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Theorem atnle 36320
Description: Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 30067 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 1185 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → 𝐾 ∈ AtLat)
2 atllat 36303 . . . . . . . . 9 (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
323ad2ant1 1127 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝐾 ∈ Lat)
4 atnle.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
5 atnle.a . . . . . . . . . 10 𝐴 = (Atoms‘𝐾)
64, 5atbase 36292 . . . . . . . . 9 (𝑃𝐴𝑃𝐵)
763ad2ant2 1128 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝑃𝐵)
8 simp3 1132 . . . . . . . 8 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝑋𝐵)
9 atnle.m . . . . . . . . 9 = (meet‘𝐾)
104, 9latmcl 17652 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋) ∈ 𝐵)
113, 7, 8, 10syl3anc 1365 . . . . . . 7 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (𝑃 𝑋) ∈ 𝐵)
1211adantr 481 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → (𝑃 𝑋) ∈ 𝐵)
13 simpr 485 . . . . . 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 36319 . . . . . 6 ((𝐾 ∈ AtLat ∧ (𝑃 𝑋) ∈ 𝐵 ∧ (𝑃 𝑋) ≠ 0 ) → ∃𝑦𝐴 𝑦 (𝑃 𝑋))
171, 12, 13, 16syl3anc 1365 . . . . 5 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) → ∃𝑦𝐴 𝑦 (𝑃 𝑋))
18 simpl1 1185 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝐾 ∈ AtLat)
1918, 2syl 17 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝐾 ∈ Lat)
204, 5atbase 36292 . . . . . . . . . 10 (𝑦𝐴𝑦𝐵)
2120adantl 482 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑦𝐵)
22 simpl2 1186 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑃𝐴)
2322, 6syl 17 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑃𝐵)
24 simpl3 1187 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑋𝐵)
254, 14, 9latlem12 17678 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑦𝐵𝑃𝐵𝑋𝐵)) → ((𝑦 𝑃𝑦 𝑋) ↔ 𝑦 (𝑃 𝑋)))
2619, 21, 23, 24, 25syl13anc 1366 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → ((𝑦 𝑃𝑦 𝑋) ↔ 𝑦 (𝑃 𝑋)))
27 simpr 485 . . . . . . . . . . 11 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → 𝑦𝐴)
2814, 5atcmp 36314 . . . . . . . . . . 11 ((𝐾 ∈ AtLat ∧ 𝑦𝐴𝑃𝐴) → (𝑦 𝑃𝑦 = 𝑃))
2918, 27, 22, 28syl3anc 1365 . . . . . . . . . 10 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → (𝑦 𝑃𝑦 = 𝑃))
30 breq1 5066 . . . . . . . . . . 11 (𝑦 = 𝑃 → (𝑦 𝑋𝑃 𝑋))
3130biimpd 230 . . . . . . . . . 10 (𝑦 = 𝑃 → (𝑦 𝑋𝑃 𝑋))
3229, 31syl6bi 254 . . . . . . . . 9 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → (𝑦 𝑃 → (𝑦 𝑋𝑃 𝑋)))
3332impd 411 . . . . . . . 8 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → ((𝑦 𝑃𝑦 𝑋) → 𝑃 𝑋))
3426, 33sylbird 261 . . . . . . 7 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ 𝑦𝐴) → (𝑦 (𝑃 𝑋) → 𝑃 𝑋))
3534adantlr 711 . . . . . 6 ((((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) ≠ 0 ) ∧ 𝑦𝐴) → (𝑦 (𝑃 𝑋) → 𝑃 𝑋))
3635rexlimdva 3289 . . . . 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 413 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → ((𝑃 𝑋) ≠ 0𝑃 𝑋))
3938necon1bd 3039 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (¬ 𝑃 𝑋 → (𝑃 𝑋) = 0 ))
4015, 5atn0 36311 . . . 4 ((𝐾 ∈ AtLat ∧ 𝑃𝐴) → 𝑃0 )
41403adant3 1126 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → 𝑃0 )
424, 14, 9latleeqm1 17679 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵) → (𝑃 𝑋 ↔ (𝑃 𝑋) = 𝑃))
433, 7, 8, 42syl3anc 1365 . . . . . . 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 2830 . . . . . . . 8 ((𝑃 𝑋) = 𝑃 → ((𝑃 𝑋) = 0𝑃 = 0 ))
4645biimpcd 250 . . . . . . 7 ((𝑃 𝑋) = 0 → ((𝑃 𝑋) = 𝑃𝑃 = 0 ))
4746adantl 482 . . . . . 6 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → ((𝑃 𝑋) = 𝑃𝑃 = 0 ))
4844, 47sylbid 241 . . . . 5 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃 𝑋𝑃 = 0 ))
4948necon3ad 3034 . . . 4 (((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) ∧ (𝑃 𝑋) = 0 ) → (𝑃0 → ¬ 𝑃 𝑋))
5049ex 413 . . 3 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → ((𝑃 𝑋) = 0 → (𝑃0 → ¬ 𝑃 𝑋)))
5141, 50mpid 44 . 2 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → ((𝑃 𝑋) = 0 → ¬ 𝑃 𝑋))
5239, 51impbid 213 1 ((𝐾 ∈ AtLat ∧ 𝑃𝐴𝑋𝐵) → (¬ 𝑃 𝑋 ↔ (𝑃 𝑋) = 0 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wrex 3144   class class class wbr 5063  cfv 6352  (class class class)co 7148  Basecbs 16473  lecple 16562  meetcmee 17545  0.cp0 17637  Latclat 17645  Atomscatm 36266  AtLatcal 36267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-proset 17528  df-poset 17546  df-plt 17558  df-lub 17574  df-glb 17575  df-join 17576  df-meet 17577  df-p0 17639  df-lat 17646  df-covers 36269  df-ats 36270  df-atl 36301
This theorem is referenced by:  atnem0  36321  iscvlat2N  36327  cvlexch3  36335  cvlexch4N  36336  cvlcvrp  36343  intnatN  36410  cvrat4  36446  dalem24  36700  cdlema2N  36795  llnexchb2lem  36871  lhpmat  37033  cdleme15b  37278  cdlemednpq  37302  cdleme20zN  37304  cdleme22cN  37345  dihmeetlem7N  38313  dihmeetlem17N  38326
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