Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lnatexN Structured version   Visualization version   GIF version

Theorem lnatexN 34083
 Description: There is an atom in a line different from any other. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnatex.b 𝐵 = (Base‘𝐾)
lnatex.l = (le‘𝐾)
lnatex.a 𝐴 = (Atoms‘𝐾)
lnatex.n 𝑁 = (Lines‘𝐾)
lnatex.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
lnatexN ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))
Distinct variable groups:   𝐴,𝑞   ,𝑞   𝑃,𝑞   𝑋,𝑞
Allowed substitution hints:   𝐵(𝑞)   𝐾(𝑞)   𝑀(𝑞)   𝑁(𝑞)

Proof of Theorem lnatexN
Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnatex.b . . . 4 𝐵 = (Base‘𝐾)
2 eqid 2610 . . . 4 (join‘𝐾) = (join‘𝐾)
3 lnatex.a . . . 4 𝐴 = (Atoms‘𝐾)
4 lnatex.n . . . 4 𝑁 = (Lines‘𝐾)
5 lnatex.m . . . 4 𝑀 = (pmap‘𝐾)
61, 2, 3, 4, 5isline3 34080 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑟𝐴𝑠𝐴 (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))))
76biimp3a 1424 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) → ∃𝑟𝐴𝑠𝐴 (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠)))
8 simpl2r 1108 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑠𝐴)
9 simpl3l 1109 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑟𝑠)
109necomd 2837 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑠𝑟)
11 simpr 476 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑟 = 𝑃)
1210, 11neeqtrd 2851 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑠𝑃)
13 simpl11 1129 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝐾 ∈ HL)
14 simpl2l 1107 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑟𝐴)
15 lnatex.l . . . . . . . . 9 = (le‘𝐾)
1615, 2, 3hlatlej2 33680 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑟𝐴𝑠𝐴) → 𝑠 (𝑟(join‘𝐾)𝑠))
1713, 14, 8, 16syl3anc 1318 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑠 (𝑟(join‘𝐾)𝑠))
18 simpl3r 1110 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑋 = (𝑟(join‘𝐾)𝑠))
1917, 18breqtrrd 4611 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → 𝑠 𝑋)
20 neeq1 2844 . . . . . . . 8 (𝑞 = 𝑠 → (𝑞𝑃𝑠𝑃))
21 breq1 4586 . . . . . . . 8 (𝑞 = 𝑠 → (𝑞 𝑋𝑠 𝑋))
2220, 21anbi12d 743 . . . . . . 7 (𝑞 = 𝑠 → ((𝑞𝑃𝑞 𝑋) ↔ (𝑠𝑃𝑠 𝑋)))
2322rspcev 3282 . . . . . 6 ((𝑠𝐴 ∧ (𝑠𝑃𝑠 𝑋)) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))
248, 12, 19, 23syl12anc 1316 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟 = 𝑃) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))
25 simpl2l 1107 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟𝑃) → 𝑟𝐴)
26 simpr 476 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟𝑃) → 𝑟𝑃)
27 simpl11 1129 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟𝑃) → 𝐾 ∈ HL)
28 simpl2r 1108 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟𝑃) → 𝑠𝐴)
2915, 2, 3hlatlej1 33679 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑟𝐴𝑠𝐴) → 𝑟 (𝑟(join‘𝐾)𝑠))
3027, 25, 28, 29syl3anc 1318 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟𝑃) → 𝑟 (𝑟(join‘𝐾)𝑠))
31 simpl3r 1110 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟𝑃) → 𝑋 = (𝑟(join‘𝐾)𝑠))
3230, 31breqtrrd 4611 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟𝑃) → 𝑟 𝑋)
33 neeq1 2844 . . . . . . . 8 (𝑞 = 𝑟 → (𝑞𝑃𝑟𝑃))
34 breq1 4586 . . . . . . . 8 (𝑞 = 𝑟 → (𝑞 𝑋𝑟 𝑋))
3533, 34anbi12d 743 . . . . . . 7 (𝑞 = 𝑟 → ((𝑞𝑃𝑞 𝑋) ↔ (𝑟𝑃𝑟 𝑋)))
3635rspcev 3282 . . . . . 6 ((𝑟𝐴 ∧ (𝑟𝑃𝑟 𝑋)) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))
3725, 26, 32, 36syl12anc 1316 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) ∧ 𝑟𝑃) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))
3824, 37pm2.61dane 2869 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑟𝐴𝑠𝐴) ∧ (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠))) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))
39383exp 1256 . . 3 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) → ((𝑟𝐴𝑠𝐴) → ((𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠)) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))))
4039rexlimdvv 3019 . 2 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) → (∃𝑟𝐴𝑠𝐴 (𝑟𝑠𝑋 = (𝑟(join‘𝐾)𝑠)) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋)))
417, 40mpd 15 1 ((𝐾 ∈ HL ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  Atomscatm 33568  HLchlt 33655  Linesclines 33798  pmapcpmap 33801 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-lines 33805  df-pmap 33808 This theorem is referenced by:  lnjatN  34084
 Copyright terms: Public domain W3C validator