Theorem tgellng 28456

Description: Property of lying on the line going through points 𝑋 and 𝑌. Definition 4.10 of [Schwabhauser] p. 36. We choose the notation 𝑍 ∈ (𝑋(LineG‘𝐺)𝑌) instead of "colinear" because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 28-Mar-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)
tglngval.z	⊢ (𝜑 → 𝑋 ≠ 𝑌)
tgellng.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)

Assertion

Ref	Expression
tgellng	⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))

Proof of Theorem tgellng

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgellng.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
2		tglngval.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
3		tglngval.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
4		tglngval.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
5		tglngval.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6		tglngval.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
7		tglngval.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
8		tglngval.z	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
9	2, 3, 4, 5, 6, 7, 8	tglngval 28454	. . . 4 ⊢ (𝜑 → (𝑋𝐿𝑌) = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
10	9	eleq2d 2814	. . 3 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ 𝑍 ∈ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))}))
11		eleq1 2816	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
12		oveq1 7376	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑧𝐼𝑌) = (𝑍𝐼𝑌))
13	12	eleq2d 2814	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑋 ∈ (𝑧𝐼𝑌) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
14		oveq2 7377	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
15	14	eleq2d 2814	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑌 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
16	11, 13, 15	3orbi123d 1437	. . . 4 ⊢ (𝑧 = 𝑍 → ((𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
17	16	elrab 3656	. . 3 ⊢ (𝑍 ∈ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ↔ (𝑍 ∈ 𝑃 ∧ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
18	10, 17	bitrdi 287	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ 𝑃 ∧ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
19	1, 18	mpbirand 707	1 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  {crab 3402  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  TarskiGcstrkg 28330  Itvcitv 28336  LineGclng 28337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-trkg 28356

This theorem is referenced by:  tgcolg  28457  hlln  28510  lnhl  28518  btwnlng1  28522  btwnlng2  28523  btwnlng3  28524  lncom  28525  lnrot1  28526  lnrot2  28527  tglineeltr  28534  colmid  28591  cgracol  28731