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Theorem dibelval1st2N 39827
Description: Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibelval1st2.b 𝐵 = (Base‘𝐾)
dibelval1st2.l = (le‘𝐾)
dibelval1st2.h 𝐻 = (LHyp‘𝐾)
dibelval1st2.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dibelval1st2.r 𝑅 = ((trL‘𝐾)‘𝑊)
dibelval1st2.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibelval1st2N (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → (𝑅‘(1st𝑌)) 𝑋)

Proof of Theorem dibelval1st2N
StepHypRef Expression
1 dibelval1st2.b . . 3 𝐵 = (Base‘𝐾)
2 dibelval1st2.l . . 3 = (le‘𝐾)
3 dibelval1st2.h . . 3 𝐻 = (LHyp‘𝐾)
4 eqid 2731 . . 3 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
5 dibelval1st2.i . . 3 𝐼 = ((DIsoB‘𝐾)‘𝑊)
61, 2, 3, 4, 5dibelval1st 39825 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → (1st𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋))
7 dibelval1st2.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
8 dibelval1st2.r . . 3 𝑅 = ((trL‘𝐾)‘𝑊)
91, 2, 3, 7, 8, 4diatrl 39720 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (1st𝑌) ∈ (((DIsoA‘𝐾)‘𝑊)‘𝑋)) → (𝑅‘(1st𝑌)) 𝑋)
106, 9syld3an3 1409 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → (𝑅‘(1st𝑌)) 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106   class class class wbr 5141  cfv 6532  1st c1st 7955  Basecbs 17126  lecple 17186  LHypclh 38660  LTrncltrn 38777  trLctrl 38834  DIsoAcdia 39704  DIsoBcdib 39814
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-1st 7957  df-disoa 39705  df-dib 39815
This theorem is referenced by: (None)
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