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Mirrors > Home > MPE Home > Th. List > colperpexlem2 | Structured version Visualization version GIF version |
Description: Lemma for colperpex 25425. 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 476 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝐴 = 𝑄) | |
3 | 2 | fveq2d 6107 | . . . . . . . . 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 2669 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝑀 = 𝐾) |
7 | 6 | fveq1d 6105 | . . . . . . 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 25357 | . . . . . . . 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 2653 | . . . . . 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 25361 | . . . . . . 7 ⊢ (𝜑 → ((𝑁‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶)) |
24 | 23 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → ((𝑁‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶)) |
25 | 20, 24 | mpbid 221 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝐵 = 𝐶) |
26 | 25 | ex 449 | . . . 4 ⊢ (𝜑 → (𝐴 = 𝑄 → 𝐵 = 𝐶)) |
27 | 26 | necon3ad 2795 | . . 3 ⊢ (𝜑 → (𝐵 ≠ 𝐶 → ¬ 𝐴 = 𝑄)) |
28 | 1, 27 | mpd 15 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝑄) |
29 | 28 | neqned 2789 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ‘cfv 5804 〈“cs3 13438 Basecbs 15695 distcds 15777 TarskiGcstrkg 25129 Itvcitv 25135 LineGclng 25136 pInvGcmir 25347 ∟Gcrag 25388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-trkgc 25147 df-trkgb 25148 df-trkgcb 25149 df-trkg 25152 df-mir 25348 |
This theorem is referenced by: colperpexlem3 25424 |
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