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Theorem mideulem2 25671
Description: Lemma for opphllem 25672, which is itself used for mideu 25675. (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 6698 . . 3 (𝑦 = 𝐵 → (𝑅𝐿𝑦) = (𝑅𝐿𝐵))
21breq1d 4695 . 2 (𝑦 = 𝐵 → ((𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵) ↔ (𝑅𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵)))
3 oveq2 6698 . . 3 (𝑦 = 𝑀 → (𝑅𝐿𝑦) = (𝑅𝐿𝑀))
43breq1d 4695 . 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 25570 . . 3 (𝜑 → (𝐴𝐿𝐵) ∈ ran 𝐿)
14 opphllem.1 . . 3 (𝜑𝑅𝑃)
1512adantr 480 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝐴𝐵)
1615neneqd 2828 . . . . 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 25651 . . . . . . . . . 10 (𝜑 → (𝐴𝐿𝑂) ∈ ran 𝐿)
215, 7, 8, 9, 10, 17, 20tglnne 25568 . . . . . . . . 9 (𝜑𝐴𝑂)
225, 6, 7, 9, 10, 17, 11, 14, 18, 21tgcgrneq 25423 . . . . . . . 8 (𝜑𝐵𝑅)
2322adantr 480 . . . . . . 7 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝐵𝑅)
2423necomd 2878 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝑅𝐵)
2524neneqd 2828 . . . . 5 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → ¬ 𝑅 = 𝐵)
2616, 25jca 553 . . . 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 25651 . . . . . . . . . . . 12 (𝜑 → (𝑄𝐿𝐵) ∈ ran 𝐿)
355, 7, 8, 9, 32, 11, 34tglnne 25568 . . . . . . . . . . 11 (𝜑𝑄𝐵)
365, 7, 8, 9, 32, 11, 35tglinerflx2 25574 . . . . . . . . . 10 (𝜑𝐵 ∈ (𝑄𝐿𝐵))
375, 6, 7, 8, 9, 13, 34, 33perpcom 25653 . . . . . . . . . . 11 (𝜑 → (𝑄𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵))
385, 7, 8, 9, 10, 11, 12tglinecom 25575 . . . . . . . . . . 11 (𝜑 → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
3937, 38breqtrd 4711 . . . . . . . . . 10 (𝜑 → (𝑄𝐿𝐵)(⟂G‘𝐺)(𝐵𝐿𝐴))
405, 6, 7, 8, 9, 32, 11, 36, 10, 39perprag 25663 . . . . . . . . 9 (𝜑 → ⟨“𝑄𝐵𝐴”⟩ ∈ (∟G‘𝐺))
41 opphllem.2 . . . . . . . . . 10 (𝜑𝑅 ∈ (𝐵𝐼𝑄))
425, 8, 7, 9, 11, 14, 32, 41btwncolg3 25497 . . . . . . . . 9 (𝜑 → (𝑄 ∈ (𝐵𝐿𝑅) ∨ 𝐵 = 𝑅))
435, 6, 7, 8, 27, 9, 32, 11, 10, 14, 40, 35, 42ragcol 25639 . . . . . . . 8 (𝜑 → ⟨“𝑅𝐵𝐴”⟩ ∈ (∟G‘𝐺))
445, 6, 7, 8, 27, 9, 14, 11, 10, 43ragcom 25638 . . . . . . 7 (𝜑 → ⟨“𝐴𝐵𝑅”⟩ ∈ (∟G‘𝐺))
4544adantr 480 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → ⟨“𝐴𝐵𝑅”⟩ ∈ (∟G‘𝐺))
46 simpr 476 . . . . . . 7 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝑅 ∈ (𝐴𝐿𝐵))
4746orcd 406 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → (𝑅 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
485, 6, 7, 8, 27, 28, 29, 30, 31, 45, 47ragflat3 25646 . . . . 5 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → (𝐴 = 𝐵𝑅 = 𝐵))
49 oran 516 . . . . 5 ((𝐴 = 𝐵𝑅 = 𝐵) ↔ ¬ (¬ 𝐴 = 𝐵 ∧ ¬ 𝑅 = 𝐵))
