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 27300

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 2737	. . 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 27120	. 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ¬ (𝑐 ∈ (𝑎𝐼𝑏) ∨ 𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑎𝐼𝑐)))
7		tglineintmo.l	. . . . . . 7 ⊢ 𝐿 = (LineG‘𝐺)
8	4	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) ∧ 𝑐 ∈ 𝑃) → 𝐺 ∈ TarskiG)
9		simpllr 774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) ∧ 𝑐 ∈ 𝑃) → 𝑎 ∈ 𝑃)
10		simplr 767	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) ∧ 𝑐 ∈ 𝑃) → 𝑏 ∈ 𝑃)
11		simpr 486	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) ∧ 𝑐 ∈ 𝑃) → 𝑐 ∈ 𝑃)
12	1, 7, 3, 8, 9, 10, 11	tgcolg 27204	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) ∧ 𝑐 ∈ 𝑃) → ((𝑐 ∈ (𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ (𝑐 ∈ (𝑎𝐼𝑏) ∨ 𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑎𝐼𝑐))))
13	12	notbid 318	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) ∧ 𝑐 ∈ 𝑃) → (¬ (𝑐 ∈ (𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ ¬ (𝑐 ∈ (𝑎𝐼𝑏) ∨ 𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑎𝐼𝑐))))
14	13	rexbidva 3170	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) → (∃𝑐 ∈ 𝑃 ¬ (𝑐 ∈ (𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ ∃𝑐 ∈ 𝑃 ¬ (𝑐 ∈ (𝑎𝐼𝑏) ∨ 𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑎𝐼𝑐))))
15	14	rexbidva 3170	. . 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 3170	. 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 257	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 397 ∨ wo 845 ∨ w3o 1086 = wceq 1541 ∈ wcel 2106 ∃wrex 3071 class class class wbr 5097 ‘cfv 6484 (class class class)co 7342 2c2 12134 Basecbs 17010 distcds 17069 TarskiGcstrkg 27077 Dim_TarskiG≥cstrkgld 27081 Itvcitv 27083 LineGclng 27084

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5234 ax-sep 5248 ax-nul 5255 ax-pow 5313 ax-pr 5377 ax-un 7655 ax-cnex 11033 ax-resscn 11034 ax-1cn 11035 ax-icn 11036 ax-addcl 11037 ax-addrcl 11038 ax-mulcl 11039 ax-mulrcl 11040 ax-mulcom 11041 ax-addass 11042 ax-mulass 11043 ax-distr 11044 ax-i2m1 11045 ax-1ne0 11046 ax-1rid 11047 ax-rnegex 11048 ax-rrecex 11049 ax-cnre 11050 ax-pre-lttri 11051 ax-pre-lttrn 11052 ax-pre-ltadd 11053 ax-pre-mulgt0 11054

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3351 df-rab 3405 df-v 3444 df-sbc 3732 df-csb 3848 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3921 df-nul 4275 df-if 4479 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4858 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5181 df-tr 5215 df-id 5523 df-eprel 5529 df-po 5537 df-so 5538 df-fr 5580 df-we 5582 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6243 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6436 df-fun 6486 df-fn 6487 df-f 6488 df-f1 6489 df-fo 6490 df-f1o 6491 df-fv 6492 df-riota 7298 df-ov 7345 df-oprab 7346 df-mpo 7347 df-om 7786 df-1st 7904 df-2nd 7905 df-frecs 8172 df-wrecs 8203 df-recs 8277 df-rdg 8316 df-er 8574 df-en 8810 df-dom 8811 df-sdom 8812 df-pnf 11117 df-mnf 11118 df-xr 11119 df-ltxr 11120 df-le 11121 df-sub 11313 df-neg 11314 df-nn 12080 df-2 12142 df-n0 12340 df-z 12426 df-uz 12689 df-fz 13346 df-fzo 13489 df-trkgc 27098 df-trkgcb 27100 df-trkgld 27102 df-trkg 27103

This theorem is referenced by: tglowdim2ln 27301

W3C validator