Theorem pmap0 40424

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 → (𝑀‘ 0 ) = ∅)

Proof of Theorem pmap0

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2769	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		pmap0.z	. . . 4 ⊢ 0 = (0.‘𝐾)
3	1, 2	atl0cl 39962	. . 3 ⊢ (𝐾 ∈ AtLat → 0 ∈ (Base‘𝐾))
4		eqid 2769	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
5		eqid 2769	. . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6		pmap0.m	. . . 4 ⊢ 𝑀 = (pmap‘𝐾)
7	1, 4, 5, 6	pmapval 40416	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 0 ∈ (Base‘𝐾)) → (𝑀‘ 0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 })
8	3, 7	mpdan 699	. 2 ⊢ (𝐾 ∈ AtLat → (𝑀‘ 0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 })
9	4, 2, 5	atnle0 39968	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑎 ∈ (Atoms‘𝐾)) → ¬ 𝑎(le‘𝐾) 0 )
10	9	nrexdv 3166	. . . 4 ⊢ (𝐾 ∈ AtLat → ¬ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 )
11		rabn0 4352	. . . 4 ⊢ ({𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅ ↔ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 )
12	10, 11	sylnibr 332	. . 3 ⊢ (𝐾 ∈ AtLat → ¬ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅)
13		nne 2968	. . 3 ⊢ (¬ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅ ↔ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } = ∅)
14	12, 13	sylib 221	. 2 ⊢ (𝐾 ∈ AtLat → {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } = ∅)
15	8, 14	eqtrd 2804	1 ⊢ (𝐾 ∈ AtLat → (𝑀‘ 0 ) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095  {crab 3423  ∅c0 4294   class class class wbr 5110  ‘cfv 6533  Basecbs 17265  lecple 17313  0.cp0 18473  Atomscatm 39922  AtLatcal 39923  pmapcpmap 40156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7365  df-proset 18346  df-poset 18365  df-plt 18380  df-glb 18397  df-p0 18475  df-lat 18484  df-covers 39925  df-ats 39926  df-atl 39957  df-pmap 40163

This theorem is referenced by:  pmapeq0  40425  pmapjat1  40512  pol1N  40569  pnonsingN  40592