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Theorem atnle 38175
Description: Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 31616 analog.) (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
atnle.b 𝐡 = (Baseβ€˜πΎ)
atnle.l ≀ = (leβ€˜πΎ)
atnle.m ∧ = (meetβ€˜πΎ)
atnle.z 0 = (0.β€˜πΎ)
atnle.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
atnle ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (Β¬ 𝑃 ≀ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 ))
Proof of Theorem atnle
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpl1 1191 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) β‰  0 ) β†’ 𝐾 ∈ AtLat)
2 atllat 38158 . . . . . . . . 9 (𝐾 ∈ AtLat β†’ 𝐾 ∈ Lat)
323ad2ant1 1133 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ 𝐾 ∈ Lat)
4 atnle.b . . . . . . . . . 10 𝐡 = (Baseβ€˜πΎ)
5 atnle.a . . . . . . . . . 10 𝐴 = (Atomsβ€˜πΎ)
64, 5atbase 38147 . . . . . . . . 9 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
763ad2ant2 1134 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ 𝑃 ∈ 𝐡)
8 simp3 1138 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
9 atnle.m . . . . . . . . 9 ∧ = (meetβ€˜πΎ)
104, 9latmcl 18389 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (𝑃 ∧ 𝑋) ∈ 𝐡)
113, 7, 8, 10syl3anc 1371 . . . . . . 7 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (𝑃 ∧ 𝑋) ∈ 𝐡)
1211adantr 481 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) β‰  0 ) β†’ (𝑃 ∧ 𝑋) ∈ 𝐡)
13 simpr 485 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) β‰  0 ) β†’ (𝑃 ∧ 𝑋) β‰  0 )
14 atnle.l . . . . . . 7 ≀ = (leβ€˜πΎ)
15 atnle.z . . . . . . 7 0 = (0.β€˜πΎ)
164, 14, 15, 5atlex 38174 . . . . . 6 ((𝐾 ∈ AtLat ∧ (𝑃 ∧ 𝑋) ∈ 𝐡 ∧ (𝑃 ∧ 𝑋) β‰  0 ) β†’ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ (𝑃 ∧ 𝑋))
171, 12, 13, 16syl3anc 1371 . . . . 5 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) β‰  0 ) β†’ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ (𝑃 ∧ 𝑋))
18 simpl1 1191 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ 𝐾 ∈ AtLat)
1918, 2syl 17 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ 𝐾 ∈ Lat)
204, 5atbase 38147 . . . . . . . . . 10 (𝑦 ∈ 𝐴 β†’ 𝑦 ∈ 𝐡)
2120adantl 482 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ 𝑦 ∈ 𝐡)
22 simpl2 1192 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ 𝑃 ∈ 𝐴)
2322, 6syl 17 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ 𝑃 ∈ 𝐡)
24 simpl3 1193 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ 𝑋 ∈ 𝐡)
254, 14, 9latlem12 18415 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑦 ∈ 𝐡 ∧ 𝑃 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡)) β†’ ((𝑦 ≀ 𝑃 ∧ 𝑦 ≀ 𝑋) ↔ 𝑦 ≀ (𝑃 ∧ 𝑋)))
2619, 21, 23, 24, 25syl13anc 1372 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ ((𝑦 ≀ 𝑃 ∧ 𝑦 ≀ 𝑋) ↔ 𝑦 ≀ (𝑃 ∧ 𝑋)))
27 simpr 485 . . . . . . . . . . 11 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ 𝑦 ∈ 𝐴)
2814, 5atcmp 38169 . . . . . . . . . . 11 ((𝐾 ∈ AtLat ∧ 𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) β†’ (𝑦 ≀ 𝑃 ↔ 𝑦 = 𝑃))
