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Theorem dihvalb 37028
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 2760 . . . 4 (join‘𝐾) = (join‘𝐾)
4 eqid 2760 . . . 4 (meet‘𝐾) = (meet‘𝐾)
5 eqid 2760 . . . 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 2760 . . . 4 ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊)
10 eqid 2760 . . . 4 ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
11 eqid 2760 . . . 4 (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊))
12 eqid 2760 . . . 4 (LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊))
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12dihval 37023 . . 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 4236 . . 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 2814 . 2 ((((𝐾𝑉𝑊𝐻) ∧ 𝑋𝐵) ∧ 𝑋 𝑊) → (𝐼𝑋) = (𝐷𝑋))
1615anasss 682 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = (𝐷𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  ifcif 4230   class class class wbr 4804  cfv 6049  crio 6773  (class class class)co 6813  Basecbs 16059  lecple 16150  joincjn 17145  meetcmee 17146  LSSumclsm 18249  LSubSpclss 19134  Atomscatm 35053  LHypclh 35773  DVecHcdvh 36869  DIsoBcdib 36929  DIsoCcdic 36963  DIsoHcdih 37019
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-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-pr 5055
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-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-riota 6774  df-ov 6816  df-dih 37020
This theorem is referenced by:  dihopelvalbN  37029  dih1dimb  37031  dih2dimb  37035  dih2dimbALTN  37036  dihvalcq2  37038  dihlss  37041  dihord6apre  37047  dihord3  37048  dihord5b  37050  dihord5apre  37053  dih0  37071  dihwN  37080  dihglblem3N  37086  dihmeetlem2N  37090  dih1dimatlem  37120  dihjatcclem4  37212
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