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Theorem atnle 38651
Description: Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 32062 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 1190 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → 𝐾 ∈ AtLat)
2 atllat 38634 . . . . . . . . 9 (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
323ad2ant1 1132 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝐾 ∈ Lat)
4 atnle.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
5 atnle.a . . . . . . . . . 10 𝐴 = (Atoms‘𝐾)
64, 5atbase 38623 . . . . . . . . 9 (𝑃𝐴𝑃𝐵)
763ad2ant2 1133 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝑃𝐵)
8 simp3 1137 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝑋𝐵)
9 atnle.m . . . . . . . . 9 = (meet‘𝐾)
104, 9latmcl 18403 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋) ∈ 𝐵)
113, 7, 8, 10syl3anc 1370 . . . . . . 7 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (𝑃 𝑋) ∈ 𝐵)
1211adantr 480 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → (𝑃 𝑋) ∈ 𝐵)
13 simpr 484 . . . . . 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 38650 . . . . . 6 ((𝐾 ∈ AtLat ∧ (𝑃 𝑋) ∈ 𝐵 ∧ (𝑃 𝑋) ≠ 0 ) → ∃𝑦𝐴 𝑦 (𝑃 𝑋))
171, 12, 13, 16syl3anc 1370 . . . . 5 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → ∃𝑦𝐴 𝑦 (𝑃 𝑋))
18 simpl1 1190 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝐾 ∈ AtLat)
1918, 2syl 17 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝐾 ∈ Lat)
204, 5atbase 38623 . . . . . . . . . 10 (𝑦𝐴𝑦𝐵)
2120adantl 481 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑦𝐵)
22 simpl2 1191 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑃𝐴)
2322, 6syl 17 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑃𝐵)
24 simpl3 1192 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑋𝐵)
254, 14, 9latlem12 18429 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑦𝐵𝑃𝐵𝑋𝐵)) → ((𝑦 𝑃𝑦 𝑋) ↔ 𝑦 (𝑃 𝑋)))
2619, 21, 23, 24, 25syl13anc 1371 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → ((𝑦 𝑃𝑦 𝑋) ↔ 𝑦 (𝑃 𝑋)))
27 simpr 484 . . . . . . . . . . 11 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑦𝐴)
2814, 5atcmp 38645 . . . . . . . . . . 11 ((𝐾 ∈ AtLat ∧ 𝑦𝐴𝑃𝐴) → (𝑦 𝑃𝑦 = 𝑃))
2918, 27, 22, 28syl3anc 1370 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → (𝑦 𝑃𝑦 = 𝑃))
30 breq1 5151 . . . . . . . . . . 11 (𝑦 = 𝑃 → (𝑦 𝑋𝑃 𝑋))
3130biimpd 228 . . . . . . . . . 10 (𝑦 = 𝑃 → (𝑦 𝑋𝑃 𝑋))
3229, 31syl6bi 253 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → (𝑦 𝑃 → (𝑦 𝑋𝑃 𝑋)))
3332impd 410 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → ((𝑦 𝑃𝑦 𝑋) → 𝑃 𝑋))
3426, 33sylbird 260 . . . . . . 7 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → (𝑦 (𝑃 𝑋) → 𝑃 𝑋))
3534adantlr 712 . . . . . 6 ((((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) ∧ 𝑦𝐴) → (𝑦 (𝑃 𝑋) → 𝑃 𝑋))
3635rexlimdva 3154 . . . . 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 412 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → ((𝑃 𝑋) ≠ 0𝑃 𝑋))
3938necon1bd 2957 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (¬ 𝑃 𝑋 → (𝑃 𝑋) = 0 ))
4015, 5atn0 38642 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝑃0 )
41403adant3 1131 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝑃0 )
424, 14, 9latleeqm1 18430 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋 ↔ (𝑃 𝑋) = 𝑃))
433, 7, 8, 42syl3anc 1370 . . . . . . 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 2735 . . . . . . . 8 ((𝑃 𝑋) = 𝑃 → ((𝑃 𝑋) = 0𝑃 = 0 ))
4645biimpcd 248 . . . . . . 7 ((𝑃 𝑋) = 0 → ((𝑃 𝑋) = 𝑃𝑃 = 0 ))
4746adantl 481 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → ((𝑃 𝑋) = 𝑃𝑃 = 0 ))
4844, 47sylbid 239 . . . . 5 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 𝑋𝑃 = 0 ))
4948necon3ad 2952 . . . 4 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃0 → ¬ 𝑃 𝑋))
5049ex 412 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → ((𝑃 𝑋) = 0 → (𝑃0 → ¬ 𝑃 𝑋)))
5141, 50mpid 44 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → ((𝑃 𝑋) = 0 → ¬ 𝑃 𝑋))
5239, 51impbid 211 1 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wne 2939  wrex 3069   class class class wbr 5148  cfv 6543  (class class class)co 7412  Basecbs 17151  lecple 17211  meetcmee 18275  0.cp0 18386  Latclat 18394  Atomscatm 38597  AtLatcal 38598
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-proset 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-lat 18395  df-covers 38600  df-ats 38601  df-atl 38632
This theorem is referenced by:  atnem0  38652  iscvlat2N  38658  cvlexch3  38666  cvlexch4N  38667  cvlcvrp  38674  intnatN  38742  cvrat4  38778  dalem24  39032  cdlema2N  39127  llnexchb2lem  39203  lhpmat  39365  cdleme15b  39610  cdlemednpq  39634  cdleme20zN  39636  cdleme22cN  39677  dihmeetlem7N  40645  dihmeetlem17N  40658
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