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Theorem colrot2 25675
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 25674 . 2 (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
101, 2, 3, 4, 5, 6, 7, 9colrot1 25674 1 (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382   = wceq 1632  wcel 2139  cfv 6049  (class class class)co 6814  Basecbs 16079  TarskiGcstrkg 25549  Itvcitv 25555  LineGclng 25556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-trkgc 25567  df-trkgb 25568  df-trkgcb 25569  df-trkg 25572
This theorem is referenced by:  ncolrot1  25677  tglineeltr  25746  ncolncol  25761  symquadlem  25804  hlpasch  25868  hphl  25883  trgcopy  25916
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