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Theorem mideulem2 25543
 Description: Lemma for opphllem 25544, which is itself used for mideu 25547. (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 6618 . . 3 (𝑦 = 𝐵 → (𝑅𝐿𝑦) = (𝑅𝐿𝐵))
21breq1d 4628 . 2 (𝑦 = 𝐵 → ((𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵) ↔ (𝑅𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵)))
3 oveq2 6618 . . 3 (𝑦 = 𝑀 → (𝑅𝐿𝑦) = (𝑅𝐿𝑀))
43breq1d 4628 . 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 25442 . . 3 (𝜑 → (𝐴𝐿𝐵) ∈ ran 𝐿)
14 opphllem.1 . . 3 (𝜑𝑅𝑃)
1512adantr 481 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝐴𝐵)
1615neneqd 2795 . . . . 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 25523 . . . . . . . . . 10 (𝜑 → (𝐴𝐿𝑂) ∈ ran 𝐿)
215, 7, 8, 9, 10, 17, 20tglnne 25440 . . . . . . . . 9 (𝜑𝐴𝑂)
225, 6, 7, 9, 10, 17, 11, 14, 18, 21tgcgrneq 25295 . . . . . . . 8 (𝜑𝐵𝑅)
2322adantr 481 . . . . . . 7 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝐵𝑅)
2423necomd 2845 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝑅𝐵)
2524neneqd 2795 . . . . 5 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → ¬ 𝑅 = 𝐵)
2616, 25jca 554 . . . 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 25523 . . . . . . . . . . . 12 (𝜑 → (𝑄𝐿𝐵) ∈ ran 𝐿)
355, 7, 8, 9, 32, 11, 34tglnne 25440 . . . . . . . . . . 11 (𝜑𝑄𝐵)
365, 7, 8, 9, 32, 11, 35tglinerflx2 25446 . . . . . . . . . 10 (𝜑𝐵 ∈ (𝑄𝐿𝐵))
375, 6, 7, 8, 9, 13, 34, 33perpcom 25525 . . . . . . . . . . 11 (𝜑 → (𝑄𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵))
385, 7, 8, 9, 10, 11, 12tglinecom 25447 . . . . . . . . . . 11 (𝜑 → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
3937, 38breqtrd 4644 . . . . . . . . . 10 (𝜑 → (𝑄𝐿𝐵)(⟂G‘𝐺)(𝐵𝐿𝐴))
405, 6, 7, 8, 9, 32, 11, 36, 10, 39perprag 25535 . . . . . . . . 9 (𝜑 → ⟨“𝑄𝐵𝐴”⟩ ∈ (∟G‘𝐺))
41 opphllem.2 . . . . . . . . . 10 (𝜑𝑅 ∈ (𝐵𝐼𝑄))
425, 8, 7, 9, 11, 14, 32, 41btwncolg3 25369 . . . . . . . . 9 (𝜑 → (𝑄 ∈ (𝐵𝐿𝑅) ∨ 𝐵 = 𝑅))
435, 6, 7, 8, 27, 9, 32, 11, 10, 14, 40, 35, 42ragcol 25511 . . . . . . . 8 (𝜑 → ⟨“𝑅𝐵𝐴”⟩ ∈ (∟G‘𝐺))
445, 6, 7, 8, 27, 9, 14, 11, 10, 43ragcom 25510 . . . . . . 7 (𝜑 → ⟨“𝐴𝐵𝑅”⟩ ∈ (∟G‘𝐺))
4544adantr 481 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → ⟨“𝐴𝐵𝑅”⟩ ∈ (∟G‘𝐺))
46 simpr 477 . . . . . . 7 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → 𝑅 ∈ (𝐴𝐿𝐵))
4746orcd 407 . . . . . 6 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → (𝑅 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
485, 6, 7, 8, 27, 28, 29, 30, 31, 45, 47ragflat3 25518 . . . . 5 ((𝜑𝑅 ∈ (𝐴𝐿𝐵)) → (𝐴 = 𝐵𝑅 = 𝐵))
