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Mirrors > Home > MPE Home > Th. List > lnrot1 | 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) |
lnrot1.1 | ⊢ (𝜑 → 𝑌 ∈ (𝑍𝐿𝑋)) |
lnrot1.2 | ⊢ (𝜑 → 𝑍 ≠ 𝑋) |
Ref | Expression |
---|---|
lnrot1 | ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lnrot1.1 | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝑍𝐿𝑋)) | |
2 | btwnlng1.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝐺) | |
3 | eqid 2824 | . . . . . 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.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
8 | btwnlng1.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
9 | 2, 3, 4, 5, 6, 7, 8 | tgbtwncomb 26278 | . . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐼𝑋) ↔ 𝑍 ∈ (𝑋𝐼𝑌))) |
10 | biidd 264 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑍𝐼𝑌))) | |
11 | 2, 3, 4, 5, 7, 6, 8 | tgbtwncomb 26278 | . . . . 5 ⊢ (𝜑 → (𝑌 ∈ (𝑍𝐼𝑋) ↔ 𝑌 ∈ (𝑋𝐼𝑍))) |
12 | 9, 10, 11 | 3orbi123d 1431 | . . . 4 ⊢ (𝜑 → ((𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑍𝐼𝑋)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
13 | 3orrot 1088 | . . . . 5 ⊢ ((𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑍𝐼𝑋))) | |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → ((𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑍𝐼𝑋)))) |
15 | btwnlng1.l | . . . . 5 ⊢ 𝐿 = (LineG‘𝐺) | |
16 | btwnlng1.d | . . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
17 | 2, 15, 4, 5, 8, 6, 16, 7 | tgellng 26342 | . . . 4 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
18 | 12, 14, 17 | 3bitr4rd 314 | . . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌)))) |
19 | lnrot1.2 | . . . 4 ⊢ (𝜑 → 𝑍 ≠ 𝑋) | |
20 | 2, 15, 4, 5, 7, 8, 19, 6 | tgellng 26342 | . . 3 ⊢ (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ↔ (𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑋 ∈ (𝑍𝐼𝑌)))) |
21 | 18, 20 | bitr4d 284 | . 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ 𝑌 ∈ (𝑍𝐿𝑋))) |
22 | 1, 21 | mpbird 259 | 1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∨ w3o 1082 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 distcds 16577 TarskiGcstrkg 26219 Itvcitv 26225 LineGclng 26226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-trkgc 26237 df-trkgb 26238 df-trkgcb 26239 df-trkg 26242 |
This theorem is referenced by: tglineelsb2 26421 tglineneq 26433 coltr3 26437 hlperpnel 26514 opphllem4 26539 lmieu 26573 |
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