Theorem pmap0 39722

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 2740	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		pmap0.z	. . . 4 ⊢ 0 = (0.‘𝐾)
3	1, 2	atl0cl 39259	. . 3 ⊢ (𝐾 ∈ AtLat → 0 ∈ (Base‘𝐾))
4		eqid 2740	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
5		eqid 2740	. . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6		pmap0.m	. . . 4 ⊢ 𝑀 = (pmap‘𝐾)
7	1, 4, 5, 6	pmapval 39714	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 0 ∈ (Base‘𝐾)) → (𝑀‘ 0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 })
8	3, 7	mpdan 686	. 2 ⊢ (𝐾 ∈ AtLat → (𝑀‘ 0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 })
9	4, 2, 5	atnle0 39265	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑎 ∈ (Atoms‘𝐾)) → ¬ 𝑎(le‘𝐾) 0 )
10	9	nrexdv 3155	. . . 4 ⊢ (𝐾 ∈ AtLat → ¬ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 )
11		rabn0 4412	. . . 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 2950	. . 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 2780	1 ⊢ (𝐾 ∈ AtLat → (𝑀‘ 0 ) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076  {crab 3443  ∅c0 4352   class class class wbr 5166  ‘cfv 6573  Basecbs 17258  lecple 17318  0.cp0 18493  Atomscatm 39219  AtLatcal 39220  pmapcpmap 39454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-proset 18365  df-poset 18383  df-plt 18400  df-glb 18417  df-p0 18495  df-lat 18502  df-covers 39222  df-ats 39223  df-atl 39254  df-pmap 39461

This theorem is referenced by:  pmapeq0  39723  pmapjat1  39810  pol1N  39867  pnonsingN  39890