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Theorem dibval 38397
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 38396 . . . 4 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })))
87adantr 484 . . 3 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })))
98fveq1d 6654 . 2 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼𝑋) = ((𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 }))‘𝑋))
10 fveq2 6652 . . . . 5 (𝑥 = 𝑋 → (𝐽𝑥) = (𝐽𝑋))
1110xpeq1d 5561 . . . 4 (𝑥 = 𝑋 → ((𝐽𝑥) × { 0 }) = ((𝐽𝑋) × { 0 }))
12 eqid 2822 . . . 4 (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })) = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 }))
13 fvex 6665 . . . . 5 (𝐽𝑋) ∈ V
14 snex 5309 . . . . 5 { 0 } ∈ V
1513, 14xpex 7461 . . . 4 ((𝐽𝑋) × { 0 }) ∈ V
1611, 12, 15fvmpt 6750 . . 3 (𝑋 ∈ dom 𝐽 → ((𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 }))‘𝑋) = ((𝐽𝑋) × { 0 }))
1716adantl 485 . 2 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → ((𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 }))‘𝑋) = ((𝐽𝑋) × { 0 }))
189, 17eqtrd 2857 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼𝑋) = ((𝐽𝑋) × { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  {csn 4539  cmpt 5122   I cid 5436   × cxp 5530  dom cdm 5532  cres 5534  cfv 6334  Basecbs 16474  LHypclh 37239  LTrncltrn 37356  DIsoAcdia 38283  DIsoBcdib 38393
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-dib 38394
This theorem is referenced by:  dibopelvalN  38398  dibval2  38399  dibvalrel  38418
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