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Theorem ncolrot2 26361
 Description: Rotating non-colinear points. (Contributed by Thierry Arnoux, 19-Oct-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)
ncolrot (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
Assertion
Ref Expression
ncolrot2 (𝜑 → ¬ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))

Proof of Theorem ncolrot2
StepHypRef Expression
1 ncolrot . 2 (𝜑 → ¬ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
2 tglngval.p . . 3 𝑃 = (Base‘𝐺)
3 tglngval.l . . 3 𝐿 = (LineG‘𝐺)
4 tglngval.i . . 3 𝐼 = (Itv‘𝐺)
5 tglngval.g . . . 4 (𝜑𝐺 ∈ TarskiG)
65adantr 484 . . 3 ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝐺 ∈ TarskiG)
7 tgcolg.z . . . 4 (𝜑𝑍𝑃)
87adantr 484 . . 3 ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝑍𝑃)
9 tglngval.x . . . 4 (𝜑𝑋𝑃)
109adantr 484 . . 3 ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝑋𝑃)
11 tglngval.y . . . 4 (𝜑𝑌𝑃)
1211adantr 484 . . 3 ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → 𝑌𝑃)
13 simpr 488 . . 3 ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
142, 3, 4, 6, 8, 10, 12, 13colrot1 26357 . 2 ((𝜑 ∧ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
151, 14mtand 815 1 (𝜑 → ¬ (𝑌 ∈ (𝑍𝐿𝑋) ∨ 𝑍 = 𝑋))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2112  ‘cfv 6328  (class class class)co 7139  Basecbs 16479  TarskiGcstrkg 26228  Itvcitv 26234  LineGclng 26235 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-trkgc 26246  df-trkgb 26247  df-trkgcb 26248  df-trkg 26251 This theorem is referenced by:  midexlem  26490  perpneq  26512  opphllem  26533  outpasch  26553  hlpasch  26554  trgcopy  26602  acopyeu  26632
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