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Theorem dihvalb 38532
Description: Value of isomorphism H for a lattice 𝐾 when 𝑋 𝑊. (Contributed by NM, 4-Mar-2014.)
Hypotheses
Ref Expression
dihvalb.b 𝐵 = (Base‘𝐾)
dihvalb.l = (le‘𝐾)
dihvalb.h 𝐻 = (LHyp‘𝐾)
dihvalb.i 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihvalb.d 𝐷 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dihvalb (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = (𝐷𝑋))

Proof of Theorem dihvalb
Dummy variables 𝑢 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dihvalb.b . . . 4 𝐵 = (Base‘𝐾)
2 dihvalb.l . . . 4 = (le‘𝐾)
3 eqid 2801 . . . 4 (join‘𝐾) = (join‘𝐾)
4 eqid 2801 . . . 4 (meet‘𝐾) = (meet‘𝐾)
5 eqid 2801 . . . 4 (Atoms‘𝐾) = (Atoms‘𝐾)
6 dihvalb.h . . . 4 𝐻 = (LHyp‘𝐾)
7 dihvalb.i . . . 4 𝐼 = ((DIsoH‘𝐾)‘𝑊)
8 dihvalb.d . . . 4 𝐷 = ((DIsoB‘𝐾)‘𝑊)
9 eqid 2801 . . . 4 ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊)
10 eqid 2801 . . . 4 ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
11 eqid 2801 . . . 4 (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊))
12 eqid 2801 . . . 4 (LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊))
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12dihval 38527 . . 3 (((𝐾𝑉𝑊𝐻) ∧ 𝑋𝐵) → (𝐼𝑋) = if(𝑋 𝑊, (𝐷𝑋), (𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊))∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑞 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) → 𝑢 = ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(𝐷‘(𝑋(meet‘𝐾)𝑊)))))))
14 iftrue 4434 . . 3 (𝑋 𝑊 → if(𝑋 𝑊, (𝐷𝑋), (𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊))∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑞 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) → 𝑢 = ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(𝐷‘(𝑋(meet‘𝐾)𝑊)))))) = (𝐷𝑋))
1513, 14sylan9eq 2856 . 2 ((((𝐾𝑉𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑋 𝑊) → (𝐼𝑋) = (𝐷𝑋))
1615anasss 470 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = (𝐷𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  ifcif 4428   class class class wbr 5033  cfv 6328  crio 7096  (class class class)co 7139  Basecbs 16479  lecple 16568  joincjn 17550  meetcmee 17551  LSSumclsm 18755  LSubSpclss 19700  Atomscatm 36558  LHypclh 37279  DVecHcdvh 38373  DIsoBcdib 38433  DIsoCcdic 38467  DIsoHcdih 38523
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-dih 38524
This theorem is referenced by:  dihopelvalbN  38533  dih1dimb  38535  dih2dimb  38539  dih2dimbALTN  38540  dihvalcq2  38542  dihlss  38545  dihord6apre  38551  dihord3  38552  dihord5b  38554  dihord5apre  38557  dih0  38575  dihwN  38584  dihglblem3N  38590  dihmeetlem2N  38594  dih1dimatlem  38624  dihjatcclem4  38716
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