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Theorem pmapval 37025
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 37024 . . 3 (𝐾𝐶𝑀 = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
65fveq1d 6665 . 2 (𝐾𝐶 → (𝑀𝑋) = ((𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥})‘𝑋))
7 breq2 5057 . . . 4 (𝑥 = 𝑋 → (𝑎 𝑥𝑎 𝑋))
87rabbidv 3465 . . 3 (𝑥 = 𝑋 → {𝑎𝐴𝑎 𝑥} = {𝑎𝐴𝑎 𝑋})
9 eqid 2824 . . 3 (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}) = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥})
103fvexi 6677 . . . 4 𝐴 ∈ V
1110rabex 5222 . . 3 {𝑎𝐴𝑎 𝑋} ∈ V
128, 9, 11fvmpt 6761 . 2 (𝑋𝐵 → ((𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥})‘𝑋) = {𝑎𝐴𝑎 𝑋})
136, 12sylan9eq 2879 1 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) = {𝑎𝐴𝑎 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {crab 3137   class class class wbr 5053  cmpt 5133  cfv 6345  Basecbs 16485  lecple 16574  Atomscatm 36531  pmapcpmap 36765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-pmap 36772
This theorem is referenced by:  elpmap  37026  pmapssat  37027  pmaple  37029  pmapat  37031  pmap0  37033  pmap1N  37035  pmapsub  37036  pmapglbx  37037  isline2  37042  linepmap  37043  polpmapN  37180  2polssN  37183  pmaplubN  37192
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