Theorem tgcolg 28622

Description: We choose the notation (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) instead of "colinear" in order to avoid defining an additional symbol for colinearity because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 25-May-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)

Assertion

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

Proof of Theorem tgcolg

Step	Hyp	Ref	Expression
1		animorr 981	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
2		tglngval.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
3		eqid 2736	. . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺)
4		tglngval.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
5		tglngval.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝐺 ∈ TarskiG)
7		tgcolg.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
8	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ 𝑃)
9		tglngval.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
10	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ 𝑃)
11	2, 3, 4, 6, 8, 10	tgbtwntriv2 28555	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑍𝐼𝑋))
12		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
13	12	oveq2d 7383	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑍𝐼𝑋) = (𝑍𝐼𝑌))
14	11, 13	eleqtrd 2838	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑍𝐼𝑌))
15	14	3mix2d 1339	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
16	1, 15	2thd 265	. 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
17		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌)
18	17	neneqd 2937	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌)
19		biorf 937	. . . . 5 ⊢ (¬ 𝑋 = 𝑌 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑋 = 𝑌 ∨ 𝑍 ∈ (𝑋𝐿𝑌))))
20	18, 19	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑋 = 𝑌 ∨ 𝑍 ∈ (𝑋𝐿𝑌))))
21		orcom 871	. . . 4 ⊢ ((𝑋 = 𝑌 ∨ 𝑍 ∈ (𝑋𝐿𝑌)) ↔ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
22	20, 21	bitrdi 287	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)))
23		tglngval.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
24	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐺 ∈ TarskiG)
25	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝑃)
26		tglngval.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
27	26	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝑃)
28	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑍 ∈ 𝑃)
29	2, 23, 4, 24, 25, 27, 17, 28	tgellng 28621	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
30	22, 29	bitr3d 281	. 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
31	16, 30	pm2.61dane 3019	1 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∨ w3o 1086   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  distcds 17229  TarskiGcstrkg 28495  Itvcitv 28501  LineGclng 28502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-trkgc 28516  df-trkgcb 28518  df-trkg 28521

This theorem is referenced by:  btwncolg1  28623  btwncolg2  28624  btwncolg3  28625  colcom  28626  colrot1  28627  lnxfr  28634  lnext  28635  tgfscgr  28636  tglowdim2l  28718  outpasch  28823