5048, 49sylib 208 . . . 4 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → ¬ (¬ 𝐴 = 𝐵 ∧ ¬ 𝑅 = 𝐵))
5126, 50pm2.65da 599 . . 3 (𝜑 → ¬ 𝑅 ∈ (𝐴𝐿𝐵))
525, 6, 7, 8, 9, 13, 14, 51foot 25659 . 2 (𝜑 → ∃!𝑦 ∈ (𝐴𝐿𝐵)(𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵))
535, 7, 8, 9, 10, 11, 12tglinerflx2 25574 . 2 (𝜑𝐵 ∈ (𝐴𝐿𝐵))
54 mideulem2.1 . . 3 (𝜑𝑋𝑃)
5512neneqd 2828 . . . . 5 (𝜑 → ¬ 𝐴 = 𝐵)
56 oveq2 6698 . . . . . . 7 (𝑦 = 𝐴 → (𝑅𝐿𝑦) = (𝑅𝐿𝐴))
5756breq1d 4695 . . . . . 6 (𝑦 = 𝐴 → ((𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵) ↔ (𝑅𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝐵)))
5852adantr 480 . . . . . 6 ((𝜑𝑋 = 𝐴) → ∃!𝑦 ∈ (𝐴𝐿𝐵)(𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵))
595, 7, 8, 9, 10, 11, 12tglinerflx1 25573 . . . . . . 7 (𝜑𝐴 ∈ (𝐴𝐿𝐵))
6059adantr 480 . . . . . 6 ((𝜑𝑋 = 𝐴) → 𝐴 ∈ (𝐴𝐿𝐵))
6153adantr 480 . . . . . 6 ((𝜑𝑋 = 𝐴) → 𝐵 ∈ (𝐴𝐿𝐵))
629adantr 480 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝐺 ∈ TarskiG)
6314adantr 480 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑅𝑃)
6410adantr 480 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝐴𝑃)
6551, 55jca 553 . . . . . . . . . . . 12 (𝜑 → (¬ 𝑅 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵))
66 pm4.56 515 . . . . . . . . . . . 12 ((¬ 𝑅 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵) ↔ ¬ (𝑅 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
6765, 66sylib 208 . . . . . . . . . . 11 (𝜑 → ¬ (𝑅 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
685, 7, 8, 9, 14, 10, 11, 67ncolne1 25565 . . . . . . . . . 10 (𝜑𝑅𝐴)
6968adantr 480 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑅𝐴)
705, 7, 8, 62, 63, 64, 69tglinecom 25575 . . . . . . . 8 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐴) = (𝐴𝐿𝑅))
7169necomd 2878 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝐴𝑅)
7217adantr 480 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑂𝑃)
7321necomd 2878 . . . . . . . . . 10 (𝜑𝑂𝐴)
7473adantr 480 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑂𝐴)
7554adantr 480 . . . . . . . . . . 11 ((𝜑𝑋 = 𝐴) → 𝑋𝑃)
76 simpr 476 . . . . . . . . . . . 12 ((𝜑𝑋 = 𝐴) → 𝑋 = 𝐴)
7776, 71eqnetrd 2890 . . . . . . . . . . 11 ((𝜑𝑋 = 𝐴) → 𝑋𝑅)
78 mideulem2.3 . . . . . . . . . . . . . . . 16 (𝜑𝑋 ∈ (𝑅𝐼𝑂))
795, 6, 7, 9, 14, 54, 17, 78tgbtwncom 25428 . . . . . . . . . . . . . . 15 (𝜑𝑋 ∈ (𝑂𝐼𝑅))
80 mideulem.4 . . . . . . . . . . . . . . . . 17 (𝜑𝑇𝑃)
81 mideulem.7 . . . . . . . . . . . . . . . . 17 (𝜑𝑇 ∈ (𝐴𝐿𝐵))
82 mideulem2.2 . . . . . . . . . . . . . . . . 17 (𝜑𝑋 ∈ (𝑇𝐼𝐵))
835, 7, 8, 9, 80, 10, 11, 54, 81, 82coltr3 25588 . . . . . . . . . . . . . . . 16 (𝜑𝑋 ∈ (𝐴𝐿𝐵))
8412necomd 2878 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐵𝐴)
