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Theorem pmap1N 35575
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 2771 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 pmap1.u . . . 4 1 = (1.‘𝐾)
31, 2op1cl 34994 . . 3 (𝐾 ∈ OP → 1 ∈ (Base‘𝐾))
4 eqid 2771 . . . 4 (le‘𝐾) = (le‘𝐾)
5 pmap1.a . . . 4 𝐴 = (Atoms‘𝐾)
6 pmap1.m . . . 4 𝑀 = (pmap‘𝐾)
71, 4, 5, 6pmapval 35565 . . 3 ((𝐾 ∈ OP ∧ 1 ∈ (Base‘𝐾)) → (𝑀1 ) = {𝑝𝐴𝑝(le‘𝐾) 1 })
83, 7mpdan 667 . 2 (𝐾 ∈ OP → (𝑀1 ) = {𝑝𝐴𝑝(le‘𝐾) 1 })
91, 5atbase 35098 . . . . 5 (𝑝𝐴𝑝 ∈ (Base‘𝐾))
101, 4, 2ople1 35000 . . . . 5 ((𝐾 ∈ OP ∧ 𝑝 ∈ (Base‘𝐾)) → 𝑝(le‘𝐾) 1 )
119, 10sylan2 580 . . . 4 ((𝐾 ∈ OP ∧ 𝑝𝐴) → 𝑝(le‘𝐾) 1 )
1211ralrimiva 3115 . . 3 (𝐾 ∈ OP → ∀𝑝𝐴 𝑝(le‘𝐾) 1 )
13 rabid2 3267 . . 3 (𝐴 = {𝑝𝐴𝑝(le‘𝐾) 1 } ↔ ∀𝑝𝐴 𝑝(le‘𝐾) 1 )
1412, 13sylibr 224 . 2 (𝐾 ∈ OP → 𝐴 = {𝑝𝐴𝑝(le‘𝐾) 1 })
158, 14eqtr4d 2808 1 (𝐾 ∈ OP → (𝑀1 ) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wral 3061  {crab 3065   class class class wbr 4786  cfv 6031  Basecbs 16064  lecple 16156  1.cp1 17246  OPcops 34981  Atomscatm 35072  pmapcpmap 35305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-lub 17182  df-p1 17248  df-oposet 34985  df-ats 35076  df-pmap 35312
This theorem is referenced by:  pmapglb2N  35579  pmapglb2xN  35580
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