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Mirrors > Home > MPE Home > Th. List > lnrot2 | 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 |
---|---|
btwnlng1.p | ⊢ 𝑃 = (Base‘𝐺) |
btwnlng1.i | ⊢ 𝐼 = (Itv‘𝐺) |
btwnlng1.l | ⊢ 𝐿 = (LineG‘𝐺) |
btwnlng1.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
btwnlng1.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
btwnlng1.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
btwnlng1.z | ⊢ (𝜑 → 𝑍 ∈ 𝑃) |
btwnlng1.d | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
lnrot2.1 | ⊢ (𝜑 → 𝑋 ∈ (𝑌𝐿𝑍)) |
lnrot2.2 | ⊢ (𝜑 → 𝑌 ≠ 𝑍) |
Ref | Expression |
---|---|
lnrot2 | ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lnrot2.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝑌𝐿𝑍)) | |
2 | btwnlng1.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝐺) | |
3 | eqid 2818 | . . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
4 | btwnlng1.i | . . . . . 6 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | btwnlng1.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | btwnlng1.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
7 | btwnlng1.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
8 | btwnlng1.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
9 | 2, 3, 4, 5, 6, 7, 8 | tgbtwncomb 26202 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑌𝐼𝑍) ↔ 𝑋 ∈ (𝑍𝐼𝑌))) |
10 | biidd 263 | . . . . 5 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ 𝑌 ∈ (𝑋𝐼𝑍))) | |
11 | 2, 3, 4, 5, 6, 8, 7 | tgbtwncomb 26202 | . . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐼𝑋) ↔ 𝑍 ∈ (𝑋𝐼𝑌))) |
12 | 9, 10, 11 | 3orbi123d 1426 | . . . 4 ⊢ (𝜑 → ((𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑌𝐼𝑋)) ↔ (𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌)))) |
13 | 3orrot 1084 | . . . 4 ⊢ ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌))) | |
14 | 12, 13 | syl6bbr 290 | . . 3 ⊢ (𝜑 → ((𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑌𝐼𝑋)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
15 | btwnlng1.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
16 | lnrot2.2 | . . . 4 ⊢ (𝜑 → 𝑌 ≠ 𝑍) | |
17 | 2, 15, 4, 5, 6, 8, 16, 7 | tgellng 26266 | . . 3 ⊢ (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ↔ (𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑌𝐼𝑋)))) |
18 | btwnlng1.d | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
19 | 2, 15, 4, 5, 7, 6, 18, 8 | tgellng 26266 | . . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
20 | 14, 17, 19 | 3bitr4d 312 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ↔ 𝑍 ∈ (𝑋𝐿𝑌))) |
21 | 1, 20 | mpbid 233 | 1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1078 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 distcds 16562 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: coltr 26360 mideulem2 26447 |
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