49 oran 517 . . . . 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 25531 . 2 (𝜑 → ∃!𝑦 ∈ (𝐴𝐿𝐵)(𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵))
535, 7, 8, 9, 10, 11, 12tglinerflx2 25446 . 2 (𝜑𝐵 ∈ (𝐴𝐿𝐵))
54 mideulem2.1 . . 3 (𝜑𝑋𝑃)
5512neneqd 2795 . . . . 5 (𝜑 → ¬ 𝐴 = 𝐵)
56 oveq2 6618 . . . . . . 7 (𝑦 = 𝐴 → (𝑅𝐿𝑦) = (𝑅𝐿𝐴))
5756breq1d 4628 . . . . . 6 (𝑦 = 𝐴 → ((𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵) ↔ (𝑅𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝐵)))
5852adantr 481 . . . . . 6 ((𝜑𝑋 = 𝐴) → ∃!𝑦 ∈ (𝐴𝐿𝐵)(𝑅𝐿𝑦)(⟂G‘𝐺)(𝐴𝐿𝐵))
595, 7, 8, 9, 10, 11, 12tglinerflx1 25445 . . . . . . 7 (𝜑𝐴 ∈ (𝐴𝐿𝐵))
6059adantr 481 . . . . . 6 ((𝜑𝑋 = 𝐴) → 𝐴 ∈ (𝐴𝐿𝐵))
6153adantr 481 . . . . . 6 ((𝜑𝑋 = 𝐴) → 𝐵 ∈ (𝐴𝐿𝐵))
629adantr 481 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝐺 ∈ TarskiG)
6314adantr 481 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑅𝑃)
6410adantr 481 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝐴𝑃)
6551, 55jca 554 . . . . . . . . . . . 12 (𝜑 → (¬ 𝑅 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝐴 = 𝐵))
66 pm4.56 516 . . . . . . . . . . . 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 25437 . . . . . . . . . 10 (𝜑𝑅𝐴)
6968adantr 481 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑅𝐴)
705, 7, 8, 62, 63, 64, 69tglinecom 25447 . . . . . . . 8 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐴) = (𝐴𝐿𝑅))
7169necomd 2845 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝐴𝑅)
7217adantr 481 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑂𝑃)
7321necomd 2845 . . . . . . . . . 10 (𝜑𝑂𝐴)
7473adantr 481 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑂𝐴)
7554adantr 481 . . . . . . . . . . 11 ((𝜑𝑋 = 𝐴) → 𝑋𝑃)
76 simpr 477 . . . . . . . . . . . 12 ((𝜑𝑋 = 𝐴) → 𝑋 = 𝐴)
7776, 71eqnetrd 2857 . . . . . . . . . . 11 ((𝜑𝑋 = 𝐴) → 𝑋𝑅)
78 mideulem2.3 . . . . . . . . . . . . . . . 16 (𝜑𝑋 ∈ (𝑅𝐼𝑂))
795, 6, 7, 9, 14, 54, 17, 78tgbtwncom 25300 . . . . . . . . . . . . . . 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 25460 . . . . . . . . . . . . . . . 16 (𝜑𝑋 ∈ (𝐴𝐿𝐵))
8412necomd 2845 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐵𝐴)
8584neneqd 2795 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ¬ 𝐵 = 𝐴)
8685adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝐵 = 𝐴)
8773neneqd 2795 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ¬ 𝑂 = 𝐴)
8887adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ¬ 𝑂 = 𝐴)
