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Theorem dibval 40841
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
dibval (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼𝑋) = ((𝐽𝑋) × { 0 }))
Distinct variable groups:   𝑓,𝐾   𝑓,𝑊
Allowed substitution hints:   𝐵(𝑓)   𝑇(𝑓)   𝐻(𝑓)   𝐼(𝑓)   𝐽(𝑓)   𝑉(𝑓)   𝑋(𝑓)   0 (𝑓)

Proof of Theorem dibval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dibval.b . . . . 5 𝐵 = (Base‘𝐾)
2 dibval.h . . . . 5 𝐻 = (LHyp‘𝐾)
3 dibval.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
4 dibval.o . . . . 5 0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
5 dibval.j . . . . 5 𝐽 = ((DIsoA‘𝐾)‘𝑊)
6 dibval.i . . . . 5 𝐼 = ((DIsoB‘𝐾)‘𝑊)
71, 2, 3, 4, 5, 6dibfval 40840 . . . 4 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })))
87adantr 479 . . 3 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })))
98fveq1d 6903 . 2 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼𝑋) = ((𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 }))‘𝑋))
10 fveq2 6901 . . . . 5 (𝑥 = 𝑋 → (𝐽𝑥) = (𝐽𝑋))
1110xpeq1d 5711 . . . 4 (𝑥 = 𝑋 → ((𝐽𝑥) × { 0 }) = ((𝐽𝑋) × { 0 }))
12 eqid 2726 . . . 4 (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })) = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 }))
13 fvex 6914 . . . . 5 (𝐽𝑋) ∈ V
14 snex 5437 . . . . 5 { 0 } ∈ V
1513, 14xpex 7761 . . . 4 ((𝐽𝑋) × { 0 }) ∈ V
1611, 12, 15fvmpt 7009 . . 3 (𝑋 ∈ dom 𝐽 → ((𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 }))‘𝑋) = ((𝐽𝑋) × { 0 }))
1716adantl 480 . 2 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → ((𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 }))‘𝑋) = ((𝐽𝑋) × { 0 }))
189, 17eqtrd 2766 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼𝑋) = ((𝐽𝑋) × { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  {csn 4633  cmpt 5236   I cid 5579   × cxp 5680  dom cdm 5682  cres 5684  cfv 6554  Basecbs 17213  LHypclh 39683  LTrncltrn 39800  DIsoAcdia 40727  DIsoBcdib 40837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-dib 40838
This theorem is referenced by:  dibopelvalN  40842  dibval2  40843  dibvalrel  40862
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