Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > colperpexlem2 | Structured version Visualization version GIF version |
Description: Lemma for colperpex 27092. 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 485 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝐴 = 𝑄) | |
3 | 2 | fveq2d 6775 | . . . . . . . . 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 2805 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝑀 = 𝐾) |
7 | 6 | fveq1d 6773 | . . . . . . 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 27021 | . . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐶)) = 𝐶) |
17 | 16 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝑀‘(𝑀‘𝐶)) = 𝐶) |
18 | colperpexlem.2 | . . . . . . . 8 ⊢ (𝜑 → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶)) | |
19 | 18 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → (𝐾‘(𝑀‘𝐶)) = (𝑁‘𝐶)) |
20 | 7, 17, 19 | 3eqtr3rd 2789 | . . . . . 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 27025 | . . . . . . 7 ⊢ (𝜑 → ((𝑁‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶)) |
24 | 23 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → ((𝑁‘𝐶) = 𝐶 ↔ 𝐵 = 𝐶)) |
25 | 20, 24 | mpbid 231 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝑄) → 𝐵 = 𝐶) |
26 | 25 | ex 413 | . . . 4 ⊢ (𝜑 → (𝐴 = 𝑄 → 𝐵 = 𝐶)) |
27 | 26 | necon3ad 2958 | . . 3 ⊢ (𝜑 → (𝐵 ≠ 𝐶 → ¬ 𝐴 = 𝑄)) |
28 | 1, 27 | mpd 15 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝑄) |
29 | 28 | neqned 2952 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ‘cfv 6432 〈“cs3 14553 Basecbs 16910 distcds 16969 TarskiGcstrkg 26786 Itvcitv 26792 LineGclng 26793 pInvGcmir 27011 ∟Gcrag 27052 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-trkgc 26807 df-trkgb 26808 df-trkgcb 26809 df-trkg 26812 df-mir 27012 |
This theorem is referenced by: colperpexlem3 27091 |
Copyright terms: Public domain | W3C validator |