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Theorem pmapval 39740
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 39739 . . 3 (𝐾𝐶𝑀 = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
65fveq1d 6909 . 2 (𝐾𝐶 → (𝑀𝑋) = ((𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥})‘𝑋))
7 breq2 5152 . . . 4 (𝑥 = 𝑋 → (𝑎 𝑥𝑎 𝑋))
87rabbidv 3441 . . 3 (𝑥 = 𝑋 → {𝑎𝐴𝑎 𝑥} = {𝑎𝐴𝑎 𝑋})
9 eqid 2735 . . 3 (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}) = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥})
103fvexi 6921 . . . 4 𝐴 ∈ V
1110rabex 5345 . . 3 {𝑎𝐴𝑎 𝑋} ∈ V
128, 9, 11fvmpt 7016 . 2 (𝑋𝐵 → ((𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥})‘𝑋) = {𝑎𝐴𝑎 𝑋})
136, 12sylan9eq 2795 1 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) = {𝑎𝐴𝑎 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {crab 3433   class class class wbr 5148  cmpt 5231  cfv 6563  Basecbs 17245  lecple 17305  Atomscatm 39245  pmapcpmap 39480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-pmap 39487
This theorem is referenced by:  elpmap  39741  pmapssat  39742  pmaple  39744  pmapat  39746  pmap0  39748  pmap1N  39750  pmapsub  39751  pmapglbx  39752  isline2  39757  linepmap  39758  polpmapN  39895  2polssN  39898  pmaplubN  39907
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