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 28671

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 2734	. . 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 28491	. 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ¬ (𝑐 ∈ (𝑎𝐼𝑏) ∨ 𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑎𝐼𝑐)))
7		tglineintmo.l	. . . . . . 7 ⊢ 𝐿 = (LineG‘𝐺)
8	4	ad3antrrr 730	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) ∧ 𝑐 ∈ 𝑃) → 𝐺 ∈ TarskiG)
9		simpllr 775	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) ∧ 𝑐 ∈ 𝑃) → 𝑎 ∈ 𝑃)
10		simplr 768	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) ∧ 𝑐 ∈ 𝑃) → 𝑏 ∈ 𝑃)
11		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) ∧ 𝑐 ∈ 𝑃) → 𝑐 ∈ 𝑃)
12	1, 7, 3, 8, 9, 10, 11	tgcolg 28575	. . . . . 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 3156	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝑃) → (∃𝑐 ∈ 𝑃 ¬ (𝑐 ∈ (𝑎𝐿𝑏) ∨ 𝑎 = 𝑏) ↔ ∃𝑐 ∈ 𝑃 ¬ (𝑐 ∈ (𝑎𝐼𝑏) ∨ 𝑎 ∈ (𝑐𝐼𝑏) ∨ 𝑏 ∈ (𝑎𝐼𝑐))))
15	14	rexbidva 3156	. . 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 3156	. 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 395 ∨ wo 847 ∨ w3o 1085 = wceq 1541 ∈ wcel 2113 ∃wrex 3058 class class class wbr 5096 ‘cfv 6490 (class class class)co 7356 2c2 12198 Basecbs 17134 distcds 17184 TarskiGcstrkg 28448 Dim_TarskiG≥cstrkgld 28452 Itvcitv 28454 LineGclng 28455

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-n0 12400 df-z 12487 df-uz 12750 df-fz 13422 df-fzo 13569 df-trkgc 28469 df-trkgcb 28471 df-trkgld 28473 df-trkg 28474

This theorem is referenced by: tglowdim2ln 28672

W3C validator