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Theorem dibval 36951
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 36950 . . . 4 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })))
87adantr 472 . . 3 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })))
98fveq1d 6355 . 2 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼𝑋) = ((𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 }))‘𝑋))
10 fveq2 6353 . . . . 5 (𝑥 = 𝑋 → (𝐽𝑥) = (𝐽𝑋))
1110xpeq1d 5295 . . . 4 (𝑥 = 𝑋 → ((𝐽𝑥) × { 0 }) = ((𝐽𝑋) × { 0 }))
12 eqid 2760 . . . 4 (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })) = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 }))
13 fvex 6363 . . . . 5 (𝐽𝑋) ∈ V
14 snex 5057 . . . . 5 { 0 } ∈ V
1513, 14xpex 7128 . . . 4 ((𝐽𝑋) × { 0 }) ∈ V
1611, 12, 15fvmpt 6445 . . 3 (𝑋 ∈ dom 𝐽 → ((𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 }))‘𝑋) = ((𝐽𝑋) × { 0 }))
1716adantl 473 . 2 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → ((𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 }))‘𝑋) = ((𝐽𝑋) × { 0 }))
189, 17eqtrd 2794 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼𝑋) = ((𝐽𝑋) × { 0 }))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  {csn 4321  cmpt 4881   I cid 5173   × cxp 5264  dom cdm 5266  cres 5268  cfv 6049  Basecbs 16079  LHypclh 35791  LTrncltrn 35908  DIsoAcdia 36837  DIsoBcdib 36947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-dib 36948
This theorem is referenced by:  dibopelvalN  36952  dibval2  36953  dibvalrel  36972
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