8986, 88jca 554 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (¬ 𝐵 = 𝐴 ∧ ¬ 𝑂 = 𝐴))
909adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐺 ∈ TarskiG)
9111adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐵𝑃)
9210adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝐴𝑃)
9317adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝑂𝑃)
945, 7, 8, 9, 11, 10, 84tglinerflx2 25446 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐴 ∈ (𝐵𝐿𝐴))
9538, 19eqbrtrrd 4642 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝐵𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝑂))
965, 6, 7, 8, 9, 11, 10, 94, 17, 95perprag 25535 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
9796adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → ⟨“𝐵𝐴𝑂”⟩ ∈ (∟G‘𝐺))
98 simpr 477 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → 𝑂 ∈ (𝐵𝐿𝐴))
9998orcd 407 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (𝑂 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
1005, 6, 7, 8, 27, 90, 91, 92, 93, 97, 99ragflat3 25518 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑂 ∈ (𝐵𝐿𝐴)) → (𝐵 = 𝐴𝑂 = 𝐴))
101 oran 517 . . . . . . . . . . . . . . . . . . 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 2720 . . . . . . . . . . . . . . . 16 (𝜑 → ¬ 𝑂 ∈ (𝐴𝐿𝐵))
105 nelne2 2887 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝑂 ∈ (𝐴𝐿𝐵)) → 𝑋𝑂)
10683, 104, 105syl2anc 692 . . . . . . . . . . . . . . 15 (𝜑𝑋𝑂)
1075, 6, 7, 9, 17, 54, 14, 79, 106tgbtwnne 25302 . . . . . . . . . . . . . 14 (𝜑𝑂𝑅)
108107adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑋 = 𝐴) → 𝑂𝑅)
109108necomd 2845 . . . . . . . . . . . 12 ((𝜑𝑋 = 𝐴) → 𝑅𝑂)
11078adantr 481 . . . . . . . . . . . 12 ((𝜑𝑋 = 𝐴) → 𝑋 ∈ (𝑅𝐼𝑂))
1115, 7, 8, 62, 63, 72, 75, 109, 110btwnlng1 25431 . . . . . . . . . . 11 ((𝜑𝑋 = 𝐴) → 𝑋 ∈ (𝑅𝐿𝑂))
1125, 7, 8, 62, 75, 63, 72, 77, 111, 109lnrot2 25436 . . . . . . . . . 10 ((𝜑𝑋 = 𝐴) → 𝑂 ∈ (𝑋𝐿𝑅))
11376oveq1d 6625 . . . . . . . . . 10 ((𝜑𝑋 = 𝐴) → (𝑋𝐿𝑅) = (𝐴𝐿𝑅))
114112, 113eleqtrd 2700 . . . . . . . . 9 ((𝜑𝑋 = 𝐴) → 𝑂 ∈ (𝐴𝐿𝑅))
1155, 7, 8, 62, 64, 63, 71, 72, 74, 114tglineelsb2 25444 . . . . . . . 8 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝑅) = (𝐴𝐿𝑂))
11670, 115eqtrd 2655 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐴) = (𝐴𝐿𝑂))
1175, 6, 7, 8, 9, 13, 20, 19perpcom 25525 . . . . . . . 8 (𝜑 → (𝐴𝐿𝑂)(⟂G‘𝐺)(𝐴𝐿𝐵))
118117adantr 481 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝑂)(⟂G‘𝐺)(𝐴𝐿𝐵))
119116, 118eqbrtrd 4640 . . . . . 6 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐴)(⟂G‘𝐺)(𝐴𝐿𝐵))
