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Theorem pmap1N 35556
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 2760 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 pmap1.u . . . 4 1 = (1.‘𝐾)
31, 2op1cl 34975 . . 3 (𝐾 ∈ OP → 1 ∈ (Base‘𝐾))
4 eqid 2760 . . . 4 (le‘𝐾) = (le‘𝐾)
5 pmap1.a . . . 4 𝐴 = (Atoms‘𝐾)
6 pmap1.m . . . 4 𝑀 = (pmap‘𝐾)
71, 4, 5, 6pmapval 35546 . . 3 ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾)) → (𝑀1 ) = {𝑝𝐴𝑝(le‘𝐾) 1 })
83, 7mpdan 705 . 2 (𝐾 ∈ OP → (𝑀1 ) = {𝑝𝐴𝑝(le‘𝐾) 1 })
91, 5atbase 35079 . . . . 5 (𝑝𝐴𝑝 ∈ (Base‘𝐾))
101, 4, 2ople1 34981 . . . . 5 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 𝑝(le‘𝐾) 1 )
119, 10sylan2 492 . . . 4 ((𝐾 ∈ OP ∧ 𝑝𝐴) → 𝑝(le‘𝐾) 1 )
1211ralrimiva 3104 . . 3 (𝐾 ∈ OP → ∀𝑝𝐴 𝑝(le‘𝐾) 1 )
13 rabid2 3257 . . 3 (𝐴 = {𝑝𝐴𝑝(le‘𝐾) 1 } ↔ ∀𝑝𝐴 𝑝(le‘𝐾) 1 )
1412, 13sylibr 224 . 2 (𝐾 ∈ OP → 𝐴 = {𝑝𝐴𝑝(le‘𝐾) 1 })
158, 14eqtr4d 2797 1 (𝐾 ∈ OP → (𝑀1 ) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wral 3050  {crab 3054   class class class wbr 4804  cfv 6049  Basecbs 16059  lecple 16150  1.cp1 17239  OPcops 34962  Atomscatm 35053  pmapcpmap 35286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-lub 17175  df-p1 17241  df-oposet 34966  df-ats 35057  df-pmap 35293
This theorem is referenced by:  pmapglb2N  35560  pmapglb2xN  35561
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