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Theorem pmapval 39760
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 39759 . . 3 (𝐾𝐶𝑀 = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
65fveq1d 6907 . 2 (𝐾𝐶 → (𝑀𝑋) = ((𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥})‘𝑋))
7 breq2 5146 . . . 4 (𝑥 = 𝑋 → (𝑎 𝑥𝑎 𝑋))
87rabbidv 3443 . . 3 (𝑥 = 𝑋 → {𝑎𝐴𝑎 𝑥} = {𝑎𝐴𝑎 𝑋})
9 eqid 2736 . . 3 (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}) = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥})
103fvexi 6919 . . . 4 𝐴 ∈ V
1110rabex 5338 . . 3 {𝑎𝐴𝑎 𝑋} ∈ V
128, 9, 11fvmpt 7015 . 2 (𝑋𝐵 → ((𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥})‘𝑋) = {𝑎𝐴𝑎 𝑋})
136, 12sylan9eq 2796 1 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) = {𝑎𝐴𝑎 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  {crab 3435   class class class wbr 5142  cmpt 5224  cfv 6560  Basecbs 17248  lecple 17305  Atomscatm 39265  pmapcpmap 39500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-pmap 39507
This theorem is referenced by:  elpmap  39761  pmapssat  39762  pmaple  39764  pmapat  39766  pmap0  39768  pmap1N  39770  pmapsub  39771  pmapglbx  39772  isline2  39777  linepmap  39778  polpmapN  39915  2polssN  39918  pmaplubN  39927
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