tglowdim2l - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > tglowdim2l		Structured version Visualization version GIF version

Theorem tglowdim2l 26018

Description: Reformulation of the lower dimension axiom for dimension two. There exist three non-colinear points. Theorem 6.24 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 30-May-2019.)

**Hypotheses**
Ref	Expression
tglineintmo.p	⊢ 𝑃 = (Base‘𝐺)
tglineintmo.i	⊢ 𝐼 = (Itv‘𝐺)
tglineintmo.l	⊢ 𝐿 = (LineG‘𝐺)
tglineintmo.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglowdim2l.1	⊢ (𝜑 → 𝐺Dim_TarskiG≥2)

**Assertion**
Ref	Expression
tglowdim2l	⊢ (𝜑 → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ¬ (𝑐 ∈ (𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))

Distinct variable groups: 𝑎,𝑏,𝑐,𝐺 𝐼,𝑎,𝑏,𝑐 𝑃,𝑎,𝑏,𝑐 𝜑,𝑎,𝑏,𝑐 Allowed substitution hints: 𝐿(𝑎,𝑏,𝑐)

**Proof of Theorem tglowdim2l**
Step	Hyp	Ref	Expression
1		tglineintmo.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		eqid 2778	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
3		tglineintmo.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		tglineintmo.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		tglowdim2l.1	. . 3 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
6	1, 2, 3, 4, 5	axtglowdim2 25838	. 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ¬ (𝑐 ∈ (𝑎𝐼𝑏) ∨ 𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑎𝐼𝑐)))
7		tglineintmo.l	. . . . . . 7 ⊢ 𝐿 = (LineG‘𝐺)
8	4	ad3antrrr 720	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) ∧ 𝑐 ∈ 𝑃) → 𝐺 ∈ TarskiG)
9		simpllr 766	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) ∧ 𝑐 ∈ 𝑃) → 𝑎 ∈ 𝑃)
10		simplr 759	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) ∧ 𝑐 ∈ 𝑃) → 𝑏 ∈ 𝑃)
11		simpr 479	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) ∧ 𝑐 ∈ 𝑃) → 𝑐 ∈ 𝑃)
12	1, 7, 3, 8, 9, 10, 11	tgcolg 25922	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) ∧ 𝑐 ∈ 𝑃) → ((𝑐 ∈ (𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ (𝑐 ∈ (𝑎𝐼𝑏) ∨ 𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑎𝐼𝑐))))
13	12	notbid 310	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) ∧ 𝑐 ∈ 𝑃) → (¬ (𝑐 ∈ (𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ ¬ (𝑐 ∈ (𝑎𝐼𝑏) ∨ 𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑎𝐼𝑐))))
14	13	rexbidva 3234	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) → (∃𝑐 ∈ 𝑃 ¬ (𝑐 ∈ (𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ ∃𝑐 ∈ 𝑃 ¬ (𝑐 ∈ (𝑎𝐼𝑏) ∨ 𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑎𝐼𝑐))))
15	14	rexbidva 3234	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → (∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ¬ (𝑐 ∈ (𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ¬ (𝑐 ∈ (𝑎𝐼𝑏) ∨ 𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑎𝐼𝑐))))
16	15	rexbidva 3234	. 2 ⊢ (𝜑 → (∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ¬ (𝑐 ∈ (𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ¬ (𝑐 ∈ (𝑎𝐼𝑏) ∨ 𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑎𝐼𝑐))))
17	6, 16	mpbird 249	1 ⊢ (𝜑 → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ¬ (𝑐 ∈ (𝑎𝐿𝑏) ∨ 𝑎 = 𝑏))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∨ wo 836 ∨ w3o 1070 = wceq 1601 ∈ wcel 2107 ∃wrex 3091 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 2c2 11435 Basecbs 16266 distcds 16358 TarskiGcstrkg 25798 Dim_TarskiG≥cstrkgld 25802 Itvcitv 25804 LineGclng 25805

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-n0 11648 df-z 11734 df-uz 11998 df-fz 12649 df-fzo 12790 df-trkgc 25816 df-trkgcb 25818 df-trkgld 25820 df-trkg 25821

This theorem is referenced by: tglowdim2ln 26019

W3C validator