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Theorem colcom 25652
Description: Swapping the points defining a line keeps it unchanged. 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
colcom (𝜑 → (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))

Proof of Theorem colcom
StepHypRef Expression
1 colrot . 2 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
2 3orcomb 1079 . . . 4 ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌)))
3 tglngval.p . . . . . 6 𝑃 = (Base‘𝐺)
4 eqid 2760 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
5 tglngval.i . . . . . 6 𝐼 = (Itv‘𝐺)
6 tglngval.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
7 tglngval.x . . . . . 6 (𝜑𝑋𝑃)
8 tgcolg.z . . . . . 6 (𝜑𝑍𝑃)
9 tglngval.y . . . . . 6 (𝜑𝑌𝑃)
103, 4, 5, 6, 7, 8, 9tgbtwncomb 25583 . . . . 5 (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑌𝐼𝑋)))
113, 4, 5, 6, 7, 9, 8tgbtwncomb 25583 . . . . 5 (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ 𝑌 ∈ (𝑍𝐼𝑋)))
123, 4, 5, 6, 8, 7, 9tgbtwncomb 25583 . . . . 5 (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑌𝐼𝑍)))
1310, 11, 123orbi123d 1547 . . . 4 (𝜑 → ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍))))
142, 13syl5bb 272 . . 3 (𝜑 → ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍))))
15 tglngval.l . . . 4 𝐿 = (LineG‘𝐺)
163, 15, 5, 6, 7, 9, 8tgcolg 25648 . . 3 (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
173, 15, 5, 6, 9, 7, 8tgcolg 25648 . . 3 (𝜑 → ((𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍))))
1814, 16, 173bitr4d 300 . 2 (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)))
191, 18mpbid 222 1 (𝜑 → (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  w3o 1071   = wceq 1632  wcel 2139  cfv 6049  (class class class)co 6813  Basecbs 16059  distcds 16152  TarskiGcstrkg 25528  Itvcitv 25534  LineGclng 25535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-trkgc 25546  df-trkgb 25547  df-trkgcb 25548  df-trkg 25551
This theorem is referenced by:  ncolcom  25655  tglineeltr  25725  mirtrcgr  25777  symquadlem  25783  midexlem  25786  colperpexlem1  25821  mideulem2  25825  opphllem  25826  hlpasch  25847  colhp  25861  trgcopy  25895  cgrg3col4  25933  tgasa1  25938
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