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Theorem dihffval 38545
 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 3459 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6646 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 dihval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3eqtr4di 2851 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6646 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
6 dihval.b . . . . . 6 𝐵 = (Base‘𝐾)
75, 6eqtr4di 2851 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
8 fveq2 6646 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
9 dihval.l . . . . . . . 8 = (le‘𝐾)
108, 9eqtr4di 2851 . . . . . . 7 (𝑘 = 𝐾 → (le‘𝑘) = )
1110breqd 5042 . . . . . 6 (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑤𝑥 𝑤))
12 fveq2 6646 . . . . . . . 8 (𝑘 = 𝐾 → (DIsoB‘𝑘) = (DIsoB‘𝐾))
1312fveq1d 6648 . . . . . . 7 (𝑘 = 𝐾 → ((DIsoB‘𝑘)‘𝑤) = ((DIsoB‘𝐾)‘𝑤))
1413fveq1d 6648 . . . . . 6 (𝑘 = 𝐾 → (((DIsoB‘𝑘)‘𝑤)‘𝑥) = (((DIsoB‘𝐾)‘𝑤)‘𝑥))
15 fveq2 6646 . . . . . . . . 9 (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
1615fveq1d 6648 . . . . . . . 8 (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
1716fveq2d 6650 . . . . . . 7 (𝑘 = 𝐾 → (LSubSp‘((DVecH‘𝑘)‘𝑤)) = (LSubSp‘((DVecH‘𝐾)‘𝑤)))
18 fveq2 6646 . . . . . . . . 9 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
19 dihval.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
2018, 19eqtr4di 2851 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
2110breqd 5042 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑞(le‘𝑘)𝑤𝑞 𝑤))
2221notbid 321 . . . . . . . . . 10 (𝑘 = 𝐾 → (¬ 𝑞(le‘𝑘)𝑤 ↔ ¬ 𝑞 𝑤))
23 fveq2 6646 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
24 dihval.j . . . . . . . . . . . . 13 = (join‘𝐾)
2523, 24eqtr4di 2851 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = )
26 eqidd 2799 . . . . . . . . . . . 12 (𝑘 = 𝐾𝑞 = 𝑞)
27 fveq2 6646 . . . . . . . . . . . . . 14 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
28 dihval.m . . . . . . . . . . . . . 14 = (meet‘𝐾)
2927, 28eqtr4di 2851 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (meet‘𝑘) = )
3029oveqd 7153 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (𝑥(meet‘𝑘)𝑤) = (𝑥 𝑤))
3125, 26, 30oveq123d 7157 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = (𝑞 (𝑥 𝑤)))
3231eqeq1d 2800 . . . . . . . . . 10 (𝑘 = 𝐾 → ((𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥 ↔ (𝑞 (𝑥 𝑤)) = 𝑥))
3322, 32anbi12d 633 . . . . . . . . 9 (𝑘 = 𝐾 → ((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) ↔ (¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥)))
3416fveq2d 6650 . . . . . . . . . . 11 (𝑘 = 𝐾 → (LSSum‘((DVecH‘𝑘)‘𝑤)) = (LSSum‘((DVecH‘𝐾)‘𝑤)))
35 fveq2 6646 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (DIsoC‘𝑘) = (DIsoC‘𝐾))
3635fveq1d 6648 . . . . . . . . . . . 12 (𝑘 = 𝐾 → ((DIsoC‘𝑘)‘𝑤) = ((DIsoC‘𝐾)‘𝑤))
3736fveq1d 6648 . . . . . . . . . . 11 (𝑘 = 𝐾 → (((DIsoC‘𝑘)‘𝑤)‘𝑞) = (((DIsoC‘𝐾)‘𝑤)‘𝑞))
3813, 30fveq12d 6653 . . . . . . . . . . 11 (𝑘 = 𝐾 → (((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)) = (((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤)))
3934, 37, 38oveq123d 7157 . . . . . . . . . 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 2809 . . . . . . . . 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 348 . . . . . . . 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 3354 . . . . . . 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 7097 . . . . . 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 4452 . . . . 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 5116 . . . 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 5116 . . 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 38544 . . 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 6969 . 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 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441  ifcif 4425   class class class wbr 5031   ↦ cmpt 5111  ‘cfv 6325  ℩crio 7093  (class class class)co 7136  Basecbs 16478  lecple 16567  joincjn 17549  meetcmee 17550  LSSumclsm 18755  LSubSpclss 19700  Atomscatm 36578  LHypclh 37299  DVecHcdvh 38393  DIsoBcdib 38453  DIsoCcdic 38487  DIsoHcdih 38543 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pr 5296 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-dih 38544 This theorem is referenced by:  dihfval  38546
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