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Theorem dihffval 35537
 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 3185 . 2 (𝐾𝑉𝐾 ∈ V)
2 fveq2 6103 . . . . 5 (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3 dihval.h . . . . 5 𝐻 = (LHyp‘𝐾)
42, 3syl6eqr 2662 . . . 4 (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5 fveq2 6103 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
6 dihval.b . . . . . 6 𝐵 = (Base‘𝐾)
75, 6syl6eqr 2662 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
8 fveq2 6103 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
9 dihval.l . . . . . . . 8 = (le‘𝐾)
108, 9syl6eqr 2662 . . . . . . 7 (𝑘 = 𝐾 → (le‘𝑘) = )
1110breqd 4594 . . . . . 6 (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑤𝑥 𝑤))
12 fveq2 6103 . . . . . . . 8 (𝑘 = 𝐾 → (DIsoB‘𝑘) = (DIsoB‘𝐾))
1312fveq1d 6105 . . . . . . 7 (𝑘 = 𝐾 → ((DIsoB‘𝑘)‘𝑤) = ((DIsoB‘𝐾)‘𝑤))
1413fveq1d 6105 . . . . . 6 (𝑘 = 𝐾 → (((DIsoB‘𝑘)‘𝑤)‘𝑥) = (((DIsoB‘𝐾)‘𝑤)‘𝑥))
15 fveq2 6103 . . . . . . . . 9 (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
1615fveq1d 6105 . . . . . . . 8 (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
1716fveq2d 6107 . . . . . . 7 (𝑘 = 𝐾 → (LSubSp‘((DVecH‘𝑘)‘𝑤)) = (LSubSp‘((DVecH‘𝐾)‘𝑤)))
18 fveq2 6103 . . . . . . . . 9 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
19 dihval.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
2018, 19syl6eqr 2662 . . . . . . . 8 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
2110breqd 4594 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑞(le‘𝑘)𝑤𝑞 𝑤))
2221notbid 307 . . . . . . . . . 10 (𝑘 = 𝐾 → (¬ 𝑞(le‘𝑘)𝑤 ↔ ¬ 𝑞 𝑤))
23 fveq2 6103 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
24 dihval.j . . . . . . . . . . . . 13 = (join‘𝐾)
2523, 24syl6eqr 2662 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (join‘𝑘) = )
26 eqidd 2611 . . . . . . . . . . . 12 (𝑘 = 𝐾𝑞 = 𝑞)
27 fveq2 6103 . . . . . . . . . . . . . 14 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
28 dihval.m . . . . . . . . . . . . . 14 = (meet‘𝐾)
2927, 28syl6eqr 2662 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (meet‘𝑘) = )
3029oveqd 6566 . . . . . . . . . . . 12 (𝑘 = 𝐾 → (𝑥(meet‘𝑘)𝑤) = (𝑥 𝑤))
3125, 26, 30oveq123d 6570 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = (𝑞 (𝑥 𝑤)))
3231eqeq1d 2612 . . . . . . . . . 10 (𝑘 = 𝐾 → ((𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥 ↔ (𝑞 (𝑥 𝑤)) = 𝑥))
3322, 32anbi12d 743 . . . . . . . . 9 (𝑘 = 𝐾 → ((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) ↔ (¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥)))
3416fveq2d 6107 . . . . . . . . . . 11 (𝑘 = 𝐾 → (LSSum‘((DVecH‘𝑘)‘𝑤)) = (LSSum‘((DVecH‘𝐾)‘𝑤)))
35 fveq2 6103 . . . . . . . . . . . . 13 (𝑘 = 𝐾 → (DIsoC‘𝑘) = (DIsoC‘𝐾))
3635fveq1d 6105 . . . . . . . . . . . 12 (𝑘 = 𝐾 → ((DIsoC‘𝑘)‘𝑤) = ((DIsoC‘𝐾)‘𝑤))
3736fveq1d 6105 . . . . . . . . . . 11 (𝑘 = 𝐾 → (((DIsoC‘𝑘)‘𝑤)‘𝑞) = (((DIsoC‘𝐾)‘𝑤)‘𝑞))
3813, 30fveq12d 6109 . . . . . . . . . . 11 (𝑘 = 𝐾 → (((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)) = (((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤)))
3934, 37, 38oveq123d 6570 . . . . . . . . . 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 2620 . . . . . . . . 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 333 . . . . . . . 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 3129 . . . . . . 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 6514 . . . . . 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 4063 . . . . 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 4663 . . . 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 4663 . . 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 35536 . . 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)))))))))
48 fvex 6113 . . . . 5 (LHyp‘𝐾) ∈ V
493, 48eqeltri 2684 . . . 4 𝐻 ∈ V
5049mptex 6390 . . 3 (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)))))))) ∈ V
5146, 47, 50fvmpt 6191 . 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)))))))))
521, 51syl 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 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  LSSumclsm 17872  LSubSpclss 18753  Atomscatm 33568  LHypclh 34288  DVecHcdvh 35385  DIsoBcdib 35445  DIsoCcdic 35479  DIsoHcdih 35535 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-dih 35536 This theorem is referenced by:  dihfval  35538
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