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Mirrors > Home > MPE Home > Th. List > ncolrot2 | Structured version Visualization version GIF version |
Description: Rotating 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 |
---|---|
ncolrot2 | ⊢ (𝜑 → ¬ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 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 481 | . . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝐺 ∈ TarskiG) |
7 | tgcolg.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
8 | 7 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝑍 ∈ 𝑃) |
9 | tglngval.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
10 | 9 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝑋 ∈ 𝑃) |
11 | tglngval.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
12 | 11 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝑌 ∈ 𝑃) |
13 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) | |
14 | 2, 3, 4, 6, 8, 10, 12, 13 | colrot1 26272 | . 2 ⊢ ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
15 | 1, 14 | mtand 812 | 1 ⊢ (𝜑 → ¬ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 TarskiGcstrkg 26143 Itvcitv 26149 LineGclng 26150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-trkgc 26161 df-trkgb 26162 df-trkgcb 26163 df-trkg 26166 |
This theorem is referenced by: midexlem 26405 perpneq 26427 opphllem 26448 outpasch 26468 hlpasch 26469 trgcopy 26517 acopyeu 26547 |
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