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Theorem colrot2 28524
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 28523 . 2 (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
101, 2, 3, 4, 5, 6, 7, 9colrot1 28523 1 (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 847   = wceq 1540  wcel 2109  cfv 6486  (class class class)co 7353  Basecbs 17139  TarskiGcstrkg 28391  Itvcitv 28397  LineGclng 28398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-trkgc 28412  df-trkgb 28413  df-trkgcb 28414  df-trkg 28417
This theorem is referenced by:  ncolrot1  28526  tglineeltr  28595  ncolncol  28610  symquadlem  28653  hlpasch  28720  hphl  28735  trgcopy  28768
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