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Theorem pmapval 35564
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 35563 . . 3 (𝐾𝐶𝑀 = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))
65fveq1d 6355 . 2 (𝐾𝐶 → (𝑀𝑋) = ((𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥})‘𝑋))
7 breq2 4808 . . . 4 (𝑥 = 𝑋 → (𝑎 𝑥𝑎 𝑋))
87rabbidv 3329 . . 3 (𝑥 = 𝑋 → {𝑎𝐴𝑎 𝑥} = {𝑎𝐴𝑎 𝑋})
9 eqid 2760 . . 3 (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}) = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥})
10 fvex 6363 . . . . 5 (Atoms‘𝐾) ∈ V
113, 10eqeltri 2835 . . . 4 𝐴 ∈ V
1211rabex 4964 . . 3 {𝑎𝐴𝑎 𝑋} ∈ V
138, 9, 12fvmpt 6445 . 2 (𝑋𝐵 → ((𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥})‘𝑋) = {𝑎𝐴𝑎 𝑋})
146, 13sylan9eq 2814 1 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) = {𝑎𝐴𝑎 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340   class class class wbr 4804  cmpt 4881  cfv 6049  Basecbs 16079  lecple 16170  Atomscatm 35071  pmapcpmap 35304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-pmap 35311
This theorem is referenced by:  elpmap  35565  pmapssat  35566  pmaple  35568  pmapat  35570  pmap0  35572  pmap1N  35574  pmapsub  35575  pmapglbx  35576  isline2  35581  linepmap  35582  polpmapN  35719  2polssN  35722  pmaplubN  35731
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