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 41120
Description: The partial isomorphism B for a lattice 𝐾. (Contributed by NM, 8-Dec-2013.)
Hypotheses
Ref Expression
dibval.b 𝐵 = (Base‘𝐾)
dibval.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
dibffval (𝐾𝑉 → (DIsoB‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))))
Distinct variable groups:   𝑤,𝐻   𝑤,𝑓,𝑥,𝐾
Allowed substitution hints:   𝐵(𝑥,𝑤,𝑓)   𝐻(𝑥,𝑓)   𝑉(𝑥,𝑤,𝑓)

Proof of Theorem dibffval
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3500 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6904 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 dibval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2794 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6904 . . . . . . 7 (𝑘 = 𝐾 → (DIsoA‘𝑘) = (DIsoA‘𝐾))
65fveq1d 6906 . . . . . 6 (𝑘 = 𝐾 → ((DIsoA‘𝑘)‘𝑤) = ((DIsoA‘𝐾)‘𝑤))
76dmeqd 5914 . . . . 5 (𝑘 = 𝐾 → dom ((DIsoA‘𝑘)‘𝑤) = dom ((DIsoA‘𝐾)‘𝑤))
86fveq1d 6906 . . . . . 6 (𝑘 = 𝐾 → (((DIsoA‘𝑘)‘𝑤)‘𝑥) = (((DIsoA‘𝐾)‘𝑤)‘𝑥))
9 fveq2 6904 . . . . . . . . 9 (𝑘 = 𝐾 → (LTrn‘𝑘) = (LTrn‘𝐾))
109fveq1d 6906 . . . . . . . 8 (𝑘 = 𝐾 → ((LTrn‘𝑘)‘𝑤) = ((LTrn‘𝐾)‘𝑤))
11 fveq2 6904 . . . . . . . . . 10 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
12 dibval.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
1311, 12eqtr4di 2794 . . . . . . . . 9 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
1413reseq2d 5995 . . . . . . . 8 (𝑘 = 𝐾 → ( I ↾ (Base‘𝑘)) = ( I ↾ 𝐵))
1510, 14mpteq12dv 5231 . . . . . . 7 (𝑘 = 𝐾 → (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵)))
1615sneqd 4636 . . . . . 6 (𝑘 = 𝐾 → {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))} = {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})
178, 16xpeq12d 5714 . . . . 5 (𝑘 = 𝐾 → ((((DIsoA‘𝑘)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))}) = ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))
187, 17mpteq12dv 5231 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ dom ((DIsoA‘𝑘)‘𝑤) ↦ ((((DIsoA‘𝑘)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))})) = (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})))
194, 18mpteq12dv 5231 . . 3 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ dom ((DIsoA‘𝑘)‘𝑤) ↦ ((((DIsoA‘𝑘)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))}))) = (𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))))
20 df-dib 41119 . . 3 DIsoB = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ dom ((DIsoA‘𝑘)‘𝑤) ↦ ((((DIsoA‘𝑘)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))}))))
2119, 20, 3mptfvmpt 7246 . 2 (𝐾 ∈ V → (DIsoB‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))))
221, 21syl 17 1 (𝐾𝑉 → (DIsoB‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3479  {csn 4624  cmpt 5223   I cid 5575   × cxp 5681  dom cdm 5683  cres 5685  cfv 6559  Basecbs 17243  LHypclh 39964  LTrncltrn 40081  DIsoAcdia 41008  DIsoBcdib 41118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  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-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-dib 41119
This theorem is referenced by:  dibfval  41121
  Copyright terms: Public domain W3C validator