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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglngval.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
tglngval.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
tgcolg.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
colrot	⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))

Assertion

Ref	Expression
colrot2	⊢ (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))

Proof of Theorem colrot2

Step	Hyp	Ref	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 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		tglngval.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
6		tgcolg.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
7		tglngval.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
8		colrot	. . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
9	1, 2, 3, 4, 7, 5, 6, 8	colrot1 28523	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
10	1, 2, 3, 4, 5, 6, 7, 9	colrot1 28523	1 ⊢ (𝜑 → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))

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