MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lnrot2 Structured version   Visualization version   GIF version

Theorem lnrot2 28379
Description: Rotating the points defining a line. Part of Theorem 4.11 of [Schwabhauser] p. 34. (Contributed by Thierry Arnoux, 3-Apr-2019.)
Hypotheses
Ref Expression
btwnlng1.p 𝑃 = (Baseβ€˜πΊ)
btwnlng1.i 𝐼 = (Itvβ€˜πΊ)
btwnlng1.l 𝐿 = (LineGβ€˜πΊ)
btwnlng1.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
btwnlng1.x (πœ‘ β†’ 𝑋 ∈ 𝑃)
btwnlng1.y (πœ‘ β†’ π‘Œ ∈ 𝑃)
btwnlng1.z (πœ‘ β†’ 𝑍 ∈ 𝑃)
btwnlng1.d (πœ‘ β†’ 𝑋 β‰  π‘Œ)
lnrot2.1 (πœ‘ β†’ 𝑋 ∈ (π‘ŒπΏπ‘))
lnrot2.2 (πœ‘ β†’ π‘Œ β‰  𝑍)
Assertion
Ref Expression
lnrot2 (πœ‘ β†’ 𝑍 ∈ (π‘‹πΏπ‘Œ))

Proof of Theorem lnrot2
StepHypRef Expression
1 lnrot2.1 . 2 (πœ‘ β†’ 𝑋 ∈ (π‘ŒπΏπ‘))
2 btwnlng1.p . . . . . 6 𝑃 = (Baseβ€˜πΊ)
3 eqid 2726 . . . . . 6 (distβ€˜πΊ) = (distβ€˜πΊ)
4 btwnlng1.i . . . . . 6 𝐼 = (Itvβ€˜πΊ)
5 btwnlng1.g . . . . . 6 (πœ‘ β†’ 𝐺 ∈ TarskiG)
6 btwnlng1.y . . . . . 6 (πœ‘ β†’ π‘Œ ∈ 𝑃)
7 btwnlng1.x . . . . . 6 (πœ‘ β†’ 𝑋 ∈ 𝑃)
8 btwnlng1.z . . . . . 6 (πœ‘ β†’ 𝑍 ∈ 𝑃)
92, 3, 4, 5, 6, 7, 8tgbtwncomb 28244 . . . . 5 (πœ‘ β†’ (𝑋 ∈ (π‘ŒπΌπ‘) ↔ 𝑋 ∈ (π‘πΌπ‘Œ)))
10 biidd 262 . . . . 5 (πœ‘ β†’ (π‘Œ ∈ (𝑋𝐼𝑍) ↔ π‘Œ ∈ (𝑋𝐼𝑍)))
112, 3, 4, 5, 6, 8, 7tgbtwncomb 28244 . . . . 5 (πœ‘ β†’ (𝑍 ∈ (π‘ŒπΌπ‘‹) ↔ 𝑍 ∈ (π‘‹πΌπ‘Œ)))
129, 10, 113orbi123d 1431 . . . 4 (πœ‘ β†’ ((𝑋 ∈ (π‘ŒπΌπ‘) ∨ π‘Œ ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (π‘ŒπΌπ‘‹)) ↔ (𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (π‘‹πΌπ‘Œ))))
13 3orrot 1089 . . . 4 ((𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍)) ↔ (𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (π‘‹πΌπ‘Œ)))
1412, 13bitr4di 289 . . 3 (πœ‘ β†’ ((𝑋 ∈ (π‘ŒπΌπ‘) ∨ π‘Œ ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (π‘ŒπΌπ‘‹)) ↔ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍))))
15 btwnlng1.l . . . 4 𝐿 = (LineGβ€˜πΊ)
16 lnrot2.2 . . . 4 (πœ‘ β†’ π‘Œ β‰  𝑍)
172, 15, 4, 5, 6, 8, 16, 7tgellng 28308 . . 3 (πœ‘ β†’ (𝑋 ∈ (π‘ŒπΏπ‘) ↔ (𝑋 ∈ (π‘ŒπΌπ‘) ∨ π‘Œ ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (π‘ŒπΌπ‘‹))))
18 btwnlng1.d . . . 4 (πœ‘ β†’ 𝑋 β‰  π‘Œ)
192, 15, 4, 5, 7, 6, 18, 8tgellng 28308 . . 3 (πœ‘ β†’ (𝑍 ∈ (π‘‹πΏπ‘Œ) ↔ (𝑍 ∈ (π‘‹πΌπ‘Œ) ∨ 𝑋 ∈ (π‘πΌπ‘Œ) ∨ π‘Œ ∈ (𝑋𝐼𝑍))))
2014, 17, 193bitr4d 311 . 2 (πœ‘ β†’ (𝑋 ∈ (π‘ŒπΏπ‘) ↔ 𝑍 ∈ (π‘‹πΏπ‘Œ)))
211, 20mpbid 231 1 (πœ‘ β†’ 𝑍 ∈ (π‘‹πΏπ‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∨ w3o 1083   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  β€˜cfv 6536  (class class class)co 7404  Basecbs 17151  distcds 17213  TarskiGcstrkg 28182  Itvcitv 28188  LineGclng 28189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-trkgc 28203  df-trkgb 28204  df-trkgcb 28205  df-trkg 28208
This theorem is referenced by:  coltr  28402  mideulem2  28489
  Copyright terms: Public domain W3C validator