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Theorem pmap1N 33874
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 2609 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 pmap1.u . . . 4 1 = (1.‘𝐾)
31, 2op1cl 33293 . . 3 (𝐾 ∈ OP → 1 ∈ (Base‘𝐾))
4 eqid 2609 . . . 4 (le‘𝐾) = (le‘𝐾)
5 pmap1.a . . . 4 𝐴 = (Atoms‘𝐾)
6 pmap1.m . . . 4 𝑀 = (pmap‘𝐾)
71, 4, 5, 6pmapval 33864 . . 3 ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾)) → (𝑀1 ) = {𝑝𝐴𝑝(le‘𝐾) 1 })
83, 7mpdan 698 . 2 (𝐾 ∈ OP → (𝑀1 ) = {𝑝𝐴𝑝(le‘𝐾) 1 })
91, 5atbase 33397 . . . . 5 (𝑝𝐴𝑝 ∈ (Base‘𝐾))
101, 4, 2ople1 33299 . . . . 5 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 𝑝(le‘𝐾) 1 )
119, 10sylan2 489 . . . 4 ((𝐾 ∈ OP ∧ 𝑝𝐴) → 𝑝(le‘𝐾) 1 )
1211ralrimiva 2948 . . 3 (𝐾 ∈ OP → ∀𝑝𝐴 𝑝(le‘𝐾) 1 )
13 rabid2 3095 . . 3 (𝐴 = {𝑝𝐴𝑝(le‘𝐾) 1 } ↔ ∀𝑝𝐴 𝑝(le‘𝐾) 1 )
1412, 13sylibr 222 . 2 (𝐾 ∈ OP → 𝐴 = {𝑝𝐴𝑝(le‘𝐾) 1 })
158, 14eqtr4d 2646 1 (𝐾 ∈ OP → (𝑀1 ) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  wral 2895  {crab 2899   class class class wbr 4577  cfv 5790  Basecbs 15641  lecple 15721  1.cp1 16807  OPcops 33280  Atomscatm 33371  pmapcpmap 33604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-lub 16743  df-p1 16809  df-oposet 33284  df-ats 33375  df-pmap 33611
This theorem is referenced by:  pmapglb2N  33878  pmapglb2xN  33879
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