Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihffval Structured version   Visualization version   GIF version

Theorem dihffval 41213
Description: The isomorphism H for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)
Hypotheses
Ref Expression
dihval.b 𝐵 = (Base‘𝐾)
dihval.l = (le‘𝐾)
dihval.j = (join‘𝐾)
dihval.m = (meet‘𝐾)
dihval.a 𝐴 = (Atoms‘𝐾)
dihval.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
dihffval (𝐾𝑉 → (DIsoH‘𝐾) = (𝑤𝐻 ↦ (𝑥𝐵 ↦ if(𝑥 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞𝐴 ((¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤)))))))))
Distinct variable groups:   𝐴,𝑞   𝑤,𝐻   𝑢,𝑞,𝑤,𝑥,𝐾
Allowed substitution hints:   𝐴(𝑥,𝑤,𝑢)   𝐵(𝑥,𝑤,𝑢,𝑞)   𝐻(𝑥,𝑢,𝑞)   (𝑥,𝑤,𝑢,𝑞)   (𝑥,𝑤,𝑢,𝑞)   (𝑥,𝑤,𝑢,𝑞)   𝑉(𝑥,𝑤,𝑢,𝑞)

Proof of Theorem dihffval
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 elex 3499 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6907 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 dihval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2793 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6907 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
6 dihval.b . . . . . 6 𝐵 = (Base‘𝐾)
75, 6eqtr4di 2793 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
8 fveq2 6907 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
9 dihval.l . . . . . . . 8 = (le‘𝐾)
108, 9eqtr4di 2793 . . . . . . 7 (𝑘 = 𝐾 → (le‘𝑘) = )
1110breqd 5159 . . . . . 6 (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑤𝑥 𝑤))
12 fveq2 6907 . . . . . . . 8 (𝑘 = 𝐾 → (DIsoB‘𝑘) = (DIsoB‘𝐾))
1312fveq1d 6909 . . . . . . 7 (𝑘 = 𝐾 → ((DIsoB‘𝑘)‘𝑤) = ((DIsoB‘𝐾)‘𝑤))
1413fveq1d 6909 . . . . . 6 (𝑘 = 𝐾 → (((DIsoB‘𝑘)‘𝑤)‘𝑥) = (((DIsoB‘𝐾)‘𝑤)‘𝑥))
15 fveq2 6907 . . . . . . . . 9 (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
1615fveq1d 6909 . . . . . . . 8 (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
1716fveq2d 6911 . . . . . . 7 (𝑘 = 𝐾 → (LSubSp‘((DVecH‘𝑘)‘𝑤)) = (LSubSp‘((DVecH‘𝐾)‘𝑤)))
18 fveq2 6907 . . . . . . . . 9 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
19 dihval.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
2018, 19eqtr4di 2793 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
2110breqd 5159 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑞(le‘𝑘)𝑤𝑞 𝑤))
2221notbid 318 . . . . . . . . . 10 (𝑘 = 𝐾 → (¬ 𝑞(le‘𝑘)𝑤 ↔ ¬ 𝑞 𝑤))
23 fveq2 6907 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
24 dihval.j . . . . . . . . . . . . 13 = (join‘𝐾)
2523, 24eqtr4di 2793 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = )
26 eqidd 2736 . . . . . . . . . . . 12 (𝑘 = 𝐾𝑞 = 𝑞)
27 fveq2 6907 . . . . . . . . . . . . . 14 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
28 dihval.m . . . . . . . . . . . . . 14 = (meet‘𝐾)
2927, 28eqtr4di 2793 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (meet‘𝑘) = )
3029oveqd 7448 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (𝑥(meet‘𝑘)𝑤) = (𝑥 𝑤))
3125, 26, 30oveq123d 7452 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = (𝑞 (𝑥 𝑤)))
3231eqeq1d 2737 . . . . . . . . . 10 (𝑘 = 𝐾 → ((𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥 ↔ (𝑞 (𝑥 𝑤)) = 𝑥))
3322, 32anbi12d 632 . . . . . . . . 9 (𝑘 = 𝐾 → ((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) ↔ (¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥)))
3416fveq2d 6911 . . . . . . . . . . 11 (𝑘 = 𝐾 → (LSSum‘((DVecH‘𝑘)‘𝑤)) = (LSSum‘((DVecH‘𝐾)‘𝑤)))
35 fveq2 6907 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (DIsoC‘𝑘) = (DIsoC‘𝐾))
3635fveq1d 6909 . . . . . . . . . . . 12 (𝑘 = 𝐾 → ((DIsoC‘𝑘)‘𝑤) = ((DIsoC‘𝐾)‘𝑤))
3736fveq1d 6909 . . . . . . . . . . 11 (𝑘 = 𝐾 → (((DIsoC‘𝑘)‘𝑤)‘𝑞) = (((DIsoC‘𝐾)‘𝑤)‘𝑞))
3813, 30fveq12d 6914 . . . . . . . . . . 11 (𝑘 = 𝐾 → (((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)) = (((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤)))
3934, 37, 38oveq123d 7452 . . . . . . . . . 10 (𝑘 = 𝐾 → ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))) = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤))))
4039eqeq2d 2746 . . . . . . . . 9 (𝑘 = 𝐾 → (𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))) ↔ 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤)))))
4133, 40imbi12d 344 . . . . . . . 8 (𝑘 = 𝐾 → (((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))) ↔ ((¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤))))))
4220, 41raleqbidv 3344 . . . . . . 7 (𝑘 = 𝐾 → (∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))) ↔ ∀𝑞𝐴 ((¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤))))))
4317, 42riotaeqbidv 7391 . . . . . 6 (𝑘 = 𝐾 → (𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))))) = (𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞𝐴 ((¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤))))))
4411, 14, 43ifbieq12d 4559 . . . . 5 (𝑘 = 𝐾 → if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))))) = if(𝑥 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞𝐴 ((¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤)))))))
457, 44mpteq12dv 5239 . . . 4 (𝑘 = 𝐾 → (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))))))) = (𝑥𝐵 ↦ if(𝑥 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞𝐴 ((¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤))))))))
464, 45mpteq12dv 5239 . . 3 (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))))))) = (𝑤𝐻 ↦ (𝑥𝐵 ↦ if(𝑥 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞𝐴 ((¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤)))))))))
47 df-dih 41212 . . 3 DIsoH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))))))))
4846, 47, 3mptfvmpt 7248 . 2 (𝐾 ∈ V → (DIsoH‘𝐾) = (𝑤𝐻 ↦ (𝑥𝐵 ↦ if(𝑥 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞𝐴 ((¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤)))))))))
491, 48syl 17 1 (𝐾𝑉 → (DIsoH‘𝐾) = (𝑤𝐻 ↦ (𝑥𝐵 ↦ if(𝑥 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞𝐴 ((¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤)))))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  ifcif 4531   class class class wbr 5148  cmpt 5231  cfv 6563  crio 7387  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  LSSumclsm 19667  LSubSpclss 20947  Atomscatm 39245  LHypclh 39967  DVecHcdvh 41061  DIsoBcdib 41121  DIsoCcdic 41155  DIsoHcdih 41211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-dih 41212
This theorem is referenced by:  dihfval  41214
  Copyright terms: Public domain W3C validator