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Theorem dibval2 41146
Description: Value of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.)
Hypotheses
Ref Expression
dibval2.b 𝐵 = (Base‘𝐾)
dibval2.l = (le‘𝐾)
dibval2.h 𝐻 = (LHyp‘𝐾)
dibval2.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dibval2.o 0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
dibval2.j 𝐽 = ((DIsoA‘𝐾)‘𝑊)
dibval2.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibval2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((𝐽𝑋) × { 0 }))
Distinct variable groups:   𝑓,𝐾   𝑓,𝑊
Allowed substitution hints:   𝐵(𝑓)   𝑇(𝑓)   𝐻(𝑓)   𝐼(𝑓)   𝐽(𝑓)   (𝑓)   𝑉(𝑓)   𝑋(𝑓)   0 (𝑓)

Proof of Theorem dibval2
StepHypRef Expression
1 dibval2.b . . . 4 𝐵 = (Base‘𝐾)
2 dibval2.l . . . 4 = (le‘𝐾)
3 dibval2.h . . . 4 𝐻 = (LHyp‘𝐾)
4 dibval2.j . . . 4 𝐽 = ((DIsoA‘𝐾)‘𝑊)
51, 2, 3, 4diaeldm 41038 . . 3 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐽 ↔ (𝑋𝐵𝑋 𝑊)))
65biimpar 477 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → 𝑋 ∈ dom 𝐽)
7 dibval2.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
8 dibval2.o . . 3 0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
9 dibval2.i . . 3 𝐼 = ((DIsoB‘𝐾)‘𝑊)
101, 3, 7, 8, 4, 9dibval 41144 . 2 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼𝑋) = ((𝐽𝑋) × { 0 }))
116, 10syldan 591 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((𝐽𝑋) × { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {csn 4626   class class class wbr 5143  cmpt 5225   I cid 5577   × cxp 5683  dom cdm 5685  cres 5687  cfv 6561  Basecbs 17247  lecple 17304  LHypclh 39986  LTrncltrn 40103  DIsoAcdia 41030  DIsoBcdib 41140
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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
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 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-disoa 41031  df-dib 41141
This theorem is referenced by:  dibopelval2  41147  dibval3N  41148  dibelval3  41149  dibelval1st  41151  dibelval2nd  41154  dibn0  41155  dibord  41161  dib0  41166  dib1dim  41167  dibss  41171  diblss  41172  dihwN  41291
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