2918, 27, 22, 28syl3anc 1371 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ (𝑦 ≀ 𝑃 ↔ 𝑦 = 𝑃))
30 breq1 5150 . . . . . . . . . . 11 (𝑦 = 𝑃 β†’ (𝑦 ≀ 𝑋 ↔ 𝑃 ≀ 𝑋))
3130biimpd 228 . . . . . . . . . 10 (𝑦 = 𝑃 β†’ (𝑦 ≀ 𝑋 β†’ 𝑃 ≀ 𝑋))
3229, 31syl6bi 252 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ (𝑦 ≀ 𝑃 β†’ (𝑦 ≀ 𝑋 β†’ 𝑃 ≀ 𝑋)))
3332impd 411 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ ((𝑦 ≀ 𝑃 ∧ 𝑦 ≀ 𝑋) β†’ 𝑃 ≀ 𝑋))
3426, 33sylbird 259 . . . . . . 7 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ (𝑦 ≀ (𝑃 ∧ 𝑋) β†’ 𝑃 ≀ 𝑋))
3534adantlr 713 . . . . . 6 ((((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) β‰  0 ) ∧ 𝑦 ∈ 𝐴) β†’ (𝑦 ≀ (𝑃 ∧ 𝑋) β†’ 𝑃 ≀ 𝑋))
3635rexlimdva 3155 . . . . 5 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) β‰  0 ) β†’ (βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ (𝑃 ∧ 𝑋) β†’ 𝑃 ≀ 𝑋))
3717, 36mpd 15 . . . 4 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) β‰  0 ) β†’ 𝑃 ≀ 𝑋)
3837ex 413 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ ((𝑃 ∧ 𝑋) β‰  0 β†’ 𝑃 ≀ 𝑋))
3938necon1bd 2958 . 2 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (Β¬ 𝑃 ≀ 𝑋 β†’ (𝑃 ∧ 𝑋) = 0 ))
4015, 5atn0 38166 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ 𝑃 β‰  0 )
41403adant3 1132 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ 𝑃 β‰  0 )
424, 14, 9latleeqm1 18416 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (𝑃 ≀ 𝑋 ↔ (𝑃 ∧ 𝑋) = 𝑃))
433, 7, 8, 42syl3anc 1371 . . . . . . 7 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (𝑃 ≀ 𝑋 ↔ (𝑃 ∧ 𝑋) = 𝑃))
4443adantr 481 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) = 0 ) β†’ (𝑃 ≀ 𝑋 ↔ (𝑃 ∧ 𝑋) = 𝑃))
45 eqeq1 2736 . . . . . . . 8 ((𝑃 ∧ 𝑋) = 𝑃 β†’ ((𝑃 ∧ 𝑋) = 0 ↔ 𝑃 = 0 ))
4645biimpcd 248 . . . . . . 7 ((𝑃 ∧ 𝑋) = 0 β†’ ((𝑃 ∧ 𝑋) = 𝑃 β†’ 𝑃 = 0 ))
4746adantl 482 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) = 0 ) β†’ ((𝑃 ∧ 𝑋) = 𝑃 β†’ 𝑃 = 0 ))
4844, 47sylbid 239 . . . . 5 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) = 0 ) β†’ (𝑃 ≀ 𝑋 β†’ 𝑃 = 0 ))
4948necon3ad 2953 . . . 4 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) = 0 ) β†’ (𝑃 β‰  0 β†’ Β¬ 𝑃 ≀ 𝑋))
5049ex 413 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ ((𝑃 ∧ 𝑋) = 0 β†’ (𝑃 β‰  0 β†’ Β¬ 𝑃 ≀ 𝑋)))
5141, 50mpid 44 . 2 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ ((𝑃 ∧ 𝑋) = 0 β†’ Β¬ 𝑃 ≀ 𝑋))
5239, 51impbid 211 1 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (Β¬ 𝑃 ≀ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  meetcmee 18261  0.cp0 18372  Latclat 18380  Atomscatm 38121  AtLatcal 38122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-lat 18381  df-covers 38124  df-ats 38125  df-atl 38156
This theorem is referenced by:  atnem0  38176  iscvlat2N  38182  cvlexch3  38190  cvlexch4N  38191  cvlcvrp  38198  intnatN  38266  cvrat4  38302  dalem24  38556  cdlema2N  38651  llnexchb2lem  38727  lhpmat  38889  cdleme15b  39134  cdlemednpq  39158  cdleme20zN  39160  cdleme22cN  39201  dihmeetlem7N  40169  dihmeetlem17N  40182
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