8584neneqd 2828 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ¬ 𝐵 = 𝐴)
8685adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝐵 = 𝐴)
8773neneqd 2828 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ¬ 𝑂 = 𝐴)
8887adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝑂 = 𝐴)
8986, 88jca 553 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
909adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐺 ∈ TarskiG)
9111adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐵𝑃)
9210adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐴𝑃)
9317adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝑂𝑃)
945, 7, 8, 9, 11, 10, 84tglinerflx2 25574 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐴 ∈ (𝐵𝐿𝐴))
9538, 19eqbrtrrd 4709 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐵𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝑂))
965, 6, 7, 8, 9, 11, 10, 94, 17, 95perprag 25663 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
9796adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
98 simpr 476 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝑂 ∈ (𝐵𝐿𝐴))
9998orcd 406 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (𝑂 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
1005, 6, 7, 8, 27, 90, 91, 92, 93, 97, 99ragflat3 25646 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (𝐵 = 𝐴𝑂 = 𝐴))
101 oran 516 . . . . . . . . . . . . . . . . . . 19 ((𝐵 = 𝐴𝑂 = 𝐴) ↔ ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
102100, 101sylib 208 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
10389, 102pm2.65da 599 . . . . . . . . . . . . . . . . 17 (𝜑 → ¬ 𝑂 ∈ (𝐵𝐿𝐴))
104103, 38neleqtrrd 2752 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
105 nelne2 2920 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝑂 ∈ (𝐴𝐿𝐵)) → 𝑋𝑂)
10683, 104, 105syl2anc 694 . . . . . . . . . . . . . . 15 (𝜑𝑋𝑂)
1075, 6, 7, 9, 17, 54, 14, 79, 106tgbtwnne 25430 . . . . . . . . . . . . . 14 (𝜑𝑂𝑅)
108107adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑋 = 𝐴) → 𝑂𝑅)
109108necomd 2878 . . . . . . . . . . . 12 ((𝜑𝑋 = 𝐴) → 𝑅𝑂)
11078adantr 480 . . . . . . . . . . . 12 ((𝜑𝑋 = 𝐴) → 𝑋 ∈ (𝑅𝐼𝑂))
1115, 7, 8, 62, 63, 72, 75, 109, 110btwnlng1 25559 . . . . . . . . . . 11 ((𝜑𝑋 = 𝐴) → 𝑋 ∈ (𝑅𝐿𝑂))
1125, 7, 8, 62, 75, 63, 72, 77, 111, 109lnrot2 25564 . . . . . . . . . 10 ((𝜑𝑋 = 𝐴) → 𝑂 ∈ (𝑋𝐿𝑅))
11376oveq1d 6705 . . . . . . . . . 10 ((𝜑𝑋 = 𝐴) → (𝑋𝐿𝑅) = (𝐴𝐿𝑅))
114112, 113eleqtrd 2732 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑂 ∈ (𝐴𝐿𝑅))
1155, 7, 8, 62, 64, 63, 71, 72, 74, 114tglineelsb2 25572 . . . . . . . 8 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝑅) = (𝐴𝐿𝑂))
11670, 115eqtrd 2685 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐴) = (𝐴𝐿𝑂))
