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Theorem dibval 37301
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 37300 . . . 4 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })))
87adantr 474 . . 3 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })))
98fveq1d 6450 . 2 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼𝑋) = ((𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 }))‘𝑋))
10 fveq2 6448 . . . . 5 (𝑥 = 𝑋 → (𝐽𝑥) = (𝐽𝑋))
1110xpeq1d 5386 . . . 4 (𝑥 = 𝑋 → ((𝐽𝑥) × { 0 }) = ((𝐽𝑋) × { 0 }))
12 eqid 2778 . . . 4 (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })) = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 }))
13 fvex 6461 . . . . 5 (𝐽𝑋) ∈ V
14 snex 5142 . . . . 5 { 0 } ∈ V
1513, 14xpex 7242 . . . 4 ((𝐽𝑋) × { 0 }) ∈ V
1611, 12, 15fvmpt 6544 . . 3 (𝑋 ∈ dom 𝐽 → ((𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 }))‘𝑋) = ((𝐽𝑋) × { 0 }))
1716adantl 475 . 2 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → ((𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 }))‘𝑋) = ((𝐽𝑋) × { 0 }))
189, 17eqtrd 2814 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼𝑋) = ((𝐽𝑋) × { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  {csn 4398  cmpt 4967   I cid 5262   × cxp 5355  dom cdm 5357  cres 5359  cfv 6137  Basecbs 16259  LHypclh 36143  LTrncltrn 36260  DIsoAcdia 37187  DIsoBcdib 37297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-dib 37298
This theorem is referenced by:  dibopelvalN  37302  dibval2  37303  dibvalrel  37322
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