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

Theorem lnrot2 25875
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 = (Base‘𝐺)
btwnlng1.i 𝐼 = (Itv‘𝐺)
btwnlng1.l 𝐿 = (LineG‘𝐺)
btwnlng1.g (𝜑𝐺 ∈ TarskiG)
btwnlng1.x (𝜑𝑋𝑃)
btwnlng1.y (𝜑𝑌𝑃)
btwnlng1.z (𝜑𝑍𝑃)
btwnlng1.d (𝜑𝑋𝑌)
lnrot2.1 (𝜑𝑋 ∈ (𝑌𝐿𝑍))
lnrot2.2 (𝜑𝑌𝑍)
Assertion
Ref Expression
lnrot2 (𝜑𝑍 ∈ (𝑋𝐿𝑌))

Proof of Theorem lnrot2
StepHypRef Expression
1 lnrot2.1 . 2 (𝜑𝑋 ∈ (𝑌𝐿𝑍))
2 btwnlng1.p . . . . . 6 𝑃 = (Base‘𝐺)
3 eqid 2799 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
4 btwnlng1.i . . . . . 6 𝐼 = (Itv‘𝐺)
5 btwnlng1.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
6 btwnlng1.y . . . . . 6 (𝜑𝑌𝑃)
7 btwnlng1.x . . . . . 6 (𝜑𝑋𝑃)
8 btwnlng1.z . . . . . 6 (𝜑𝑍𝑃)
92, 3, 4, 5, 6, 7, 8tgbtwncomb 25740 . . . . 5 (𝜑 → (𝑋 ∈ (𝑌𝐼𝑍) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
10 biidd 254 . . . . 5 (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
112, 3, 4, 5, 6, 8, 7tgbtwncomb 25740 . . . . 5 (𝜑 → (𝑍 ∈ (𝑌𝐼𝑋) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
129, 10, 113orbi123d 1560 . . . 4 (𝜑 → ((𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑌𝐼𝑋)) ↔ (𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌))))
13 3orrot 1113 . . . 4 ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑋𝐼𝑌)))
1412, 13syl6bbr 281 . . 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 (𝜑𝑌𝑍)
172, 15, 4, 5, 6, 8, 16, 7tgellng 25804 . . 3 (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ↔ (𝑋 ∈ (𝑌𝐼𝑍) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑍 ∈ (𝑌𝐼𝑋))))
18 btwnlng1.d . . . 4 (𝜑𝑋𝑌)
192, 15, 4, 5, 7, 6, 18, 8tgellng 25804 . . 3 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
2014, 17, 193bitr4d 303 . 2 (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ↔ 𝑍 ∈ (𝑋𝐿𝑌)))
211, 20mpbid 224 1 (𝜑𝑍 ∈ (𝑋𝐿𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3o 1107   = wceq 1653  wcel 2157  wne 2971  cfv 6101  (class class class)co 6878  Basecbs 16184  distcds 16276  TarskiGcstrkg 25681  Itvcitv 25687  LineGclng 25688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-trkgc 25699  df-trkgb 25700  df-trkgcb 25701  df-trkg 25704
This theorem is referenced by:  coltr  25898  mideulem2  25982
  Copyright terms: Public domain W3C validator