1175, 6, 7, 8, 9, 13, 20, 19perpcom 25653 . . . . . . . 8 (𝜑 → (𝐴𝐿𝑂)(⟂G‘𝐺)(𝐴𝐿𝐵))
118117adantr 480 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝑂)(⟂G‘𝐺)(𝐴𝐿𝐵))
119116, 118eqbrtrd 4707 . . . . . 6 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝐵))
12013adantr 480 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝐵) ∈ ran 𝐿)
12122necomd 2878 . . . . . . . . 9 (𝜑𝑅𝐵)
1225, 7, 8, 9, 14, 11, 121tgelrnln 25570 . . . . . . . 8 (𝜑 → (𝑅𝐿𝐵) ∈ ran 𝐿)
123122adantr 480 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐵) ∈ ran 𝐿)
1245, 7, 8, 9, 14, 11, 121tglinerflx2 25574 . . . . . . . . . 10 (𝜑𝐵 ∈ (𝑅𝐿𝐵))
12553, 124elind 3831 . . . . . . . . 9 (𝜑𝐵 ∈ ((𝐴𝐿𝐵) ∩ (𝑅𝐿𝐵)))
1265, 7, 8, 9, 14, 11, 121tglinerflx1 25573 . . . . . . . . 9 (𝜑𝑅 ∈ (𝑅𝐿𝐵))
1275, 6, 7, 8, 9, 13, 122, 125, 59, 126, 12, 121, 44ragperp 25657 . . . . . . . 8 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑅𝐿𝐵))
128127adantr 480 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑅𝐿𝐵))
1295, 6, 7, 8, 62, 120, 123, 128perpcom 25653 . . . . . 6 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵))
13057, 2, 58, 60, 61, 119, 129reu2eqd 3436 . . . . 5 ((𝜑𝑋 = 𝐴) → 𝐴 = 𝐵)
13155, 130mtand 692 . . . 4 (𝜑 → ¬ 𝑋 = 𝐴)
132131neqned 2830 . . 3 (𝜑𝑋𝐴)
133 mideulem2.7 . . 3 (𝜑𝑀𝑃)
134132necomd 2878 . . . 4 (𝜑𝐴𝑋)
135 eqid 2651 . . . . 5 (𝑆𝐴) = (𝑆𝐴)
136 eqid 2651 . . . . 5 (𝑆𝑀) = (𝑆𝑀)
1375, 6, 7, 8, 27, 9, 10, 135, 17mircl 25601 . . . . 5 (𝜑 → ((𝑆𝐴)‘𝑂) ∈ 𝑃)
138 mideulem2.4 . . . . 5 (𝜑𝑍𝑃)
139 mideulem2.5 . . . . 5 (𝜑𝑋 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))
14083orcd 406 . . . . . . . . 9 (𝜑 → (𝑋 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
1415, 8, 7, 9, 10, 11, 54, 140colcom 25498 . . . . . . . 8 (𝜑 → (𝑋 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
1425, 8, 7, 9, 11, 10, 54, 141colrot1 25499 . . . . . . 7 (𝜑 → (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
1435, 6, 7, 8, 27, 9, 11, 10, 17, 54, 96, 84, 142ragcol 25639 . . . . . 6 (𝜑 → ⟨“𝑋𝐴𝑂”⟩ ∈ (∟G‘𝐺))
1445, 6, 7, 8, 27, 9, 54, 10, 17israg 25637 . . . . . 6 (𝜑 → (⟨“𝑋𝐴𝑂”⟩ ∈ (∟G‘𝐺) ↔ (𝑋 𝑂) = (𝑋 ((𝑆𝐴)‘𝑂))))
145143, 144mpbid 222 . . . . 5 (𝜑 → (𝑋 𝑂) = (𝑋 ((𝑆𝐴)‘𝑂)))
146 mideulem2.6 . . . . . 6 (𝜑 → (𝑋 𝑍) = (𝑋 𝑅))
147146eqcomd 2657 . . . . 5 (𝜑 → (𝑋 𝑅) = (𝑋 𝑍))
148 eqidd 2652 . . . . 5 (𝜑 → ((𝑆𝐴)‘𝑂) = ((𝑆𝐴)‘𝑂))
149 mideulem2.8 . . . . . . . 8 (𝜑𝑅 = ((𝑆𝑀)‘𝑍))
150149eqcomd 2657 . . . . . . 7 (𝜑 → ((𝑆𝑀)‘𝑍) = 𝑅)
1515, 6, 7, 8, 27, 9, 133, 136, 138, 150mircom 25603 . . . . . 6 (𝜑 → ((𝑆𝑀)‘𝑅) = 𝑍)
152151eqcomd 2657 . . . . 5 (𝜑𝑍 = ((𝑆𝑀)‘𝑅))
1535, 6, 7, 8, 27, 9, 135, 136, 17, 137, 54, 14, 138, 10, 133, 79, 139, 145, 147, 148, 152krippen 25631 . . . 4 (𝜑𝑋 ∈ (𝐴𝐼𝑀))
