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Theorem dibfval 39604
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 39603 . . . 4 (𝐾𝑉 → (DIsoB‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))))
54fveq1d 6844 . . 3 (𝐾𝑉 → ((DIsoB‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})))‘𝑊))
61, 5eqtrid 2788 . 2 (𝐾𝑉𝐼 = ((𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})))‘𝑊))
7 fveq2 6842 . . . . . 6 (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = ((DIsoA‘𝐾)‘𝑊))
8 dibval.j . . . . . 6 𝐽 = ((DIsoA‘𝐾)‘𝑊)
97, 8eqtr4di 2794 . . . . 5 (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = 𝐽)
109dmeqd 5861 . . . 4 (𝑤 = 𝑊 → dom ((DIsoA‘𝐾)‘𝑤) = dom 𝐽)
119fveq1d 6844 . . . . 5 (𝑤 = 𝑊 → (((DIsoA‘𝐾)‘𝑤)‘𝑥) = (𝐽𝑥))
12 fveq2 6842 . . . . . . . . 9 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
13 dibval.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
1412, 13eqtr4di 2794 . . . . . . . 8 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
15 eqidd 2737 . . . . . . . 8 (𝑤 = 𝑊 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
1614, 15mpteq12dv 5196 . . . . . . 7 (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵)) = (𝑓𝑇 ↦ ( I ↾ 𝐵)))
17 dibval.o . . . . . . 7 0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
1816, 17eqtr4di 2794 . . . . . 6 (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵)) = 0 )
1918sneqd 4598 . . . . 5 (𝑤 = 𝑊 → {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))} = { 0 })
2011, 19xpeq12d 5664 . . . 4 (𝑤 = 𝑊 → ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}) = ((𝐽𝑥) × { 0 }))
2110, 20mpteq12dv 5196 . . 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 2736 . . 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 6856 . . . . 5 𝐽 ∈ V
2423dmex 7848 . . . 4 dom 𝐽 ∈ V
2524mptex 7173 . . 3 (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })) ∈ V
2621, 22, 25fvmpt 6948 . 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 2796 1 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {csn 4586  cmpt 5188   I cid 5530   × cxp 5631  dom cdm 5633  cres 5635  cfv 6496  Basecbs 17083  LHypclh 38447  LTrncltrn 38564  DIsoAcdia 39491  DIsoBcdib 39601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-dib 39602
This theorem is referenced by:  dibval  39605  dibfna  39617
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