Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dibffval Structured version   Visualization version   GIF version

Theorem dibffval 40550
Description: The partial isomorphism B for a lattice 𝐾. (Contributed by NM, 8-Dec-2013.)
Hypotheses
Ref Expression
dibval.b 𝐡 = (Baseβ€˜πΎ)
dibval.h 𝐻 = (LHypβ€˜πΎ)
Assertion
Ref Expression
dibffval (𝐾 ∈ 𝑉 β†’ (DIsoBβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ dom ((DIsoAβ€˜πΎ)β€˜π‘€) ↦ ((((DIsoAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡))}))))
Distinct variable groups:   𝑀,𝐻   𝑀,𝑓,π‘₯,𝐾
Allowed substitution hints:   𝐡(π‘₯,𝑀,𝑓)   𝐻(π‘₯,𝑓)   𝑉(π‘₯,𝑀,𝑓)

Proof of Theorem dibffval
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 elex 3488 . 2 (𝐾 ∈ 𝑉 β†’ 𝐾 ∈ V)
2 fveq2 6891 . . . . 5 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = (LHypβ€˜πΎ))
3 dibval.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3eqtr4di 2785 . . . 4 (π‘˜ = 𝐾 β†’ (LHypβ€˜π‘˜) = 𝐻)
5 fveq2 6891 . . . . . . 7 (π‘˜ = 𝐾 β†’ (DIsoAβ€˜π‘˜) = (DIsoAβ€˜πΎ))
65fveq1d 6893 . . . . . 6 (π‘˜ = 𝐾 β†’ ((DIsoAβ€˜π‘˜)β€˜π‘€) = ((DIsoAβ€˜πΎ)β€˜π‘€))
76dmeqd 5902 . . . . 5 (π‘˜ = 𝐾 β†’ dom ((DIsoAβ€˜π‘˜)β€˜π‘€) = dom ((DIsoAβ€˜πΎ)β€˜π‘€))
86fveq1d 6893 . . . . . 6 (π‘˜ = 𝐾 β†’ (((DIsoAβ€˜π‘˜)β€˜π‘€)β€˜π‘₯) = (((DIsoAβ€˜πΎ)β€˜π‘€)β€˜π‘₯))
9 fveq2 6891 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (LTrnβ€˜π‘˜) = (LTrnβ€˜πΎ))
109fveq1d 6893 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ((LTrnβ€˜π‘˜)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘€))
11 fveq2 6891 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = (Baseβ€˜πΎ))
12 dibval.b . . . . . . . . . 10 𝐡 = (Baseβ€˜πΎ)
1311, 12eqtr4di 2785 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = 𝐡)
1413reseq2d 5979 . . . . . . . 8 (π‘˜ = 𝐾 β†’ ( I β†Ύ (Baseβ€˜π‘˜)) = ( I β†Ύ 𝐡))
1510, 14mpteq12dv 5233 . . . . . . 7 (π‘˜ = 𝐾 β†’ (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ ( I β†Ύ (Baseβ€˜π‘˜))) = (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡)))
1615sneqd 4636 . . . . . 6 (π‘˜ = 𝐾 β†’ {(𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ ( I β†Ύ (Baseβ€˜π‘˜)))} = {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡))})
178, 16xpeq12d 5703 . . . . 5 (π‘˜ = 𝐾 β†’ ((((DIsoAβ€˜π‘˜)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ ( I β†Ύ (Baseβ€˜π‘˜)))}) = ((((DIsoAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡))}))
187, 17mpteq12dv 5233 . . . 4 (π‘˜ = 𝐾 β†’ (π‘₯ ∈ dom ((DIsoAβ€˜π‘˜)β€˜π‘€) ↦ ((((DIsoAβ€˜π‘˜)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ ( I β†Ύ (Baseβ€˜π‘˜)))})) = (π‘₯ ∈ dom ((DIsoAβ€˜πΎ)β€˜π‘€) ↦ ((((DIsoAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡))})))
194, 18mpteq12dv 5233 . . 3 (π‘˜ = 𝐾 β†’ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ (π‘₯ ∈ dom ((DIsoAβ€˜π‘˜)β€˜π‘€) ↦ ((((DIsoAβ€˜π‘˜)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ ( I β†Ύ (Baseβ€˜π‘˜)))}))) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ dom ((DIsoAβ€˜πΎ)β€˜π‘€) ↦ ((((DIsoAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡))}))))
20 df-dib 40549 . . 3 DIsoB = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ (π‘₯ ∈ dom ((DIsoAβ€˜π‘˜)β€˜π‘€) ↦ ((((DIsoAβ€˜π‘˜)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ ( I β†Ύ (Baseβ€˜π‘˜)))}))))
2119, 20, 3mptfvmpt 7234 . 2 (𝐾 ∈ V β†’ (DIsoBβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ dom ((DIsoAβ€˜πΎ)β€˜π‘€) ↦ ((((DIsoAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡))}))))
221, 21syl 17 1 (𝐾 ∈ 𝑉 β†’ (DIsoBβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (π‘₯ ∈ dom ((DIsoAβ€˜πΎ)β€˜π‘€) ↦ ((((DIsoAβ€˜πΎ)β€˜π‘€)β€˜π‘₯) Γ— {(𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ( I β†Ύ 𝐡))}))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099  Vcvv 3469  {csn 4624   ↦ cmpt 5225   I cid 5569   Γ— cxp 5670  dom cdm 5672   β†Ύ cres 5674  β€˜cfv 6542  Basecbs 17171  LHypclh 39394  LTrncltrn 39511  DIsoAcdia 40438  DIsoBcdib 40548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-dib 40549
This theorem is referenced by:  dibfval  40551
  Copyright terms: Public domain W3C validator