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Theorem pmap0 37023
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 2822 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 pmap0.z . . . 4 0 = (0.‘𝐾)
31, 2atl0cl 36561 . . 3 (𝐾 ∈ AtLat → 0 ∈ (Base‘𝐾))
4 eqid 2822 . . . 4 (le‘𝐾) = (le‘𝐾)
5 eqid 2822 . . . 4 (Atoms‘𝐾) = (Atoms‘𝐾)
6 pmap0.m . . . 4 𝑀 = (pmap‘𝐾)
71, 4, 5, 6pmapval 37015 . . 3 ((𝐾 ∈ AtLat ∧ 0 ∈ (Base‘𝐾)) → (𝑀0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 })
83, 7mpdan 686 . 2 (𝐾 ∈ AtLat → (𝑀0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 })
94, 2, 5atnle0 36567 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑎 ∈ (Atoms‘𝐾)) → ¬ 𝑎(le‘𝐾) 0 )
109nrexdv 3256 . . . 4 (𝐾 ∈ AtLat → ¬ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 )
11 rabn0 4311 . . . 4 ({𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅ ↔ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 )
1210, 11sylnibr 332 . . 3 (𝐾 ∈ AtLat → ¬ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅)
13 nne 3015 . . 3 (¬ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅ ↔ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } = ∅)
1412, 13sylib 221 . 2 (𝐾 ∈ AtLat → {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } = ∅)
158, 14eqtrd 2857 1 (𝐾 ∈ AtLat → (𝑀0 ) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2114  wne 3011  wrex 3131  {crab 3134  c0 4265   class class class wbr 5042  cfv 6334  Basecbs 16474  lecple 16563  0.cp0 17638  Atomscatm 36521  AtLatcal 36522  pmapcpmap 36755
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-proset 17529  df-poset 17547  df-plt 17559  df-glb 17576  df-p0 17640  df-lat 17647  df-covers 36524  df-ats 36525  df-atl 36556  df-pmap 36762
This theorem is referenced by:  pmapeq0  37024  pmapjat1  37111  pol1N  37168  pnonsingN  37191
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