1545, 7, 8, 9, 10, 54, 133, 134, 153btwnlng3 25561 . . 3 (𝜑𝑀 ∈ (𝐴𝐿𝑋))
1555, 7, 8, 9, 10, 11, 12, 54, 132, 83, 133, 154tglineeltr 25571 . 2 (𝜑𝑀 ∈ (𝐴𝐿𝐵))
1565, 6, 7, 8, 9, 13, 122, 127perpcom 25653 . 2 (𝜑 → (𝑅𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵))
157 nelne2 2920 . . . . . 6 ((𝑀 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝑅 ∈ (𝐴𝐿𝐵)) → 𝑀𝑅)
158155, 51, 157syl2anc 694 . . . . 5 (𝜑𝑀𝑅)
159158necomd 2878 . . . 4 (𝜑𝑅𝑀)
1605, 7, 8, 9, 14, 133, 159tgelrnln 25570 . . 3 (𝜑 → (𝑅𝐿𝑀) ∈ ran 𝐿)
1615, 7, 8, 9, 14, 133, 159tglinerflx2 25574 . . . . 5 (𝜑𝑀 ∈ (𝑅𝐿𝑀))
162155, 161elind 3831 . . . 4 (𝜑𝑀 ∈ ((𝐴𝐿𝐵) ∩ (𝑅𝐿𝑀)))
1635, 7, 8, 9, 14, 133, 159tglinerflx1 25573 . . . 4 (𝜑𝑅 ∈ (𝑅𝐿𝑀))
164 simpr 476 . . . . . . . 8 ((𝜑𝑀 = 𝑋) → 𝑀 = 𝑋)
1659adantr 480 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → 𝐺 ∈ TarskiG)
166133adantr 480 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → 𝑀𝑃)
16710adantr 480 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → 𝐴𝑃)
16817adantr 480 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → 𝑂𝑃)
169137adantr 480 . . . . . . . . . . 11 ((𝜑𝑀 = 𝑋) → ((𝑆𝐴)‘𝑂) ∈ 𝑃)
170145adantr 480 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → (𝑋 𝑂) = (𝑋 ((𝑆𝐴)‘𝑂)))
171164oveq1d 6705 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → (𝑀 𝑂) = (𝑋 𝑂))
172164oveq1d 6705 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → (𝑀 ((𝑆𝐴)‘𝑂)) = (𝑋 ((𝑆𝐴)‘𝑂)))
173170, 171, 1723eqtr4rd 2696 . . . . . . . . . . 11 ((𝜑𝑀 = 𝑋) → (𝑀 ((𝑆𝐴)‘𝑂)) = (𝑀 𝑂))
174138adantr 480 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑍𝑃)
17514adantr 480 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑅𝑃)
176149adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑀 = 𝑋) → 𝑅 = ((𝑆𝑀)‘𝑍))
177176oveq2d 6706 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → (𝑀 𝑅) = (𝑀 ((𝑆𝑀)‘𝑍)))
1785, 6, 7, 8, 27, 165, 166, 136, 174mircgr 25597 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → (𝑀 ((𝑆𝑀)‘𝑍)) = (𝑀 𝑍))
179177, 178eqtrd 2685 . . . . . . . . . . . . . 14 ((𝜑𝑀 = 𝑋) → (𝑀 𝑅) = (𝑀 𝑍))
1805, 6, 7, 165, 166, 175, 166, 174, 179tgcgrcomlr 25420 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → (𝑅 𝑀) = (𝑍 𝑀))
18183adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑀 = 𝑋) → 𝑋 ∈ (𝐴𝐿𝐵))
182164, 181eqeltrd 2730 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝐴𝐿𝐵))
18351adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → ¬ 𝑅 ∈ (𝐴𝐿𝐵))
184182, 183, 157syl2anc 694 . . . . . . . . . . . . . 14 ((𝜑𝑀 = 𝑋) → 𝑀𝑅)
185184necomd 2878 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑅𝑀)
1865, 6, 7, 165, 175, 166, 174, 166, 180, 185tgcgrneq 25423 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑍𝑀)
