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Theorem ncolcom 26932
Description: Swapping 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
ncolcom (𝜑 → ¬ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))

Proof of Theorem ncolcom
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 481 . . 3 ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → 𝐺 ∈ TarskiG)
7 tglngval.y . . . 4 (𝜑𝑌𝑃)
87adantr 481 . . 3 ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → 𝑌𝑃)
9 tglngval.x . . . 4 (𝜑𝑋𝑃)
109adantr 481 . . 3 ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → 𝑋𝑃)
11 tgcolg.z . . . 4 (𝜑𝑍𝑃)
1211adantr 481 . . 3 ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → 𝑍𝑃)
13 simpr 485 . . 3 ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))
142, 3, 4, 6, 8, 10, 12, 13colcom 26929 . 2 ((𝜑 ∧ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
151, 14mtand 813 1 (𝜑 → ¬ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 844   = wceq 1539  wcel 2106  cfv 6426  (class class class)co 7267  Basecbs 16922  TarskiGcstrkg 26798  Itvcitv 26804  LineGclng 26805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-sbc 3716  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-iota 6384  df-fun 6428  df-fv 6434  df-ov 7270  df-oprab 7271  df-mpo 7272  df-trkgc 26819  df-trkgb 26820  df-trkgcb 26821  df-trkg 26824
This theorem is referenced by:  ncolne2  26997  symquadlem  27060  midexlem  27063  outpasch  27126  acopyeu  27205  cgrg3col4  27224  tgasa1  27229
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