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 → (𝑀‘ 0 ) = ∅)

Proof of Theorem pmap0

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		pmap0.z	. . . 4 ⊢ 0 = (0.‘𝐾)
3	1, 2	atl0cl 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)
7	1, 4, 5, 6	pmapval 39759	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 0 ∈ (Base‘𝐾)) → (𝑀‘ 0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 })
8	3, 7	mpdan 687	. 2 ⊢ (𝐾 ∈ AtLat → (𝑀‘ 0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 })
9	4, 2, 5	atnle0 39310	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑎 ∈ (Atoms‘𝐾)) → ¬ 𝑎(le‘𝐾) 0 )
10	9	nrexdv 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 )
12	10, 11	sylnibr 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 } = ∅)
14	12, 13	sylib 218	. 2 ⊢ (𝐾 ∈ AtLat → {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } = ∅)
15	8, 14	eqtrd 2777	1 ⊢ (𝐾 ∈ AtLat → (𝑀‘ 0 ) = ∅)

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