![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ncolne1 | Structured version Visualization version GIF version |
Description: Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.) |
Ref | Expression |
---|---|
tglineelsb2.p | ⊢ 𝐵 = (Base‘𝐺) |
tglineelsb2.i | ⊢ 𝐼 = (Itv‘𝐺) |
tglineelsb2.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglineelsb2.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
ncolne.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ncolne.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
ncolne.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
ncolne.2 | ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) |
Ref | Expression |
---|---|
ncolne1 | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ncolne.2 | . . 3 ⊢ (𝜑 → ¬ (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) | |
2 | tglineelsb2.p | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
3 | tglineelsb2.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
4 | tglineelsb2.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | tglineelsb2.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | 5 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝐺 ∈ TarskiG) |
7 | ncolne.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
8 | 7 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑌 ∈ 𝐵) |
9 | ncolne.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
10 | 9 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ 𝐵) |
11 | ncolne.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
12 | 11 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ 𝐵) |
13 | eqid 2798 | . . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
14 | 2, 13, 4, 6, 12, 10 | tgbtwntriv1 26285 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑋𝐼𝑍)) |
15 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌) | |
16 | 15 | oveq1d 7150 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋𝐼𝑍) = (𝑌𝐼𝑍)) |
17 | 14, 16 | eleqtrd 2892 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑌𝐼𝑍)) |
18 | 2, 3, 4, 6, 8, 10, 12, 17 | btwncolg1 26349 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) |
19 | 1, 18 | mtand 815 | . 2 ⊢ (𝜑 → ¬ 𝑋 = 𝑌) |
20 | 19 | neqned 2994 | 1 ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 distcds 16566 TarskiGcstrkg 26224 Itvcitv 26230 LineGclng 26231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-trkgc 26242 df-trkgb 26243 df-trkgcb 26244 df-trkg 26247 |
This theorem is referenced by: ncolne2 26420 tglineneq 26438 midexlem 26486 mideulem2 26528 outpasch 26549 hlpasch 26550 trgcopy 26598 trgcopyeulem 26599 acopy 26627 acopyeu 26628 cgrg3col4 26647 tgasa1 26652 |
Copyright terms: Public domain | W3C validator |