Theorem lncom 28948

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 1110	. . . 4 ⊢ ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌)))
3		btwnlng1.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
4		eqid 2765	. . . . . 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 28811	. . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑌𝐼𝑋)))
11	3, 4, 5, 6, 7, 9, 8	tgbtwncomb 28811	. . . . 5 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ 𝑌 ∈ (𝑍𝐼𝑋)))
12	3, 4, 5, 6, 8, 7, 9	tgbtwncomb 28811	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑌𝐼𝑍)))
13	10, 11, 12	3orbi123d 1463	. . . 4 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍))))
14	2, 13	bitrid 286	. . 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 28875	. . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
18	16	necomd 3015	. . . 4 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
19	3, 15, 5, 6, 9, 7, 18, 8	tgellng 28875	. . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐿𝑋) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍))))
20	14, 17, 19	3bitr4d 314	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ 𝑍 ∈ (𝑌𝐿𝑋)))
21	1, 20	mpbird 260	1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

This proof depends on syntax axioms:   → wi 4   ∨ w3o 1102   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ‘cfv 6540  (class class class)co 7419  Basecbs 17293  distcds 17343  TarskiGcstrkg 28749  Itvcitv 28755  LineGclng 28756

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-trkgc 28770  df-trkgb 28771  df-trkgcb 28772  df-trkg 28775

This theorem is used by:  tglineelsb2  28958  tglinecom  28961  ncolncol  28973  coltr  28974  midexlem  29022  footexALT  29051  footexlem1  29052  footexlem2  29053  opphllem1  29081  opphllem2  29082  outpasch  29090  hlpasch  29091  lnincplng  29119  lnssplnglem  29126  trgcopy  29168  trgcopyeulem  29169  cgracgr  29182  tgaaddcpbllem3  29207  tgasa1  29232  prlngmid2  29268