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Theorem dibfval 38430
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 38429 . . . 4 (𝐾𝑉 → (DIsoB‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))))
54fveq1d 6651 . . 3 (𝐾𝑉 → ((DIsoB‘𝐾)‘𝑊) = ((𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})))‘𝑊))
61, 5syl5eq 2848 . 2 (𝐾𝑉𝐼 = ((𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})))‘𝑊))
7 fveq2 6649 . . . . . 6 (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = ((DIsoA‘𝐾)‘𝑊))
8 dibval.j . . . . . 6 𝐽 = ((DIsoA‘𝐾)‘𝑊)
97, 8eqtr4di 2854 . . . . 5 (𝑤 = 𝑊 → ((DIsoA‘𝐾)‘𝑤) = 𝐽)
109dmeqd 5742 . . . 4 (𝑤 = 𝑊 → dom ((DIsoA‘𝐾)‘𝑤) = dom 𝐽)
119fveq1d 6651 . . . . 5 (𝑤 = 𝑊 → (((DIsoA‘𝐾)‘𝑤)‘𝑥) = (𝐽𝑥))
12 fveq2 6649 . . . . . . . . 9 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
13 dibval.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
1412, 13eqtr4di 2854 . . . . . . . 8 (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = 𝑇)
15 eqidd 2802 . . . . . . . 8 (𝑤 = 𝑊 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
1614, 15mpteq12dv 5118 . . . . . . 7 (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵)) = (𝑓𝑇 ↦ ( I ↾ 𝐵)))
17 dibval.o . . . . . . 7 0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
1816, 17eqtr4di 2854 . . . . . 6 (𝑤 = 𝑊 → (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵)) = 0 )
1918sneqd 4540 . . . . 5 (𝑤 = 𝑊 → {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))} = { 0 })
2011, 19xpeq12d 5554 . . . 4 (𝑤 = 𝑊 → ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}) = ((𝐽𝑥) × { 0 }))
2110, 20mpteq12dv 5118 . . 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 2801 . . 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 6663 . . . . 5 𝐽 ∈ V
2423dmex 7602 . . . 4 dom 𝐽 ∈ V
2524mptex 6967 . . 3 (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })) ∈ V
2621, 22, 25fvmpt 6749 . 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 2856 1 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  {csn 4528  cmpt 5113   I cid 5427   × cxp 5521  dom cdm 5523  cres 5525  cfv 6328  Basecbs 16478  LHypclh 37273  LTrncltrn 37390  DIsoAcdia 38317  DIsoBcdib 38427
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-dib 38428
This theorem is referenced by:  dibval  38431  dibfna  38443
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