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Theorem dibfval 38157
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)
dibval.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dibval.o 0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
dibval.j 𝐽 = ((DIsoA‘𝐾)‘𝑊)
dibval.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibfval ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })))
Distinct variable groups:   𝑥,𝑓,𝐾   𝑥,𝐽   𝑓,𝑊,𝑥
Allowed substitution hints:   𝐵(𝑥,𝑓)   𝑇(𝑥,𝑓)   𝐻(𝑥,𝑓)   𝐼(𝑥,𝑓)   𝐽(𝑓)   𝑉(𝑥,𝑓)   0 (𝑥,𝑓)

Proof of Theorem dibfval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 dibval.i . . 3 𝐼 = ((DIsoB‘𝐾)‘𝑊)
2 dibval.b . . . . 5 𝐵 = (Base‘𝐾)
3 dibval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3dibffval 38156 . . . 4 (𝐾𝑉 → (DIsoB‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))))
54fveq1d 6665 . . 3 (𝐾𝑉 → ((DIsoB‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})))‘𝑊))
61, 5syl5eq 2865 . 2 (𝐾𝑉𝐼 = ((𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})))‘𝑊))
7 fveq2 6663 . . . . . 6 (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = ((DIsoA‘𝐾)‘𝑊))
8 dibval.j . . . . . 6 𝐽 = ((DIsoA‘𝐾)‘𝑊)
97, 8syl6eqr 2871 . . . . 5 (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = 𝐽)
109dmeqd 5767 . . . 4 (𝑤 = 𝑊 → dom ((DIsoA‘𝐾)‘𝑤) = dom 𝐽)
119fveq1d 6665 . . . . 5 (𝑤 = 𝑊 → (((DIsoA‘𝐾)‘𝑤)‘𝑥) = (𝐽𝑥))
12 fveq2 6663 . . . . . . . . 9 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
13 dibval.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
1412, 13syl6eqr 2871 . . . . . . . 8 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
15 eqidd 2819 . . . . . . . 8 (𝑤 = 𝑊 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
1614, 15mpteq12dv 5142 . . . . . . 7 (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵)) = (𝑓𝑇 ↦ ( I ↾ 𝐵)))
17 dibval.o . . . . . . 7 0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
1816, 17syl6eqr 2871 . . . . . 6 (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵)) = 0 )
1918sneqd 4569 . . . . 5 (𝑤 = 𝑊 → {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))} = { 0 })
2011, 19xpeq12d 5579 . . . 4 (𝑤 = 𝑊 → ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}) = ((𝐽𝑥) × { 0 }))
2110, 20mpteq12dv 5142 . . 3 (𝑤 = 𝑊 → (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})) = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })))
22 eqid 2818 . . 3 (𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))) = (𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})))
238fvexi 6677 . . . . 5 𝐽 ∈ V
2423dmex 7605 . . . 4 dom 𝐽 ∈ V
2524mptex 6977 . . 3 (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })) ∈ V
2621, 22, 25fvmpt 6761 . 2 (𝑊𝐻 → ((𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})))‘𝑊) = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })))
276, 26sylan9eq 2873 1 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  {csn 4557  cmpt 5137   I cid 5452   × cxp 5546  dom cdm 5548  cres 5550  cfv 6348  Basecbs 16471  LHypclh 37000  LTrncltrn 37117  DIsoAcdia 38044  DIsoBcdib 38154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-dib 38155
This theorem is referenced by:  dibval  38158  dibfna  38170
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