Theorem lncom 27873

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
btwnlng1.p	⊢ 𝑃 = (Base‘𝐺)
btwnlng1.i	⊢ 𝐼 = (Itv‘𝐺)
btwnlng1.l	⊢ 𝐿 = (LineG‘𝐺)
btwnlng1.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
btwnlng1.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
btwnlng1.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
btwnlng1.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
btwnlng1.d	⊢ (𝜑 → 𝑋 ≠ 𝑌)
lncom.1	⊢ (𝜑 → 𝑍 ∈ (𝑌𝐿𝑋))

Assertion

Ref	Expression
lncom	⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

Proof of Theorem lncom

Step	Hyp	Ref	Expression
1		lncom.1	. 2 ⊢ (𝜑 → 𝑍 ∈ (𝑌𝐿𝑋))
2		3orcomb 1095	. . . 4 ⊢ ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌)))
3		btwnlng1.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
4		eqid 2733	. . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺)
5		btwnlng1.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
6		btwnlng1.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		btwnlng1.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
8		btwnlng1.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
9		btwnlng1.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
10	3, 4, 5, 6, 7, 8, 9	tgbtwncomb 27740	. . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑌𝐼𝑋)))
11	3, 4, 5, 6, 7, 9, 8	tgbtwncomb 27740	. . . . 5 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ 𝑌 ∈ (𝑍𝐼𝑋)))
12	3, 4, 5, 6, 8, 7, 9	tgbtwncomb 27740	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑌𝐼𝑍)))
13	10, 11, 12	3orbi123d 1436	. . . 4 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍))))
14	2, 13	bitrid 283	. . 3 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍))))
15		btwnlng1.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
16		btwnlng1.d	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
17	3, 15, 5, 6, 7, 9, 16, 8	tgellng 27804	. . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
18	16	necomd 2997	. . . 4 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
19	3, 15, 5, 6, 9, 7, 18, 8	tgellng 27804	. . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐿𝑋) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍))))
20	14, 17, 19	3bitr4d 311	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ 𝑍 ∈ (𝑌𝐿𝑋)))
21	1, 20	mpbird 257	1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

Syntax hints:   → wi 4   ∨ w3o 1087   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  distcds 17206  TarskiGcstrkg 27678  Itvcitv 27684  LineGclng 27685

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-trkgc 27699  df-trkgb 27700  df-trkgcb 27701  df-trkg 27704

This theorem is referenced by:  tglineelsb2  27883  tglinecom  27886  ncolncol  27897  coltr  27898  midexlem  27943  footexALT  27969  footexlem1  27970  footexlem2  27971  opphllem1  27998  opphllem2  27999  outpasch  28006  hlpasch  28007  trgcopy  28055  trgcopyeulem  28056  cgracgr  28069  tgasa1  28109