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Theorem pmap0 34528
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 2621 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 pmap0.z . . . 4 0 = (0.‘𝐾)
31, 2atl0cl 34067 . . 3 (𝐾 ∈ AtLat → 0 ∈ (Base‘𝐾))
4 eqid 2621 . . . 4 (le‘𝐾) = (le‘𝐾)
5 eqid 2621 . . . 4 (Atoms‘𝐾) = (Atoms‘𝐾)
6 pmap0.m . . . 4 𝑀 = (pmap‘𝐾)
71, 4, 5, 6pmapval 34520 . . 3 ((𝐾 ∈ AtLat ∧ 0 ∈ (Base‘𝐾)) → (𝑀0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 })
83, 7mpdan 701 . 2 (𝐾 ∈ AtLat → (𝑀0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 })
94, 2, 5atnle0 34073 . . . . 5 ((𝐾 ∈ AtLat ∧ 𝑎 ∈ (Atoms‘𝐾)) → ¬ 𝑎(le‘𝐾) 0 )
109nrexdv 2995 . . . 4 (𝐾 ∈ AtLat → ¬ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 )
11 rabn0 3932 . . . 4 ({𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅ ↔ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 )
1210, 11sylnibr 319 . . 3 (𝐾 ∈ AtLat → ¬ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅)
13 nne 2794 . . 3 (¬ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅ ↔ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } = ∅)
1412, 13sylib 208 . 2 (𝐾 ∈ AtLat → {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } = ∅)
158, 14eqtrd 2655 1 (𝐾 ∈ AtLat → (𝑀0 ) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wcel 1987  wne 2790  wrex 2908  {crab 2911  c0 3891   class class class wbr 4613  cfv 5847  Basecbs 15781  lecple 15869  0.cp0 16958  Atomscatm 34027  AtLatcal 34028  pmapcpmap 34260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-preset 16849  df-poset 16867  df-plt 16879  df-glb 16896  df-p0 16960  df-lat 16967  df-covers 34030  df-ats 34031  df-atl 34062  df-pmap 34267
This theorem is referenced by:  pmapeq0  34529  pmapjat1  34616  pol1N  34673  pnonsingN  34696
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