1875, 6, 7, 8, 27, 9, 133, 136, 138mirbtwn 25598 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ (((𝑆𝑀)‘𝑍)𝐼𝑍))
188149oveq1d 6705 . . . . . . . . . . . . . . 15 (𝜑 → (𝑅𝐼𝑍) = (((𝑆𝑀)‘𝑍)𝐼𝑍))
189187, 188eleqtrrd 2733 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (𝑅𝐼𝑍))
190189adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑅𝐼𝑍))
1915, 6, 7, 165, 175, 166, 174, 190tgbtwncom 25428 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑍𝐼𝑅))
192139adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑀 = 𝑋) → 𝑋 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))
193164, 192eqeltrd 2730 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))
1945, 6, 7, 165, 169, 166, 174, 193tgbtwncom 25428 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑍𝐼((𝑆𝐴)‘𝑂)))
19578adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑋 ∈ (𝑅𝐼𝑂))
196164, 195eqeltrd 2730 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑅𝐼𝑂))
1975, 7, 165, 174, 166, 175, 169, 168, 186, 185, 191, 194, 196tgbtwnconn22 25519 . . . . . . . . . . 11 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑂))
1985, 6, 7, 8, 27, 165, 166, 136, 168, 169, 173, 197ismir 25599 . . . . . . . . . 10 ((𝜑𝑀 = 𝑋) → ((𝑆𝐴)‘𝑂) = ((𝑆𝑀)‘𝑂))
199198eqcomd 2657 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → ((𝑆𝑀)‘𝑂) = ((𝑆𝐴)‘𝑂))
2005, 6, 7, 8, 27, 165, 166, 167, 168, 199miduniq1 25626 . . . . . . . 8 ((𝜑𝑀 = 𝑋) → 𝑀 = 𝐴)
201164, 200eqtr3d 2687 . . . . . . 7 ((𝜑𝑀 = 𝑋) → 𝑋 = 𝐴)
202131, 201mtand 692 . . . . . 6 (𝜑 → ¬ 𝑀 = 𝑋)
203202neqned 2830 . . . . 5 (𝜑𝑀𝑋)
204203necomd 2878 . . . 4 (𝜑𝑋𝑀)
205151oveq2d 6706 . . . . . 6 (𝜑 → (𝑋 ((𝑆𝑀)‘𝑅)) = (𝑋 𝑍))
206205, 146eqtr2d 2686 . . . . 5 (𝜑 → (𝑋 𝑅) = (𝑋 ((𝑆𝑀)‘𝑅)))
2075, 6, 7, 8, 27, 9, 54, 133, 14israg 25637 . . . . 5 (𝜑 → (⟨“𝑋𝑀𝑅”⟩ ∈ (∟G‘𝐺) ↔ (𝑋 𝑅) = (𝑋 ((𝑆𝑀)‘𝑅))))
208206, 207mpbird 247 . . . 4 (𝜑 → ⟨“𝑋𝑀𝑅”⟩ ∈ (∟G‘𝐺))
2095, 6, 7, 8, 9, 13, 160, 162, 83, 163, 204, 159, 208ragperp 25657 . . 3 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑅𝐿𝑀))
2105, 6, 7, 8, 9, 13, 160, 209perpcom 25653 . 2 (𝜑 → (𝑅𝐿𝑀)(⟂G‘𝐺)(𝐴𝐿𝐵))
2112, 4, 52, 53, 155, 156, 210reu2eqd 3436 1 (𝜑𝐵 = 𝑀)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  ∃!wreu 2943   class class class wbr 4685  ran crn 5144  cfv 5926  (class class class)co 6690  ⟨“cs3 13633  Basecbs 15904  distcds 15997  TarskiGcstrkg 25374  Itvcitv 25380  LineGclng 25381  pInvGcmir 25592  ∟Gcrag 25633  ⟂Gcperpg 25635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-s2 13639  df-s3 13640  df-trkgc 25392  df-trkgb 25393  df-trkgcb 25394  df-trkg 25397  df-cgrg 25451  df-leg 25523  df-mir 25593  df-rag 25634  df-perpg 25636
This theorem is referenced by:  opphllem  25672
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