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Theorem pmapval 33859
Description: Value of the projective map of a Hilbert lattice. Definition in Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 2-Oct-2011.)
Hypotheses
Ref Expression
pmapfval.b 𝐵 = (Base‘𝐾)
pmapfval.l = (le‘𝐾)
pmapfval.a 𝐴 = (Atoms‘𝐾)
pmapfval.m 𝑀 = (pmap‘𝐾)
Assertion
Ref Expression
pmapval ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) = {𝑎𝐴𝑎 𝑋})
Distinct variable groups:   𝐴,𝑎   𝐾,𝑎   𝑋,𝑎
Allowed substitution hints:   𝐵(𝑎)   𝐶(𝑎)   (𝑎)   𝑀(𝑎)

Proof of Theorem pmapval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pmapfval.b . . . 4 𝐵 = (Base‘𝐾)
2 pmapfval.l . . . 4 = (le‘𝐾)
3 pmapfval.a . . . 4 𝐴 = (Atoms‘𝐾)
4 pmapfval.m . . . 4 𝑀 = (pmap‘𝐾)
51, 2, 3, 4pmapfval 33858 . . 3 (𝐾𝐶𝑀 = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
65fveq1d 6085 . 2 (𝐾𝐶 → (𝑀𝑋) = ((𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥})‘𝑋))
7 breq2 4576 . . . 4 (𝑥 = 𝑋 → (𝑎 𝑥𝑎 𝑋))
87rabbidv 3158 . . 3 (𝑥 = 𝑋 → {𝑎𝐴𝑎 𝑥} = {𝑎𝐴𝑎 𝑋})
9 eqid 2604 . . 3 (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}) = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥})
10 fvex 6093 . . . . 5 (Atoms‘𝐾) ∈ V
113, 10eqeltri 2678 . . . 4 𝐴 ∈ V
1211rabex 4730 . . 3 {𝑎𝐴𝑎 𝑋} ∈ V
138, 9, 12fvmpt 6171 . 2 (𝑋𝐵 → ((𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥})‘𝑋) = {𝑎𝐴𝑎 𝑋})
146, 13sylan9eq 2658 1 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) = {𝑎𝐴𝑎 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  {crab 2894  Vcvv 3167   class class class wbr 4572  cmpt 4632  cfv 5785  Basecbs 15636  lecple 15716  Atomscatm 33366  pmapcpmap 33599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pr 4823
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 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-pmap 33606
This theorem is referenced by:  elpmap  33860  pmapssat  33861  pmaple  33863  pmapat  33865  pmap0  33867  pmap1N  33869  pmapsub  33870  pmapglbx  33871  isline2  33876  linepmap  33877  polpmapN  34014  2polssN  34017  pmaplubN  34026
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