝑋) → ((𝑆𝐴)‘𝑂) = ((𝑆𝑀)‘𝑂))
197196eqcomd 2746 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → ((𝑆𝑀)‘𝑂) = ((𝑆𝐴)‘𝑂))
1985, 6, 7, 8, 27, 163, 164, 165, 166, 197miduniq1 28712 . . . . . . . 8 ((𝜑𝑀 = 𝑋) → 𝑀 = 𝐴)
199162, 198eqtr3d 2782 . . . . . . 7 ((𝜑𝑀 = 𝑋) → 𝑋 = 𝐴)
200129, 199mtand 815 . . . . . 6 (𝜑 → ¬ 𝑀 = 𝑋)
201200neqned 2953 . . . . 5 (𝜑𝑀𝑋)
202201necomd 3002 . . . 4 (𝜑𝑋𝑀)
203149oveq2d 7464 . . . . . 6 (𝜑 → (𝑋 ((𝑆𝑀)‘𝑅)) = (𝑋 𝑍))
204203, 144eqtr2d 2781 . . . . 5 (𝜑 → (𝑋 𝑅) = (𝑋 ((𝑆𝑀)‘𝑅)))
2055, 6, 7, 8, 27, 9, 53, 131, 14israg 28723 . . . . 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 28743 . . 3 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑅𝐿𝑀))
2085, 6, 7, 8, 9, 13, 158, 207perpcom 28739 . 2 (𝜑 → (𝑅𝐿𝑀)(⟂G‘𝐺)(𝐴𝐿𝐵))
2092, 4, 51, 52, 153, 154, 208reu2eqd 3758 1 (𝜑𝐵 = 𝑀)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 846   = wceq 1537  wcel 2108  wne 2946  ∃!wreu 3386   class class class wbr 5166  ran crn 5701  cfv 6573  (class class class)co 7448  ⟨“cs3 14891  Basecbs 17258  distcds 17320  TarskiGcstrkg 28453  Itvcitv 28459  LineGclng 28460  pInvGcmir 28678  ∟Gcrag 28719  ⟂Gcperpg 28721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380  df-word 14563  df-concat 14619  df-s1 14644  df-s2 14897  df-s3 14898  df-trkgc 28474  df-trkgb 28475  df-trkgcb 28476  df-trkg 28479  df-cgrg 28537  df-leg 28609  df-mir 28679  df-rag 28720  df-perpg 28722
This theorem is referenced by:  opphllem  28761
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