Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isline4N Structured version   Visualization version   GIF version

Theorem isline4N 39290
Description: Definition of line in terms of original lattice elements. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
isline4.b 𝐡 = (Baseβ€˜πΎ)
isline4.c 𝐢 = ( β‹– β€˜πΎ)
isline4.a 𝐴 = (Atomsβ€˜πΎ)
isline4.n 𝑁 = (Linesβ€˜πΎ)
isline4.m 𝑀 = (pmapβ€˜πΎ)
Assertion
Ref Expression
isline4N ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ ((π‘€β€˜π‘‹) ∈ 𝑁 ↔ βˆƒπ‘ ∈ 𝐴 𝑝𝐢𝑋))
Distinct variable groups:   𝐴,𝑝   𝐡,𝑝   𝐾,𝑝   𝑀,𝑝   𝑋,𝑝
Allowed substitution hints:   𝐢(𝑝)   𝑁(𝑝)

Proof of Theorem isline4N
Dummy variable π‘ž is distinct from all other variables.
StepHypRef Expression
1 isline4.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 eqid 2728 . . 3 (joinβ€˜πΎ) = (joinβ€˜πΎ)
3 isline4.a . . 3 𝐴 = (Atomsβ€˜πΎ)
4 isline4.n . . 3 𝑁 = (Linesβ€˜πΎ)
5 isline4.m . . 3 𝑀 = (pmapβ€˜πΎ)
61, 2, 3, 4, 5isline3 39289 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ ((π‘€β€˜π‘‹) ∈ 𝑁 ↔ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž))))
7 simpll 765 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) ∧ 𝑝 ∈ 𝐴) β†’ 𝐾 ∈ HL)
81, 3atbase 38801 . . . . . 6 (𝑝 ∈ 𝐴 β†’ 𝑝 ∈ 𝐡)
98adantl 480 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) ∧ 𝑝 ∈ 𝐴) β†’ 𝑝 ∈ 𝐡)
10 simplr 767 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) ∧ 𝑝 ∈ 𝐴) β†’ 𝑋 ∈ 𝐡)
11 eqid 2728 . . . . . 6 (leβ€˜πΎ) = (leβ€˜πΎ)
12 isline4.c . . . . . 6 𝐢 = ( β‹– β€˜πΎ)
131, 11, 2, 12, 3cvrval3 38926 . . . . 5 ((𝐾 ∈ HL ∧ 𝑝 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (𝑝𝐢𝑋 ↔ βˆƒπ‘ž ∈ 𝐴 (Β¬ π‘ž(leβ€˜πΎ)𝑝 ∧ (𝑝(joinβ€˜πΎ)π‘ž) = 𝑋)))
147, 9, 10, 13syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) ∧ 𝑝 ∈ 𝐴) β†’ (𝑝𝐢𝑋 ↔ βˆƒπ‘ž ∈ 𝐴 (Β¬ π‘ž(leβ€˜πΎ)𝑝 ∧ (𝑝(joinβ€˜πΎ)π‘ž) = 𝑋)))
15 hlatl 38872 . . . . . . . . 9 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
1615ad3antrrr 728 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) ∧ 𝑝 ∈ 𝐴) ∧ π‘ž ∈ 𝐴) β†’ 𝐾 ∈ AtLat)
17 simpr 483 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) ∧ 𝑝 ∈ 𝐴) ∧ π‘ž ∈ 𝐴) β†’ π‘ž ∈ 𝐴)
18 simplr 767 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) ∧ 𝑝 ∈ 𝐴) ∧ π‘ž ∈ 𝐴) β†’ 𝑝 ∈ 𝐴)
1911, 3atncmp 38824 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ π‘ž ∈ 𝐴 ∧ 𝑝 ∈ 𝐴) β†’ (Β¬ π‘ž(leβ€˜πΎ)𝑝 ↔ π‘ž β‰  𝑝))
2016, 17, 18, 19syl3anc 1368 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) ∧ 𝑝 ∈ 𝐴) ∧ π‘ž ∈ 𝐴) β†’ (Β¬ π‘ž(leβ€˜πΎ)𝑝 ↔ π‘ž β‰  𝑝))
21 necom 2991 . . . . . . 7 (π‘ž β‰  𝑝 ↔ 𝑝 β‰  π‘ž)
2220, 21bitrdi 286 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) ∧ 𝑝 ∈ 𝐴) ∧ π‘ž ∈ 𝐴) β†’ (Β¬ π‘ž(leβ€˜πΎ)𝑝 ↔ 𝑝 β‰  π‘ž))
23 eqcom 2735 . . . . . . 7 ((𝑝(joinβ€˜πΎ)π‘ž) = 𝑋 ↔ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž))
2423a1i 11 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) ∧ 𝑝 ∈ 𝐴) ∧ π‘ž ∈ 𝐴) β†’ ((𝑝(joinβ€˜πΎ)π‘ž) = 𝑋 ↔ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž)))
2522, 24anbi12d 630 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) ∧ 𝑝 ∈ 𝐴) ∧ π‘ž ∈ 𝐴) β†’ ((Β¬ π‘ž(leβ€˜πΎ)𝑝 ∧ (𝑝(joinβ€˜πΎ)π‘ž) = 𝑋) ↔ (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž))))
2625rexbidva 3174 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) ∧ 𝑝 ∈ 𝐴) β†’ (βˆƒπ‘ž ∈ 𝐴 (Β¬ π‘ž(leβ€˜πΎ)𝑝 ∧ (𝑝(joinβ€˜πΎ)π‘ž) = 𝑋) ↔ βˆƒπ‘ž ∈ 𝐴 (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž))))
2714, 26bitrd 278 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) ∧ 𝑝 ∈ 𝐴) β†’ (𝑝𝐢𝑋 ↔ βˆƒπ‘ž ∈ 𝐴 (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž))))
2827rexbidva 3174 . 2 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ (βˆƒπ‘ ∈ 𝐴 𝑝𝐢𝑋 ↔ βˆƒπ‘ ∈ 𝐴 βˆƒπ‘ž ∈ 𝐴 (𝑝 β‰  π‘ž ∧ 𝑋 = (𝑝(joinβ€˜πΎ)π‘ž))))
296, 28bitr4d 281 1 ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ ((π‘€β€˜π‘‹) ∈ 𝑁 ↔ βˆƒπ‘ ∈ 𝐴 𝑝𝐢𝑋))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2937  βˆƒwrex 3067   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  Basecbs 17189  lecple 17249  joincjn 18312   β‹– ccvr 38774  Atomscatm 38775  AtLatcal 38776  HLchlt 38862  Linesclines 39007  pmapcpmap 39010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-proset 18296  df-poset 18314  df-plt 18331  df-lub 18347  df-glb 18348  df-join 18349  df-meet 18350  df-p0 18426  df-lat 18433  df-clat 18500  df-oposet 38688  df-ol 38690  df-oml 38691  df-covers 38778  df-ats 38779  df-atl 38810  df-cvlat 38834  df-hlat 38863  df-lines 39014  df-pmap 39017
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator