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Theorem atnle 38187
Description: Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 31629 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 1192 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) β‰  0 ) β†’ 𝐾 ∈ AtLat)
2 atllat 38170 . . . . . . . . 9 (𝐾 ∈ AtLat β†’ 𝐾 ∈ Lat)
323ad2ant1 1134 . . . . . . . 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 38159 . . . . . . . . 9 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ 𝐡)
763ad2ant2 1135 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ 𝑃 ∈ 𝐡)
8 simp3 1139 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
9 atnle.m . . . . . . . . 9 ∧ = (meetβ€˜πΎ)
104, 9latmcl 18393 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (𝑃 ∧ 𝑋) ∈ 𝐡)
113, 7, 8, 10syl3anc 1372 . . . . . . 7 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (𝑃 ∧ 𝑋) ∈ 𝐡)
1211adantr 482 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) β‰  0 ) β†’ (𝑃 ∧ 𝑋) ∈ 𝐡)
13 simpr 486 . . . . . 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 38186 . . . . . 6 ((𝐾 ∈ AtLat ∧ (𝑃 ∧ 𝑋) ∈ 𝐡 ∧ (𝑃 ∧ 𝑋) β‰  0 ) β†’ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ (𝑃 ∧ 𝑋))
171, 12, 13, 16syl3anc 1372 . . . . 5 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) β‰  0 ) β†’ βˆƒπ‘¦ ∈ 𝐴 𝑦 ≀ (𝑃 ∧ 𝑋))
18 simpl1 1192 . . . . . . . . . 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 38159 . . . . . . . . . 10 (𝑦 ∈ 𝐴 β†’ 𝑦 ∈ 𝐡)
2120adantl 483 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ 𝑦 ∈ 𝐡)
22 simpl2 1193 . . . . . . . . . 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 1194 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ 𝑋 ∈ 𝐡)
254, 14, 9latlem12 18419 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑦 ∈ 𝐡 ∧ 𝑃 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡)) β†’ ((𝑦 ≀ 𝑃 ∧ 𝑦 ≀ 𝑋) ↔ 𝑦 ≀ (𝑃 ∧ 𝑋)))
2619, 21, 23, 24, 25syl13anc 1373 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ ((𝑦 ≀ 𝑃 ∧ 𝑦 ≀ 𝑋) ↔ 𝑦 ≀ (𝑃 ∧ 𝑋)))
27 simpr 486 . . . . . . . . . . 11 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ 𝑦 ∈ 𝐴)
2814, 5atcmp 38181 . . . . . . . . . . 11 ((𝐾 ∈ AtLat ∧ 𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) β†’ (𝑦 ≀ 𝑃 ↔ 𝑦 = 𝑃))
2918, 27, 22, 28syl3anc 1372 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ (𝑦 ≀ 𝑃 ↔ 𝑦 = 𝑃))
30 breq1 5152 . . . . . . . . . . 11 (𝑦 = 𝑃 β†’ (𝑦 ≀ 𝑋 ↔ 𝑃 ≀ 𝑋))
3130biimpd 228 . . . . . . . . . 10 (𝑦 = 𝑃 β†’ (𝑦 ≀ 𝑋 β†’ 𝑃 ≀ 𝑋))
3229, 31syl6bi 253 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ (𝑦 ≀ 𝑃 β†’ (𝑦 ≀ 𝑋 β†’ 𝑃 ≀ 𝑋)))
3332impd 412 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ ((𝑦 ≀ 𝑃 ∧ 𝑦 ≀ 𝑋) β†’ 𝑃 ≀ 𝑋))
3426, 33sylbird 260 . . . . . . 7 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ 𝑦 ∈ 𝐴) β†’ (𝑦 ≀ (𝑃 ∧ 𝑋) β†’ 𝑃 ≀ 𝑋))
3534adantlr 714 . . . . . 6 ((((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) β‰  0 ) ∧ 𝑦 ∈ 𝐴) β†’ (𝑦 ≀ (𝑃 ∧ 𝑋) β†’ 𝑃 ≀ 𝑋))
3635rexlimdva 3156 . . . . 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 414 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ ((𝑃 ∧ 𝑋) β‰  0 β†’ 𝑃 ≀ 𝑋))
3938necon1bd 2959 . 2 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (Β¬ 𝑃 ≀ 𝑋 β†’ (𝑃 ∧ 𝑋) = 0 ))
4015, 5atn0 38178 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) β†’ 𝑃 β‰  0 )
41403adant3 1133 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ 𝑃 β‰  0 )
424, 14, 9latleeqm1 18420 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐡 ∧ 𝑋 ∈ 𝐡) β†’ (𝑃 ≀ 𝑋 ↔ (𝑃 ∧ 𝑋) = 𝑃))
433, 7, 8, 42syl3anc 1372 . . . . . . 7 ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ (𝑃 ≀ 𝑋 ↔ (𝑃 ∧ 𝑋) = 𝑃))
4443adantr 482 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) = 0 ) β†’ (𝑃 ≀ 𝑋 ↔ (𝑃 ∧ 𝑋) = 𝑃))
45 eqeq1 2737 . . . . . . . 8 ((𝑃 ∧ 𝑋) = 𝑃 β†’ ((𝑃 ∧ 𝑋) = 0 ↔ 𝑃 = 0 ))
4645biimpcd 248 . . . . . . 7 ((𝑃 ∧ 𝑋) = 0 β†’ ((𝑃 ∧ 𝑋) = 𝑃 β†’ 𝑃 = 0 ))
4746adantl 483 . . . . . 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 2954 . . . 4 (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ∧ 𝑋) = 0 ) β†’ (𝑃 β‰  0 β†’ Β¬ 𝑃 ≀ 𝑋))
5049ex 414 . . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆƒwrex 3071   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  lecple 17204  meetcmee 18265  0.cp0 18376  Latclat 18384  Atomscatm 38133  AtLatcal 38134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-lat 18385  df-covers 38136  df-ats 38137  df-atl 38168
This theorem is referenced by:  atnem0  38188  iscvlat2N  38194  cvlexch3  38202  cvlexch4N  38203  cvlcvrp  38210  intnatN  38278  cvrat4  38314  dalem24  38568  cdlema2N  38663  llnexchb2lem  38739  lhpmat  38901  cdleme15b  39146  cdlemednpq  39170  cdleme20zN  39172  cdleme22cN  39213  dihmeetlem7N  40181  dihmeetlem17N  40194
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