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Theorem pmap0 39368
Description: Value of the projective map of a Hilbert lattice at lattice zero. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.)
Hypotheses
Ref Expression
pmap0.z 0 = (0.‘𝐾)
pmap0.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
pmap0 (𝐾 ∈ AtLat → (𝑀0 ) = ∅)

Proof of Theorem pmap0
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 eqid 2725 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 pmap0.z . . . 4 0 = (0.‘𝐾)
31, 2atl0cl 38905 . . 3 (𝐾 ∈ AtLat → 0 ∈ (Base‘𝐾))
4 eqid 2725 . . . 4 (le‘𝐾) = (le‘𝐾)
5 eqid 2725 . . . 4 (Atoms‘𝐾) = (Atoms‘𝐾)
6 pmap0.m . . . 4 𝑀 = (pmap‘𝐾)
71, 4, 5, 6pmapval 39360 . . 3 ((𝐾 ∈ AtLat ∧ 0 ∈ (Base‘𝐾)) → (𝑀0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 })
83, 7mpdan 685 . 2 (𝐾 ∈ AtLat → (𝑀0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 })
94, 2, 5atnle0 38911 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑎 ∈ (Atoms‘𝐾)) → ¬ 𝑎(le‘𝐾) 0 )
109nrexdv 3138 . . . 4 (𝐾 ∈ AtLat → ¬ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 )
11 rabn0 4387 . . . 4 ({𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅ ↔ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 )
1210, 11sylnibr 328 . . 3 (𝐾 ∈ AtLat → ¬ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅)
13 nne 2933 . . 3 (¬ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅ ↔ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } = ∅)
1412, 13sylib 217 . 2 (𝐾 ∈ AtLat → {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } = ∅)
158, 14eqtrd 2765 1 (𝐾 ∈ AtLat → (𝑀0 ) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533  wcel 2098  wne 2929  wrex 3059  {crab 3418  c0 4322   class class class wbr 5149  cfv 6549  Basecbs 17183  lecple 17243  0.cp0 18418  Atomscatm 38865  AtLatcal 38866  pmapcpmap 39100
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-proset 18290  df-poset 18308  df-plt 18325  df-glb 18342  df-p0 18420  df-lat 18427  df-covers 38868  df-ats 38869  df-atl 38900  df-pmap 39107
This theorem is referenced by:  pmapeq0  39369  pmapjat1  39456  pol1N  39513  pnonsingN  39536
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