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

Theorem lnrot2 23149
 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
btwnlng1.i Itv
btwnlng1.l LineG
btwnlng1.g TarskiG
btwnlng1.x
btwnlng1.y
btwnlng1.z
btwnlng1.d
lnrot2.1
lnrot2.2
Assertion
Ref Expression
lnrot2

Proof of Theorem lnrot2
StepHypRef Expression
1 lnrot2.1 . 2
2 btwnlng1.p . . . . . 6
3 eqid 2451 . . . . . 6
4 btwnlng1.i . . . . . 6 Itv
5 btwnlng1.g . . . . . 6 TarskiG
6 btwnlng1.y . . . . . 6
7 btwnlng1.x . . . . . 6
8 btwnlng1.z . . . . . 6
92, 3, 4, 5, 6, 7, 8tgbtwncomb 23057 . . . . 5
10 biidd 237 . . . . 5
112, 3, 4, 5, 6, 8, 7tgbtwncomb 23057 . . . . 5
129, 10, 113orbi123d 1289 . . . 4
13 3orrot 971 . . . 4
1412, 13syl6bbr 263 . . 3
15 btwnlng1.l . . . 4 LineG
16 lnrot2.2 . . . 4
172, 15, 4, 5, 6, 8, 16, 7tgellng 23103 . . 3
18 btwnlng1.d . . . 4
192, 15, 4, 5, 7, 6, 18, 8tgellng 23103 . . 3
2014, 17, 193bitr4d 285 . 2
211, 20mpbid 210 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   w3o 964   wceq 1370   wcel 1758   wne 2642  cfv 5513  (class class class)co 6187  cbs 14273  cds 14346  TarskiGcstrkg 23002  Itvcitv 23009  LineGclng 23010 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-iota 5476  df-fun 5515  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-trkgc 23021  df-trkgb 23022  df-trkgcb 23023  df-trkg 23027 This theorem is referenced by:  coltr  23171  mideulem  23241
 Copyright terms: Public domain W3C validator