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Theorem pmap0 35839
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 2825 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 pmap0.z . . . 4 0 = (0.‘𝐾)
31, 2atl0cl 35377 . . 3 (𝐾 ∈ AtLat → 0 ∈ (Base‘𝐾))
4 eqid 2825 . . . 4 (le‘𝐾) = (le‘𝐾)
5 eqid 2825 . . . 4 (Atoms‘𝐾) = (Atoms‘𝐾)
6 pmap0.m . . . 4 𝑀 = (pmap‘𝐾)
71, 4, 5, 6pmapval 35831 . . 3 ((𝐾 ∈ AtLat ∧ 0 ∈ (Base‘𝐾)) → (𝑀0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 })
83, 7mpdan 678 . 2 (𝐾 ∈ AtLat → (𝑀0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 })
94, 2, 5atnle0 35383 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑎 ∈ (Atoms‘𝐾)) → ¬ 𝑎(le‘𝐾) 0 )
109nrexdv 3209 . . . 4 (𝐾 ∈ AtLat → ¬ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 )
11 rabn0 4189 . . . 4 ({𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅ ↔ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 )
1210, 11sylnibr 321 . . 3 (𝐾 ∈ AtLat → ¬ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅)
13 nne 3003 . . 3 (¬ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅ ↔ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } = ∅)
1412, 13sylib 210 . 2 (𝐾 ∈ AtLat → {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } = ∅)
158, 14eqtrd 2861 1 (𝐾 ∈ AtLat → (𝑀0 ) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1656  wcel 2164  wne 2999  wrex 3118  {crab 3121  c0 4146   class class class wbr 4875  cfv 6127  Basecbs 16229  lecple 16319  0.cp0 17397  Atomscatm 35337  AtLatcal 35338  pmapcpmap 35571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-proset 17288  df-poset 17306  df-plt 17318  df-glb 17335  df-p0 17399  df-lat 17406  df-covers 35340  df-ats 35341  df-atl 35372  df-pmap 35578
This theorem is referenced by:  pmapeq0  35840  pmapjat1  35927  pol1N  35984  pnonsingN  36007
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