tgdim01ln - Metamath Proof Explorer

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Theorem tgdim01ln 29027

Description: In geometries of dimension less than two, then any three points are colinear. (Contributed by Thierry Arnoux, 27-Aug-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)
tgdim01ln.1	⊢ (𝜑 → ¬ 𝐺Dim_TarskiG≥2)

**Assertion**
Ref	Expression
tgdim01ln	⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))

**Proof of Theorem tgdim01ln**
Step	Hyp	Ref	Expression
1		tglngval.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		tglngval.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
3		tglngval.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		tglngval.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝐺 ∈ TarskiG)
6		tglngval.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
7	6	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑋 ∈ 𝑃)
8		tglngval.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
9	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑌 ∈ 𝑃)
10		tgcolg.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
11	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ 𝑃)
12		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → 𝑍 ∈ (𝑋𝐼𝑌))
13	1, 2, 3, 5, 7, 9, 11, 12	btwncolg1 29018	. 2 ⊢ ((𝜑 ∧ 𝑍 ∈ (𝑋𝐼𝑌)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
14	4	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → 𝐺 ∈ TarskiG)
15	6	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → 𝑋 ∈ 𝑃)
16	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → 𝑌 ∈ 𝑃)
17	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → 𝑍 ∈ 𝑃)
18		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → 𝑋 ∈ (𝑍𝐼𝑌))
19	1, 2, 3, 14, 15, 16, 17, 18	btwncolg2 29019	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑍𝐼𝑌)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
20	4	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐺 ∈ TarskiG)
21	6	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑋 ∈ 𝑃)
22	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ∈ 𝑃)
23	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑍 ∈ 𝑃)
24		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ∈ (𝑋𝐼𝑍))
25	1, 2, 3, 20, 21, 22, 23, 24	btwncolg3 29020	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
26		tgdim01ln.1	. . 3 ⊢ (𝜑 → ¬ 𝐺Dim_TarskiG≥2)
27	1, 3, 4, 26, 6, 8, 10	tgdim01 28970	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
28	13, 19, 25, 27	mpjao3dan 1459	1 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 class class class wbr 5103 ‘cfv 6538 (class class class)co 7420 2c2 12397 Basecbs 17387 TarskiGcstrkg 28889 Dim_TarskiG≥cstrkgld 28893 Itvcitv 28895 LineGclng 28896

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 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-er 8717 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-2 12405 df-n0 12607 df-z 12694 df-uz 12966 df-fz 13640 df-fzo 13789 df-trkgc 28910 df-trkgcb 28912 df-trkgld 28914 df-trkg 28915

This theorem is used by: ncoltgdim2 29028

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