12013adantr 481 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝐵) ∈ ran 𝐿)
12122necomd 2845 . . . . . . . . 9 (𝜑𝑅𝐵)
1225, 7, 8, 9, 14, 11, 121tgelrnln 25442 . . . . . . . 8 (𝜑 → (𝑅𝐿𝐵) ∈ ran 𝐿)
123122adantr 481 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐵) ∈ ran 𝐿)
1245, 7, 8, 9, 14, 11, 121tglinerflx2 25446 . . . . . . . . . 10 (𝜑𝐵 ∈ (𝑅𝐿𝐵))
12553, 124elind 3781 . . . . . . . . 9 (𝜑𝐵 ∈ ((𝐴𝐿𝐵) ∩ (𝑅𝐿𝐵)))
1265, 7, 8, 9, 14, 11, 121tglinerflx1 25445 . . . . . . . . 9 (𝜑𝑅 ∈ (𝑅𝐿𝐵))
1275, 6, 7, 8, 9, 13, 122, 125, 59, 126, 12, 121, 44ragperp 25529 . . . . . . . 8 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑅𝐿𝐵))
128127adantr 481 . . . . . . 7 ((𝜑𝑋 = 𝐴) → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑅𝐿𝐵))
1295, 6, 7, 8, 62, 120, 123, 128perpcom 25525 . . . . . 6 ((𝜑𝑋 = 𝐴) → (𝑅𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵))
13057, 2, 58, 60, 61, 119, 129reu2eqd 3389 . . . . 5 ((𝜑𝑋 = 𝐴) → 𝐴 = 𝐵)
13155, 130mtand 690 . . . 4 (𝜑 → ¬ 𝑋 = 𝐴)
132131neqned 2797 . . 3 (𝜑𝑋𝐴)
133 mideulem2.7 . . 3 (𝜑𝑀𝑃)
134132necomd 2845 . . . 4 (𝜑𝐴𝑋)
135 eqid 2621 . . . . 5 (𝑆𝐴) = (𝑆𝐴)
136 eqid 2621 . . . . 5 (𝑆𝑀) = (𝑆𝑀)
1375, 6, 7, 8, 27, 9, 10, 135, 17mircl 25473 . . . . 5 (𝜑 → ((𝑆𝐴)‘𝑂) ∈ 𝑃)
138 mideulem2.4 . . . . 5 (𝜑𝑍𝑃)
139 mideulem2.5 . . . . 5 (𝜑𝑋 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))
14083orcd 407 . . . . . . . . 9 (𝜑 → (𝑋 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
1415, 8, 7, 9, 10, 11, 54, 140colcom 25370 . . . . . . . 8 (𝜑 → (𝑋 ∈ (𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
1425, 8, 7, 9, 11, 10, 54, 141colrot1 25371 . . . . . . 7 (𝜑 → (𝐵 ∈ (𝐴𝐿𝑋) ∨ 𝐴 = 𝑋))
1435, 6, 7, 8, 27, 9, 11, 10, 17, 54, 96, 84, 142ragcol 25511 . . . . . 6 (𝜑 → ⟨“𝑋𝐴𝑂”⟩ ∈ (∟G‘𝐺))
1445, 6, 7, 8, 27, 9, 54, 10, 17israg 25509 . . . . . 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 2627 . . . . 5 (𝜑 → (𝑋 𝑅) = (𝑋 𝑍))
148 eqidd 2622 . . . . 5 (𝜑 → ((𝑆𝐴)‘𝑂) = ((𝑆𝐴)‘𝑂))
149 mideulem2.8 . . . . . . . 8 (𝜑𝑅 = ((𝑆𝑀)‘𝑍))
150149eqcomd 2627 . . . . . . 7 (𝜑 → ((𝑆𝑀)‘𝑍) = 𝑅)
1515, 6, 7, 8, 27, 9, 133, 136, 138, 150mircom 25475 . . . . . 6 (𝜑 → ((𝑆𝑀)‘𝑅) = 𝑍)
152151eqcomd 2627 . . . . 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 25503 . . . 4 (𝜑𝑋 ∈ (𝐴𝐼𝑀))
1545, 7, 8, 9, 10, 54, 133, 134, 153btwnlng3 25433 . . 3 (𝜑𝑀 ∈ (𝐴𝐿𝑋))
1555, 7, 8, 9, 10, 11, 12, 54, 132, 83, 133, 154tglineeltr 25443 . 2 (𝜑𝑀 ∈ (𝐴𝐿𝐵))
1565, 6, 7, 8, 9, 13, 122, 127perpcom 25525 . 2 (𝜑 → (𝑅𝐿𝐵)(⟂G‘𝐺)(𝐴𝐿𝐵))
157 nelne2 2887 . . . . . 6 ((𝑀 ∈ (𝐴𝐿𝐵) ∧ ¬ 𝑅 ∈ (𝐴𝐿𝐵)) → 𝑀𝑅)
158155, 51, 157syl2anc 692 . . . . 5 (𝜑𝑀𝑅)
