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Mirrors > Home > MPE Home > Th. List > colrot2 | Structured version Visualization version GIF version |
Description: Rotating the points defining a line. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-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) |
colrot | ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
Ref | Expression |
---|---|
colrot2 | ⊢ (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglngval.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tglngval.l | . 2 ⊢ 𝐿 = (LineG‘𝐺) | |
3 | tglngval.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tglngval.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | tglngval.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
6 | tgcolg.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
7 | tglngval.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
8 | colrot | . . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) | |
9 | 1, 2, 3, 4, 7, 5, 6, 8 | colrot1 26272 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍)) |
10 | 1, 2, 3, 4, 5, 6, 7, 9 | colrot1 26272 | 1 ⊢ (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ 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: ncolrot1 26275 tglineeltr 26344 ncolncol 26359 symquadlem 26402 hlpasch 26469 hphl 26484 trgcopy 26517 |
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