Theorem lncom 25235

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 1040	. . . 4 ⊢ ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌)))
3		btwnlng1.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
4		eqid 2609	. . . . . 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 25101	. . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑌𝐼𝑋)))
11	3, 4, 5, 6, 7, 9, 8	tgbtwncomb 25101	. . . . 5 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ 𝑌 ∈ (𝑍𝐼𝑋)))
12	3, 4, 5, 6, 8, 7, 9	tgbtwncomb 25101	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ (𝑍𝐼𝑌) ↔ 𝑋 ∈ (𝑌𝐼𝑍)))
13	10, 11, 12	3orbi123d 1389	. . . 4 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌)) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍))))
14	2, 13	syl5bb 270	. . 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 25166	. . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
18	16	necomd 2836	. . . 4 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
19	3, 15, 5, 6, 9, 7, 18, 8	tgellng 25166	. . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐿𝑋) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍))))
20	14, 17, 19	3bitr4d 298	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ 𝑍 ∈ (𝑌𝐿𝑋)))
21	1, 20	mpbird 245	1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

Syntax hints:   → wi 4   ∨ w3o 1029   = wceq 1474   ∈ wcel 1976   ≠ wne 2779  ‘cfv 5790  (class class class)co 6527  Basecbs 15641  distcds 15723  TarskiGcstrkg 25046  Itvcitv 25052  LineGclng 25053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-trkgc 25064  df-trkgb 25065  df-trkgcb 25066  df-trkg 25069

This theorem is referenced by:  tglineelsb2  25245  tglinecom  25248  ncolncol  25259  coltr  25260  midexlem  25305  footex  25331  opphllem1  25357  opphllem2  25358  outpasch  25365  hlpasch  25366  trgcopy  25414  trgcopyeulem  25415  cgracgr  25428  tgasa1  25457