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Theorem pmap1N 34071
 Description: Value of the projective map of a Hilbert lattice at lattice unit. 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 → (𝑀1 ) = 𝐴)

Proof of Theorem pmap1N
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 pmap1.u . . . 4 1 = (1.‘𝐾)
31, 2op1cl 33490 . . 3 (𝐾 ∈ OP → 1 ∈ (Base‘𝐾))
4 eqid 2610 . . . 4 (le‘𝐾) = (le‘𝐾)
5 pmap1.a . . . 4 𝐴 = (Atoms‘𝐾)
6 pmap1.m . . . 4 𝑀 = (pmap‘𝐾)
71, 4, 5, 6pmapval 34061 . . 3 ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾)) → (𝑀1 ) = {𝑝𝐴𝑝(le‘𝐾) 1 })
83, 7mpdan 699 . 2 (𝐾 ∈ OP → (𝑀1 ) = {𝑝𝐴𝑝(le‘𝐾) 1 })
91, 5atbase 33594 . . . . 5 (𝑝𝐴𝑝 ∈ (Base‘𝐾))
101, 4, 2ople1 33496 . . . . 5 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 𝑝(le‘𝐾) 1 )
119, 10sylan2 490 . . . 4 ((𝐾 ∈ OP ∧ 𝑝𝐴) → 𝑝(le‘𝐾) 1 )
1211ralrimiva 2949 . . 3 (𝐾 ∈ OP → ∀𝑝𝐴 𝑝(le‘𝐾) 1 )
13 rabid2 3096 . . 3 (𝐴 = {𝑝𝐴𝑝(le‘𝐾) 1 } ↔ ∀𝑝𝐴 𝑝(le‘𝐾) 1 )
1412, 13sylibr 223 . 2 (𝐾 ∈ OP → 𝐴 = {𝑝𝐴𝑝(le‘𝐾) 1 })
158, 14eqtr4d 2647 1 (𝐾 ∈ OP → (𝑀1 ) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  lecple 15775  1.cp1 16861  OPcops 33477  Atomscatm 33568  pmapcpmap 33801 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-lub 16797  df-p1 16863  df-oposet 33481  df-ats 33572  df-pmap 33808 This theorem is referenced by:  pmapglb2N  34075  pmapglb2xN  34076
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