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Theorem atnle 34119
Description: Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 29105 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 1062 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → 𝐾 ∈ AtLat)
2 atllat 34102 . . . . . . . . 9 (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
323ad2ant1 1080 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝐾 ∈ Lat)
4 atnle.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
5 atnle.a . . . . . . . . . 10 𝐴 = (Atoms‘𝐾)
64, 5atbase 34091 . . . . . . . . 9 (𝑃𝐴𝑃𝐵)
763ad2ant2 1081 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝑃𝐵)
8 simp3 1061 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝑋𝐵)
9 atnle.m . . . . . . . . 9 = (meet‘𝐾)
104, 9latmcl 16984 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋) ∈ 𝐵)
113, 7, 8, 10syl3anc 1323 . . . . . . 7 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (𝑃 𝑋) ∈ 𝐵)
1211adantr 481 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → (𝑃 𝑋) ∈ 𝐵)
13 simpr 477 . . . . . 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 34118 . . . . . 6 ((𝐾 ∈ AtLat ∧ (𝑃 𝑋) ∈ 𝐵 ∧ (𝑃 𝑋) ≠ 0 ) → ∃𝑦𝐴 𝑦 (𝑃 𝑋))
171, 12, 13, 16syl3anc 1323 . . . . 5 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → ∃𝑦𝐴 𝑦 (𝑃 𝑋))
18 simpl1 1062 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝐾 ∈ AtLat)
1918, 2syl 17 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝐾 ∈ Lat)
204, 5atbase 34091 . . . . . . . . . 10 (𝑦𝐴𝑦𝐵)
2120adantl 482 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑦𝐵)
22 simpl2 1063 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑃𝐴)
2322, 6syl 17 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑃𝐵)
24 simpl3 1064 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑋𝐵)
254, 14, 9latlem12 17010 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑦𝐵𝑃𝐵𝑋𝐵)) → ((𝑦 𝑃𝑦 𝑋) ↔ 𝑦 (𝑃 𝑋)))
2619, 21, 23, 24, 25syl13anc 1325 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → ((𝑦 𝑃𝑦 𝑋) ↔ 𝑦 (𝑃 𝑋)))
27 simpr 477 . . . . . . . . . . 11 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑦𝐴)
2814, 5atcmp 34113 . . . . . . . . . . 11 ((𝐾 ∈ AtLat ∧ 𝑦𝐴𝑃𝐴) → (𝑦 𝑃𝑦 = 𝑃))
2918, 27, 22, 28syl3anc 1323 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → (𝑦 𝑃𝑦 = 𝑃))
30 breq1 4621 . . . . . . . . . . 11 (𝑦 = 𝑃 → (𝑦 𝑋𝑃 𝑋))
3130biimpd 219 . . . . . . . . . 10 (𝑦 = 𝑃 → (𝑦 𝑋𝑃 𝑋))
3229, 31syl6bi 243 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → (𝑦 𝑃 → (𝑦 𝑋𝑃 𝑋)))
3332impd 447 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → ((𝑦 𝑃𝑦 𝑋) → 𝑃 𝑋))
3426, 33sylbird 250 . . . . . . 7 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → (𝑦 (𝑃 𝑋) → 𝑃 𝑋))
3534adantlr 750 . . . . . 6 ((((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) ∧ 𝑦𝐴) → (𝑦 (𝑃 𝑋) → 𝑃 𝑋))
3635rexlimdva 3025 . . . . 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 450 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → ((𝑃 𝑋) ≠ 0𝑃 𝑋))
3938necon1bd 2808 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (¬ 𝑃 𝑋 → (𝑃 𝑋) = 0 ))
4015, 5atn0 34110 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝑃0 )
41403adant3 1079 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝑃0 )
424, 14, 9latleeqm1 17011 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋 ↔ (𝑃 𝑋) = 𝑃))
433, 7, 8, 42syl3anc 1323 . . . . . . 7 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (𝑃 𝑋 ↔ (𝑃 𝑋) = 𝑃))
4443adantr 481 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 𝑋 ↔ (𝑃 𝑋) = 𝑃))
45 eqeq1 2625 . . . . . . . 8 ((𝑃 𝑋) = 𝑃 → ((𝑃 𝑋) = 0𝑃 = 0 ))
4645biimpcd 239 . . . . . . 7 ((𝑃 𝑋) = 0 → ((𝑃 𝑋) = 𝑃𝑃 = 0 ))
4746adantl 482 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → ((𝑃 𝑋) = 𝑃𝑃 = 0 ))
4844, 47sylbid 230 . . . . 5 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 𝑋𝑃 = 0 ))
4948necon3ad 2803 . . . 4 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃0 → ¬ 𝑃 𝑋))
5049ex 450 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → ((𝑃 𝑋) = 0 → (𝑃0 → ¬ 𝑃 𝑋)))
5141, 50mpid 44 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → ((𝑃 𝑋) = 0 → ¬ 𝑃 𝑋))
5239, 51impbid 202 1 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908   class class class wbr 4618  cfv 5852  (class class class)co 6610  Basecbs 15792  lecple 15880  meetcmee 16877  0.cp0 16969  Latclat 16977  Atomscatm 34065  AtLatcal 34066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-preset 16860  df-poset 16878  df-plt 16890  df-lub 16906  df-glb 16907  df-join 16908  df-meet 16909  df-p0 16971  df-lat 16978  df-covers 34068  df-ats 34069  df-atl 34100
This theorem is referenced by:  atnem0  34120  iscvlat2N  34126  cvlexch3  34134  cvlexch4N  34135  cvlcvrp  34142  intnatN  34208  cvrat4  34244  dalem24  34498  cdlema2N  34593  llnexchb2lem  34669  lhpmat  34831  ltrnmwOLD  34953  cdleme15b  35077  cdlemednpq  35101  cdleme20zN  35103  cdleme20yOLD  35105  cdleme22cN  35145  dihmeetlem7N  36114  dihmeetlem17N  36127
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