159158necomd 2845 . . . 4 (𝜑𝑅𝑀)
1605, 7, 8, 9, 14, 133, 159tgelrnln 25442 . . 3 (𝜑 → (𝑅𝐿𝑀) ∈ ran 𝐿)
1615, 7, 8, 9, 14, 133, 159tglinerflx2 25446 . . . . 5 (𝜑𝑀 ∈ (𝑅𝐿𝑀))
162155, 161elind 3781 . . . 4 (𝜑𝑀 ∈ ((𝐴𝐿𝐵) ∩ (𝑅𝐿𝑀)))
1635, 7, 8, 9, 14, 133, 159tglinerflx1 25445 . . . 4 (𝜑𝑅 ∈ (𝑅𝐿𝑀))
164 simpr 477 . . . . . . . 8 ((𝜑𝑀 = 𝑋) → 𝑀 = 𝑋)
1659adantr 481 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → 𝐺 ∈ TarskiG)
166133adantr 481 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → 𝑀𝑃)
16710adantr 481 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → 𝐴𝑃)
16817adantr 481 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → 𝑂𝑃)
169137adantr 481 . . . . . . . . . . 11 ((𝜑𝑀 = 𝑋) → ((𝑆𝐴)‘𝑂) ∈ 𝑃)
170145adantr 481 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → (𝑋 𝑂) = (𝑋 ((𝑆𝐴)‘𝑂)))
171164oveq1d 6625 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → (𝑀 𝑂) = (𝑋 𝑂))
172164oveq1d 6625 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → (𝑀 ((𝑆𝐴)‘𝑂)) = (𝑋 ((𝑆𝐴)‘𝑂)))
173170, 171, 1723eqtr4rd 2666 . . . . . . . . . . 11 ((𝜑𝑀 = 𝑋) → (𝑀 ((𝑆𝐴)‘𝑂)) = (𝑀 𝑂))
174138adantr 481 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑍𝑃)
17514adantr 481 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑅𝑃)
176149adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑀 = 𝑋) → 𝑅 = ((𝑆𝑀)‘𝑍))
177176oveq2d 6626 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → (𝑀 𝑅) = (𝑀 ((𝑆𝑀)‘𝑍)))
1785, 6, 7, 8, 27, 165, 166, 136, 174mircgr 25469 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → (𝑀 ((𝑆𝑀)‘𝑍)) = (𝑀 𝑍))
179177, 178eqtrd 2655 . . . . . . . . . . . . . 14 ((𝜑𝑀 = 𝑋) → (𝑀 𝑅) = (𝑀 𝑍))
1805, 6, 7, 165, 166, 175, 166, 174, 179tgcgrcomlr 25292 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → (𝑅 𝑀) = (𝑍 𝑀))
18183adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑀 = 𝑋) → 𝑋 ∈ (𝐴𝐿𝐵))
182164, 181eqeltrd 2698 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝐴𝐿𝐵))
18351adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑀 = 𝑋) → ¬ 𝑅 ∈ (𝐴𝐿𝐵))
184182, 183, 157syl2anc 692 . . . . . . . . . . . . . 14 ((𝜑𝑀 = 𝑋) → 𝑀𝑅)
185184necomd 2845 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑅𝑀)
1865, 6, 7, 165, 175, 166, 174, 166, 180, 185tgcgrneq 25295 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑍𝑀)
1875, 6, 7, 8, 27, 9, 133, 136, 138mirbtwn 25470 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ (((𝑆𝑀)‘𝑍)𝐼𝑍))
188149oveq1d 6625 . . . . . . . . . . . . . . 15 (𝜑 → (𝑅𝐼𝑍) = (((𝑆𝑀)‘𝑍)𝐼𝑍))
189187, 188eleqtrrd 2701 . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ (𝑅𝐼𝑍))
