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Theorem colrot2 26273
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
tglngval.p 𝑃 = (Base‘𝐺)
tglngval.l 𝐿 = (LineG‘𝐺)
tglngval.i 𝐼 = (Itv‘𝐺)
tglngval.g (𝜑𝐺 ∈ TarskiG)
tglngval.x (𝜑𝑋𝑃)
tglngval.y (𝜑𝑌𝑃)
tgcolg.z (𝜑𝑍𝑃)
colrot (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
Assertion
Ref Expression
colrot2 (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))

Proof of Theorem colrot2
StepHypRef Expression
1 tglngval.p . 2 𝑃 = (Base‘𝐺)
2 tglngval.l . 2 𝐿 = (LineG‘𝐺)
3 tglngval.i . 2 𝐼 = (Itv‘𝐺)
4 tglngval.g . 2 (𝜑𝐺 ∈ TarskiG)
5 tglngval.y . 2 (𝜑𝑌𝑃)
6 tgcolg.z . 2 (𝜑𝑍𝑃)
7 tglngval.x . 2 (𝜑𝑋𝑃)
8 colrot . . 3 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
91, 2, 3, 4, 7, 5, 6, 8colrot1 26272 . 2 (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
101, 2, 3, 4, 5, 6, 7, 9colrot1 26272 1 (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 841   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  Basecbs 16471  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:  ncolrot1  26275  tglineeltr  26344  ncolncol  26359  symquadlem  26402  hlpasch  26469  hphl  26484  trgcopy  26517
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