Theorem tgcolg 28899

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 994	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
2		tglngval.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
3		eqid 2760	. . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺)
4		tglngval.i	. . . . . 6 ⊢ 𝐼 = (Itv‘𝐺)
5		tglngval.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6	5	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝐺 ∈ TarskiG)
7		tgcolg.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
8	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑍 ∈ 𝑃)
9		tglngval.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
10	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ 𝑃)
11	2, 3, 4, 6, 8, 10	tgbtwntriv2 28832	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑍𝐼𝑋))
12		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
13	12	oveq2d 7430	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑍𝐼𝑋) = (𝑍𝐼𝑌))
14	11, 13	eleqtrd 2862	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 ∈ (𝑍𝐼𝑌))
15	14	3mix2d 1356	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
16	1, 15	2thd 268	. 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
17		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌)
18	17	neneqd 2960	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌)
19		biorf 950	. . . . 5 ⊢ (¬ 𝑋 = 𝑌 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑋 = 𝑌 ∨ 𝑍 ∈ (𝑋𝐿𝑌))))
20	18, 19	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑋 = 𝑌 ∨ 𝑍 ∈ (𝑋𝐿𝑌))))
21		orcom 884	. . . 4 ⊢ ((𝑋 = 𝑌 ∨ 𝑍 ∈ (𝑋𝐿𝑌)) ↔ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
22	20, 21	bitrdi 290	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)))
23		tglngval.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
24	5	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐺 ∈ TarskiG)
25	9	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝑃)
26		tglngval.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
27	26	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝑃)
28	7	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑍 ∈ 𝑃)
29	2, 23, 4, 24, 25, 27, 17, 28	tgellng 28898	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
30	22, 29	bitr3d 284	. 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
31	16, 30	pm2.61dane 3042	1 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ‘cfv 6533  (class class class)co 7414  Basecbs 17304  distcds 17354  TarskiGcstrkg 28771  Itvcitv 28777  LineGclng 28778

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-trkgc 28792  df-trkgcb 28794  df-trkg 28797

This theorem is used by:  btwncolg1  28900  btwncolg2  28901  btwncolg3  28902  colcom  28903  colrot1  28904  lnxfr  28911  lnext  28912  tgfscgr  28913  tglowdim2l  29001  outpasch  29115