190189adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑅𝐼𝑍))
1915, 6, 7, 165, 175, 166, 174, 190tgbtwncom 25300 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑍𝐼𝑅))
192139adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑀 = 𝑋) → 𝑋 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))
193164, 192eqeltrd 2698 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑍))
1945, 6, 7, 165, 169, 166, 174, 193tgbtwncom 25300 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑍𝐼((𝑆𝐴)‘𝑂)))
19578adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑀 = 𝑋) → 𝑋 ∈ (𝑅𝐼𝑂))
196164, 195eqeltrd 2698 . . . . . . . . . . . 12 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (𝑅𝐼𝑂))
1975, 7, 165, 174, 166, 175, 169, 168, 186, 185, 191, 194, 196tgbtwnconn22 25391 . . . . . . . . . . 11 ((𝜑𝑀 = 𝑋) → 𝑀 ∈ (((𝑆𝐴)‘𝑂)𝐼𝑂))
1985, 6, 7, 8, 27, 165, 166, 136, 168, 169, 173, 197ismir 25471 . . . . . . . . . 10 ((𝜑𝑀 = 𝑋) → ((𝑆𝐴)‘𝑂) = ((𝑆𝑀)‘𝑂))
199198eqcomd 2627 . . . . . . . . 9 ((𝜑𝑀 = 𝑋) → ((𝑆𝑀)‘𝑂) = ((𝑆𝐴)‘𝑂))
2005, 6, 7, 8, 27, 165, 166, 167, 168, 199miduniq1 25498 . . . . . . . 8 ((𝜑𝑀 = 𝑋) → 𝑀 = 𝐴)
201164, 200eqtr3d 2657 . . . . . . 7 ((𝜑𝑀 = 𝑋) → 𝑋 = 𝐴)
202131, 201mtand 690 . . . . . 6 (𝜑 → ¬ 𝑀 = 𝑋)
203202neqned 2797 . . . . 5 (𝜑𝑀𝑋)
204203necomd 2845 . . . 4 (𝜑𝑋𝑀)
205151oveq2d 6626 . . . . . 6 (𝜑 → (𝑋 ((𝑆𝑀)‘𝑅)) = (𝑋 𝑍))
206205, 146eqtr2d 2656 . . . . 5 (𝜑 → (𝑋 𝑅) = (𝑋 ((𝑆𝑀)‘𝑅)))
2075, 6, 7, 8, 27, 9, 54, 133, 14israg 25509 . . . . 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 25529 . . 3 (𝜑 → (𝐴𝐿𝐵)(⟂G‘𝐺)(𝑅𝐿𝑀))
2105, 6, 7, 8, 9, 13, 160, 209perpcom 25525 . 2 (𝜑 → (𝑅𝐿𝑀)(⟂G‘𝐺)(𝐴𝐿𝐵))
2112, 4, 52, 53, 155, 156, 210reu2eqd 3389 1 (𝜑𝐵 = 𝑀)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∃!wreu 2909   class class class wbr 4618  ran crn 5080  ‘cfv 5852  (class class class)co 6610  ⟨“cs3 13532  Basecbs 15792  distcds 15882  TarskiGcstrkg 25246  Itvcitv 25252  LineGclng 25253  pInvGcmir 25464  ∟Gcrag 25505  ⟂Gcperpg 25507 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-card 8717  df-cda 8942  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-n0 11245  df-xnn0 11316  df-z 11330  df-uz 11640  df-fz 12277  df-fzo 12415  df-hash 13066  df-word 13246  df-concat 13248  df-s1 13249  df-s2 13538  df-s3 13539  df-trkgc 25264  df-trkgb 25265  df-trkgcb 25266  df-trkg 25269  df-cgrg 25323  df-leg 25395  df-mir 25465  df-rag 25506  df-perpg 25508 This theorem is referenced by:  opphllem  25544
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