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Theorem colrot2 26675
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 26674 . 2 (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
101, 2, 3, 4, 5, 6, 7, 9colrot1 26674 1 (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 847   = wceq 1543  wcel 2111  cfv 6397  (class class class)co 7231  Basecbs 16784  TarskiGcstrkg 26545  Itvcitv 26551  LineGclng 26552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pr 5336
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3422  df-sbc 3709  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-pw 4529  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-br 5068  df-opab 5130  df-id 5469  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-iota 6355  df-fun 6399  df-fv 6405  df-ov 7234  df-oprab 7235  df-mpo 7236  df-trkgc 26563  df-trkgb 26564  df-trkgcb 26565  df-trkg 26568
This theorem is referenced by:  ncolrot1  26677  tglineeltr  26746  ncolncol  26761  symquadlem  26804  hlpasch  26871  hphl  26886  trgcopy  26919
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