Theorem pmap1N 40465

Description: Value of the projective map of a Hilbert lattice at lattice unity. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pmap1.u	⊢ 1 = (1.‘𝐾)
pmap1.a	⊢ 𝐴 = (Atoms‘𝐾)
pmap1.m	⊢ 𝑀 = (pmap‘𝐾)

Assertion

Ref	Expression
pmap1N	⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = 𝐴)

Proof of Theorem pmap1N

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2769	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		pmap1.u	. . . 4 ⊢ 1 = (1.‘𝐾)
3	1, 2	op1cl 39883	. . 3 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾))
4		eqid 2769	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
5		pmap1.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
6		pmap1.m	. . . 4 ⊢ 𝑀 = (pmap‘𝐾)
7	1, 4, 5, 6	pmapval 40455	. . 3 ⊢ ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾)) → (𝑀‘ 1 ) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 })
8	3, 7	mpdan 699	. 2 ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 })
9	1, 5	atbase 39987	. . . . 5 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾))
10	1, 4, 2	ople1 39889	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 𝑝(le‘𝐾) 1 )
11	9, 10	sylan2 604	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑝 ∈ 𝐴) → 𝑝(le‘𝐾) 1 )
12	11	ralrimiva 3163	. . 3 ⊢ (𝐾 ∈ OP → ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾) 1 )
13		rabid2 3456	. . 3 ⊢ (𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 } ↔ ∀𝑝 ∈ 𝐴 𝑝(le‘𝐾) 1 )
14	12, 13	sylibr 237	. 2 ⊢ (𝐾 ∈ OP → 𝐴 = {𝑝 ∈ 𝐴 ∣ 𝑝(le‘𝐾) 1 })
15	8, 14	eqtr4d 2807	1 ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = 𝐴)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3423   class class class wbr 5113  ‘cfv 6537  Basecbs 17269  lecple 17317  1.cp1 18478  OPcops 39870  Atomscatm 39961  pmapcpmap 40195

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 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405

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 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-lub 18400  df-p1 18480  df-oposet 39874  df-ats 39965  df-pmap 40202

This theorem is referenced by:  pmapglb2N  40469  pmapglb2xN  40470