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Theorem pmap0 39767
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 2737 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 pmap0.z . . . 4 0 = (0.‘𝐾)
31, 2atl0cl 39304 . . 3 (𝐾 ∈ AtLat → 0 ∈ (Base‘𝐾))
4 eqid 2737 . . . 4 (le‘𝐾) = (le‘𝐾)
5 eqid 2737 . . . 4 (Atoms‘𝐾) = (Atoms‘𝐾)
6 pmap0.m . . . 4 𝑀 = (pmap‘𝐾)
71, 4, 5, 6pmapval 39759 . . 3 ((𝐾 ∈ AtLat ∧ 0 ∈ (Base‘𝐾)) → (𝑀0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 })
83, 7mpdan 687 . 2 (𝐾 ∈ AtLat → (𝑀0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 })
94, 2, 5atnle0 39310 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑎 ∈ (Atoms‘𝐾)) → ¬ 𝑎(le‘𝐾) 0 )
109nrexdv 3149 . . . 4 (𝐾 ∈ AtLat → ¬ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 )
11 rabn0 4389 . . . 4 ({𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅ ↔ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 )
1210, 11sylnibr 329 . . 3 (𝐾 ∈ AtLat → ¬ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅)
13 nne 2944 . . 3 (¬ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅ ↔ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } = ∅)
1412, 13sylib 218 . 2 (𝐾 ∈ AtLat → {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } = ∅)
158, 14eqtrd 2777 1 (𝐾 ∈ AtLat → (𝑀0 ) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  wne 2940  wrex 3070  {crab 3436  c0 4333   class class class wbr 5143  cfv 6561  Basecbs 17247  lecple 17304  0.cp0 18468  Atomscatm 39264  AtLatcal 39265  pmapcpmap 39499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-proset 18340  df-poset 18359  df-plt 18375  df-glb 18392  df-p0 18470  df-lat 18477  df-covers 39267  df-ats 39268  df-atl 39299  df-pmap 39506
This theorem is referenced by:  pmapeq0  39768  pmapjat1  39855  pol1N  39912  pnonsingN  39935
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