Theorem colcom 29033

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglngval.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
tglngval.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
tgcolg.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
colrot	⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))

Assertion

Ref	Expression
colcom	⊢ (𝜑 → (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))

Proof of Theorem colcom

Step	Hyp	Ref	Expression
1		colrot	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
2		3orcomb 1110	. . . 4 ⊢ ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍) ∨ 𝑋 ∈ (𝑍𝐼𝑌)))
3		tglngval.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
4		eqid 2763	. . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺)
5		tglngval.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
6		tglngval.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		tglngval.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
8		tgcolg.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
9		tglngval.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
10	3, 4, 5, 6, 7, 8, 9	tgbtwncomb 28964	. . . . 5 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑌𝐼𝑋)))
11	3, 4, 5, 6, 7, 9, 8	tgbtwncomb 28964	. . . . 5 ⊢ (𝜑 → (𝑌 ∈ (𝑋𝐼𝑍) ↔ 𝑌 ∈ (𝑍𝐼𝑋)))
12	3, 4, 5, 6, 8, 7, 9	tgbtwncomb 28964	. . . . 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		tglngval.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
16	3, 15, 5, 6, 7, 9, 8	tgcolg 29029	. . 3 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
17	3, 15, 5, 6, 9, 7, 8	tgcolg 29029	. . 3 ⊢ (𝜑 → ((𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋) ↔ (𝑍 ∈ (𝑌𝐼𝑋) ∨ 𝑌 ∈ (𝑍𝐼𝑋) ∨ 𝑋 ∈ (𝑌𝐼𝑍))))
18	14, 16, 17	3bitr4d 314	. 2 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋)))
19	1, 18	mpbid 235	1 ⊢ (𝜑 → (𝑍 ∈ (𝑌𝐿𝑋) ∨ 𝑌 = 𝑋))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   ∨ w3o 1102   = wceq 1569   ∈ wcel 2146  ‘cfv 6546  (class class class)co 7427  Basecbs 17399  distcds 17449  TarskiGcstrkg 28901  Itvcitv 28907  LineGclng 28908

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-sbc 3742  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6500  df-fun 6548  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-trkgc 28922  df-trkgb 28923  df-trkgcb 28924  df-trkg 28927

This theorem is used by:  ncolcom  29036  tglineeltr  29111  mirtrcgr  29167  symquadlem  29173  midexlem  29176  symquadprlnglem  29177  colperpexlem1  29218  mideulem2  29222  opphllem  29223  hlpasch  29246  colhp  29260  trgcopy  29323  tgaaddcpbllem1  29361  tgaaddcpbl  29364  tgaaddcpbl2  29365  cgrg3col4  29384  tgasa1  29415  symquadprlng  29452