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Theorem dibfval 39367
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 39366 . . . 4 (𝐾𝑉 → (DIsoB‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))))
54fveq1d 6811 . . 3 (𝐾𝑉 → ((DIsoB‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})))‘𝑊))
61, 5eqtrid 2789 . 2 (𝐾𝑉𝐼 = ((𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})))‘𝑊))
7 fveq2 6809 . . . . . 6 (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = ((DIsoA‘𝐾)‘𝑊))
8 dibval.j . . . . . 6 𝐽 = ((DIsoA‘𝐾)‘𝑊)
97, 8eqtr4di 2795 . . . . 5 (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = 𝐽)
109dmeqd 5832 . . . 4 (𝑤 = 𝑊 → dom ((DIsoA‘𝐾)‘𝑤) = dom 𝐽)
119fveq1d 6811 . . . . 5 (𝑤 = 𝑊 → (((DIsoA‘𝐾)‘𝑤)‘𝑥) = (𝐽𝑥))
12 fveq2 6809 . . . . . . . . 9 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
13 dibval.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
1412, 13eqtr4di 2795 . . . . . . . 8 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
15 eqidd 2738 . . . . . . . 8 (𝑤 = 𝑊 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
1614, 15mpteq12dv 5176 . . . . . . 7 (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵)) = (𝑓𝑇 ↦ ( I ↾ 𝐵)))
17 dibval.o . . . . . . 7 0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
1816, 17eqtr4di 2795 . . . . . 6 (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵)) = 0 )
1918sneqd 4581 . . . . 5 (𝑤 = 𝑊 → {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))} = { 0 })
2011, 19xpeq12d 5636 . . . 4 (𝑤 = 𝑊 → ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}) = ((𝐽𝑥) × { 0 }))
2110, 20mpteq12dv 5176 . . 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 2737 . . 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 6823 . . . . 5 𝐽 ∈ V
2423dmex 7801 . . . 4 dom 𝐽 ∈ V
2524mptex 7136 . . 3 (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })) ∈ V
2621, 22, 25fvmpt 6912 . 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 2797 1 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  {csn 4569  cmpt 5168   I cid 5504   × cxp 5603  dom cdm 5605  cres 5607  cfv 6463  Basecbs 16979  LHypclh 38210  LTrncltrn 38327  DIsoAcdia 39254  DIsoBcdib 39364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5222  ax-sep 5236  ax-nul 5243  ax-pr 5365  ax-un 7626
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-iun 4937  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-dib 39365
This theorem is referenced by:  dibval  39368  dibfna  39380
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