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Mirrors > Home > MPE Home > Th. List > colperpexlem2 | Structured version Visualization version GIF version |
Description: Lemma for colperpex 28756. Second part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 10-Nov-2019.) |
Ref | Expression |
---|---|
colperpex.p | ⊢ 𝑃 = (Base‘𝐺) |
colperpex.d | ⊢ − = (dist‘𝐺) |
colperpex.i | ⊢ 𝐼 = (Itv‘𝐺) |
colperpex.l | ⊢ 𝐿 = (LineG‘𝐺) |
colperpex.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
colperpexlem.s | ⊢ 𝑆 = (pInvG‘𝐺) |
colperpexlem.m | ⊢ 𝑀 = (𝑆‘𝐴) |
colperpexlem.n | ⊢ 𝑁 = (𝑆‘𝐵) |
colperpexlem.k | ⊢ 𝐾 = (𝑆‘𝑄) |
colperpexlem.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
colperpexlem.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
colperpexlem.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
colperpexlem.q | ⊢ (𝜑 → 𝑄 ∈ 𝑃) |
colperpexlem.1 | ⊢ (𝜑 → 〈“𝐴𝐵𝐶”〉 ∈ (∟G‘𝐺)) |
colperpexlem.2 | ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶)) |
colperpexlem2.e | ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
Ref | Expression |
---|---|
colperpexlem2 | ⊢ (𝜑 → 𝐴 ≠ 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | colperpexlem2.e | . . 3 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | |
2 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝐴 = 𝑄) | |
3 | 2 | fveq2d 6911 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝑆‘𝐴) = (𝑆‘𝑄)) |
4 | colperpexlem.m | . . . . . . . . 9 ⊢ 𝑀 = (𝑆‘𝐴) | |
5 | colperpexlem.k | . . . . . . . . 9 ⊢ 𝐾 = (𝑆‘𝑄) | |
6 | 3, 4, 5 | 3eqtr4g 2800 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝑀 = 𝐾) |
7 | 6 | fveq1d 6909 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝑀‘(𝑀‘𝐶)) = (𝐾‘(𝑀‘𝐶))) |
8 | colperpex.p | . . . . . . . . 9 ⊢ 𝑃 = (Base‘𝐺) | |
9 | colperpex.d | . . . . . . . . 9 ⊢ − = (dist‘𝐺) | |
10 | colperpex.i | . . . . . . . . 9 ⊢ 𝐼 = (Itv‘𝐺) | |
11 | colperpex.l | . . . . . . . . 9 ⊢ 𝐿 = (LineG‘𝐺) | |
12 | colperpexlem.s | . . . . . . . . 9 ⊢ 𝑆 = (pInvG‘𝐺) | |
13 | colperpex.g | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
14 | colperpexlem.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
15 | colperpexlem.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
16 | 8, 9, 10, 11, 12, 13, 14, 4, 15 | mirmir 28685 | . . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐶)) = 𝐶) |
17 | 16 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝑀‘(𝑀‘𝐶)) = 𝐶) |
18 | colperpexlem.2 | . . . . . . . 8 ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶)) | |
19 | 18 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶)) |
20 | 7, 17, 19 | 3eqtr3rd 2784 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝑁‘𝐶) = 𝐶) |
21 | colperpexlem.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
22 | colperpexlem.n | . . . . . . . 8 ⊢ 𝑁 = (𝑆‘𝐵) | |
23 | 8, 9, 10, 11, 12, 13, 21, 22, 15 | mirinv 28689 | . . . . . . 7 ⊢ (𝜑 → ((𝑁‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶)) |
24 | 23 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → ((𝑁‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶)) |
25 | 20, 24 | mpbid 232 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝐵 = 𝐶) |
26 | 25 | ex 412 | . . . 4 ⊢ (𝜑 → (𝐴 = 𝑄 → 𝐵 = 𝐶)) |
27 | 26 | necon3ad 2951 | . . 3 ⊢ (𝜑 → (𝐵 ≠ 𝐶 → ¬ 𝐴 = 𝑄)) |
28 | 1, 27 | mpd 15 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝑄) |
29 | 28 | neqned 2945 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ‘cfv 6563 〈“cs3 14878 Basecbs 17245 distcds 17307 TarskiGcstrkg 28450 Itvcitv 28456 LineGclng 28457 pInvGcmir 28675 ∟Gcrag 28716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-trkgc 28471 df-trkgb 28472 df-trkgcb 28473 df-trkg 28476 df-mir 28676 |
This theorem is referenced by: colperpexlem3 28755 |
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