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Theorem colrot2 28632
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 28631 . 2 (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
101, 2, 3, 4, 5, 6, 7, 9colrot1 28631 1 (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 847   = wceq 1541  wcel 2113  cfv 6492  (class class class)co 7358  Basecbs 17136  TarskiGcstrkg 28499  Itvcitv 28505  LineGclng 28506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-trkgc 28520  df-trkgb 28521  df-trkgcb 28522  df-trkg 28525
This theorem is referenced by:  ncolrot1  28634  tglineeltr  28703  ncolncol  28718  symquadlem  28761  hlpasch  28828  hphl  28843  trgcopy  28876
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