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

Theorem atnle 35392
Description: Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 29790 analog.) (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
atnle.b 𝐵 = (Base‘𝐾)
atnle.l = (le‘𝐾)
atnle.m = (meet‘𝐾)
atnle.z 0 = (0.‘𝐾)
atnle.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
atnle ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = 0 ))

Proof of Theorem atnle
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpl1 1248 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → 𝐾 ∈ AtLat)
2 atllat 35375 . . . . . . . . 9 (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
323ad2ant1 1169 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝐾 ∈ Lat)
4 atnle.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
5 atnle.a . . . . . . . . . 10 𝐴 = (Atoms‘𝐾)
64, 5atbase 35364 . . . . . . . . 9 (𝑃𝐴𝑃𝐵)
763ad2ant2 1170 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝑃𝐵)
8 simp3 1174 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝑋𝐵)
9 atnle.m . . . . . . . . 9 = (meet‘𝐾)
104, 9latmcl 17405 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋) ∈ 𝐵)
113, 7, 8, 10syl3anc 1496 . . . . . . 7 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (𝑃 𝑋) ∈ 𝐵)
1211adantr 474 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → (𝑃 𝑋) ∈ 𝐵)
13 simpr 479 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → (𝑃 𝑋) ≠ 0 )
14 atnle.l . . . . . . 7 = (le‘𝐾)
15 atnle.z . . . . . . 7 0 = (0.‘𝐾)
164, 14, 15, 5atlex 35391 . . . . . 6 ((𝐾 ∈ AtLat ∧ (𝑃 𝑋) ∈ 𝐵 ∧ (𝑃 𝑋) ≠ 0 ) → ∃𝑦𝐴 𝑦 (𝑃 𝑋))
171, 12, 13, 16syl3anc 1496 . . . . 5 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → ∃𝑦𝐴 𝑦 (𝑃 𝑋))
18 simpl1 1248 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝐾 ∈ AtLat)
1918, 2syl 17 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝐾 ∈ Lat)
204, 5atbase 35364 . . . . . . . . . 10 (𝑦𝐴𝑦𝐵)
2120adantl 475 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑦𝐵)
22 simpl2 1250 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑃𝐴)
2322, 6syl 17 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑃𝐵)
24 simpl3 1252 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑋𝐵)
254, 14, 9latlem12 17431 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑦𝐵𝑃𝐵𝑋𝐵)) → ((𝑦 𝑃𝑦 𝑋) ↔ 𝑦 (𝑃 𝑋)))
2619, 21, 23, 24, 25syl13anc 1497 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → ((𝑦 𝑃𝑦 𝑋) ↔ 𝑦 (𝑃 𝑋)))
27 simpr 479 . . . . . . . . . . 11 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑦𝐴)
2814, 5atcmp 35386 . . . . . . . . . . 11 ((𝐾 ∈ AtLat ∧ 𝑦𝐴𝑃𝐴) → (𝑦 𝑃𝑦 = 𝑃))
2918, 27, 22, 28syl3anc 1496 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → (𝑦 𝑃𝑦 = 𝑃))
30 breq1 4876 . . . . . . . . . . 11 (𝑦 = 𝑃 → (𝑦 𝑋𝑃 𝑋))
3130biimpd 221 . . . . . . . . . 10 (𝑦 = 𝑃 → (𝑦 𝑋𝑃 𝑋))
3229, 31syl6bi 245 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → (𝑦 𝑃 → (𝑦 𝑋𝑃 𝑋)))
3332impd 400 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → ((𝑦 𝑃𝑦 𝑋) → 𝑃 𝑋))
3426, 33sylbird 252 . . . . . . 7 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → (𝑦 (𝑃 𝑋) → 𝑃 𝑋))
3534adantlr 708 . . . . . 6 ((((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) ∧ 𝑦𝐴) → (𝑦 (𝑃 𝑋) → 𝑃 𝑋))
3635rexlimdva 3240 . . . . 5 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → (∃𝑦𝐴 𝑦 (𝑃 𝑋) → 𝑃 𝑋))
3717, 36mpd 15 . . . 4 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → 𝑃 𝑋)
3837ex 403 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → ((𝑃 𝑋) ≠ 0𝑃 𝑋))
3938necon1bd 3017 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (¬ 𝑃 𝑋 → (𝑃 𝑋) = 0 ))
4015, 5atn0 35383 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝑃0 )
41403adant3 1168 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝑃0 )
424, 14, 9latleeqm1 17432 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋 ↔ (𝑃 𝑋) = 𝑃))
433, 7, 8, 42syl3anc 1496 . . . . . . 7 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (𝑃 𝑋 ↔ (𝑃 𝑋) = 𝑃))
4443adantr 474 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 𝑋 ↔ (𝑃 𝑋) = 𝑃))
45 eqeq1 2829 . . . . . . . 8 ((𝑃 𝑋) = 𝑃 → ((𝑃 𝑋) = 0𝑃 = 0 ))
4645biimpcd 241 . . . . . . 7 ((𝑃 𝑋) = 0 → ((𝑃 𝑋) = 𝑃𝑃 = 0 ))
4746adantl 475 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → ((𝑃 𝑋) = 𝑃𝑃 = 0 ))
4844, 47sylbid 232 . . . . 5 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 𝑋𝑃 = 0 ))
4948necon3ad 3012 . . . 4 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃0 → ¬ 𝑃 𝑋))
5049ex 403 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → ((𝑃 𝑋) = 0 → (𝑃0 → ¬ 𝑃 𝑋)))
5141, 50mpid 44 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → ((𝑃 𝑋) = 0 → ¬ 𝑃 𝑋))
5239, 51impbid 204 1 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  wne 2999  wrex 3118   class class class wbr 4873  cfv 6123  (class class class)co 6905  Basecbs 16222  lecple 16312  meetcmee 17298  0.cp0 17390  Latclat 17398  Atomscatm 35338  AtLatcal 35339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-proset 17281  df-poset 17299  df-plt 17311  df-lub 17327  df-glb 17328  df-join 17329  df-meet 17330  df-p0 17392  df-lat 17399  df-covers 35341  df-ats 35342  df-atl 35373
This theorem is referenced by:  atnem0  35393  iscvlat2N  35399  cvlexch3  35407  cvlexch4N  35408  cvlcvrp  35415  intnatN  35482  cvrat4  35518  dalem24  35772  cdlema2N  35867  llnexchb2lem  35943  lhpmat  36105  cdleme15b  36350  cdlemednpq  36374  cdleme20zN  36376  cdleme22cN  36417  dihmeetlem7N  37385  dihmeetlem17N  37398
  Copyright terms: Public domain W3C validator