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Theorem atnle 33622
 Description: Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 28619 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 1057 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → 𝐾 ∈ AtLat)
2 atllat 33605 . . . . . . . . 9 (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
323ad2ant1 1075 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝐾 ∈ Lat)
4 atnle.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
5 atnle.a . . . . . . . . . 10 𝐴 = (Atoms‘𝐾)
64, 5atbase 33594 . . . . . . . . 9 (𝑃𝐴𝑃𝐵)
763ad2ant2 1076 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝑃𝐵)
8 simp3 1056 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝑋𝐵)
9 atnle.m . . . . . . . . 9 = (meet‘𝐾)
104, 9latmcl 16875 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋) ∈ 𝐵)
113, 7, 8, 10syl3anc 1318 . . . . . . 7 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (𝑃 𝑋) ∈ 𝐵)
1211adantr 480 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → (𝑃 𝑋) ∈ 𝐵)
13 simpr 476 . . . . . 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 33621 . . . . . 6 ((𝐾 ∈ AtLat ∧ (𝑃 𝑋) ∈ 𝐵 ∧ (𝑃 𝑋) ≠ 0 ) → ∃𝑦𝐴 𝑦 (𝑃 𝑋))
171, 12, 13, 16syl3anc 1318 . . . . 5 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → ∃𝑦𝐴 𝑦 (𝑃 𝑋))
18 simpl1 1057 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝐾 ∈ AtLat)
1918, 2syl 17 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝐾 ∈ Lat)
204, 5atbase 33594 . . . . . . . . . 10 (𝑦𝐴𝑦𝐵)
2120adantl 481 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑦𝐵)
22 simpl2 1058 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑃𝐴)
2322, 6syl 17 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑃𝐵)
24 simpl3 1059 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑋𝐵)
254, 14, 9latlem12 16901 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑦𝐵𝑃𝐵𝑋𝐵)) → ((𝑦 𝑃𝑦 𝑋) ↔ 𝑦 (𝑃 𝑋)))
2619, 21, 23, 24, 25syl13anc 1320 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → ((𝑦 𝑃𝑦 𝑋) ↔ 𝑦 (𝑃 𝑋)))
27 simpr 476 . . . . . . . . . . 11 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑦𝐴)
2814, 5atcmp 33616 . . . . . . . . . . 11 ((𝐾 ∈ AtLat ∧ 𝑦𝐴𝑃𝐴) → (𝑦 𝑃𝑦 = 𝑃))
2918, 27, 22, 28syl3anc 1318 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → (𝑦 𝑃𝑦 = 𝑃))
30 breq1 4586 . . . . . . . . . . 11 (𝑦 = 𝑃 → (𝑦 𝑋𝑃 𝑋))
3130biimpd 218 . . . . . . . . . 10 (𝑦 = 𝑃 → (𝑦 𝑋𝑃 𝑋))
3229, 31syl6bi 242 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → (𝑦 𝑃 → (𝑦 𝑋𝑃 𝑋)))
3332impd 446 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → ((𝑦 𝑃𝑦 𝑋) → 𝑃 𝑋))
3426, 33sylbird 249 . . . . . . 7 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → (𝑦 (𝑃 𝑋) → 𝑃 𝑋))
3534adantlr 747 . . . . . 6 ((((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) ∧ 𝑦𝐴) → (𝑦 (𝑃 𝑋) → 𝑃 𝑋))
3635rexlimdva 3013 . . . . 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 449 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → ((𝑃 𝑋) ≠ 0𝑃 𝑋))
3938necon1bd 2800 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (¬ 𝑃 𝑋 → (𝑃 𝑋) = 0 ))
4015, 5atn0 33613 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝑃0 )
41403adant3 1074 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝑃0 )
424, 14, 9latleeqm1 16902 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋 ↔ (𝑃 𝑋) = 𝑃))
433, 7, 8, 42syl3anc 1318 . . . . . . 7 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (𝑃 𝑋 ↔ (𝑃 𝑋) = 𝑃))
4443adantr 480 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 𝑋 ↔ (𝑃 𝑋) = 𝑃))
45 eqeq1 2614 . . . . . . . 8 ((𝑃 𝑋) = 𝑃 → ((𝑃 𝑋) = 0𝑃 = 0 ))
4645biimpcd 238 . . . . . . 7 ((𝑃 𝑋) = 0 → ((𝑃 𝑋) = 𝑃𝑃 = 0 ))
4746adantl 481 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → ((𝑃 𝑋) = 𝑃𝑃 = 0 ))
4844, 47sylbid 229 . . . . 5 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 𝑋𝑃 = 0 ))
4948necon3ad 2795 . . . 4 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃0 → ¬ 𝑃 𝑋))
5049ex 449 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → ((𝑃 𝑋) = 0 → (𝑃0 → ¬ 𝑃 𝑋)))
5141, 50mpid 43 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → ((𝑃 𝑋) = 0 → ¬ 𝑃 𝑋))
5239, 51impbid 201 1 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = 0 ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ 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  meetcmee 16768  0.cp0 16860  Latclat 16868  Atomscatm 33568  AtLatcal 33569 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-covers 33571  df-ats 33572  df-atl 33603 This theorem is referenced by:  atnem0  33623  iscvlat2N  33629  cvlexch3  33637  cvlexch4N  33638  cvlcvrp  33645  intnatN  33711  cvrat4  33747  dalem24  34001  cdlema2N  34096  llnexchb2lem  34172  lhpmat  34334  ltrnmwOLD  34456  cdleme15b  34580  cdlemednpq  34604  cdleme20zN  34606  cdleme20yOLD  34608  cdleme22cN  34648  dihmeetlem7N  35617  dihmeetlem17N  35630
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