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Mirrors > Home > MPE Home > Th. List > ncolcom | Structured version Visualization version GIF version |
Description: Swapping non-colinear points. (Contributed by Thierry Arnoux, 19-Oct-2019.) |
Ref | Expression |
---|---|
tglngval.p | ⊢ 𝑃 = (Base‘𝐺) |
tglngval.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglngval.i | ⊢ 𝐼 = (Itv‘𝐺) |
tglngval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tglngval.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
tglngval.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
tgcolg.z | ⊢ (𝜑 → 𝑍 ∈ 𝑃) |
ncolrot | ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
Ref | Expression |
---|---|
ncolcom | ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ncolrot | . 2 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) | |
2 | tglngval.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
3 | tglngval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
4 | tglngval.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | tglngval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | 5 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → 𝐺 ∈ TarskiG) |
7 | tglngval.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
8 | 7 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → 𝑌 ∈ 𝑃) |
9 | tglngval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
10 | 9 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → 𝑋 ∈ 𝑃) |
11 | tgcolg.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
12 | 11 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → 𝑍 ∈ 𝑃) |
13 | simpr 479 | . . 3 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) | |
14 | 2, 3, 4, 6, 8, 10, 12, 13 | colcom 25909 | . 2 ⊢ ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
15 | 1, 14 | mtand 806 | 1 ⊢ (𝜑 → ¬ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∨ wo 836 = wceq 1601 ∈ wcel 2106 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 TarskiGcstrkg 25781 Itvcitv 25787 LineGclng 25788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-trkgc 25799 df-trkgb 25800 df-trkgcb 25801 df-trkg 25804 |
This theorem is referenced by: ncolne2 25977 symquadlem 26040 midexlem 26043 outpasch 26103 acopyeu 26183 cgrg3col4 26202 tgasa1 26207 |
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