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Theorem dihfval 38247
Description: 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)
dihval.i 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihval.d 𝐷 = ((DIsoB‘𝐾)‘𝑊)
dihval.c 𝐶 = ((DIsoC‘𝐾)‘𝑊)
dihval.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihval.s 𝑆 = (LSubSp‘𝑈)
dihval.p = (LSSum‘𝑈)
Assertion
Ref Expression
dihfval ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))))))
Distinct variable groups:   𝐴,𝑞   𝑢,𝑞,𝑥,𝐾   𝑥,𝐵   𝑢,𝑆   𝑊,𝑞,𝑢,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑢)   𝐵(𝑢,𝑞)   𝐶(𝑥,𝑢,𝑞)   𝐷(𝑥,𝑢,𝑞)   (𝑥,𝑢,𝑞)   𝑆(𝑥,𝑞)   𝑈(𝑥,𝑢,𝑞)   𝐻(𝑥,𝑢,𝑞)   𝐼(𝑥,𝑢,𝑞)   (𝑥,𝑢,𝑞)   (𝑥,𝑢,𝑞)   (𝑥,𝑢,𝑞)   𝑉(𝑥,𝑢,𝑞)

Proof of Theorem dihfval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 dihval.i . . 3 𝐼 = ((DIsoH‘𝐾)‘𝑊)
2 dihval.b . . . . 5 𝐵 = (Base‘𝐾)
3 dihval.l . . . . 5 = (le‘𝐾)
4 dihval.j . . . . 5 = (join‘𝐾)
5 dihval.m . . . . 5 = (meet‘𝐾)
6 dihval.a . . . . 5 𝐴 = (Atoms‘𝐾)
7 dihval.h . . . . 5 𝐻 = (LHyp‘𝐾)
82, 3, 4, 5, 6, 7dihffval 38246 . . . 4 (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)))))))))
98fveq1d 6665 . . 3 (𝐾𝑉 → ((DIsoH‘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))))))))‘𝑊))
101, 9syl5eq 2865 . 2 (𝐾𝑉𝐼 = ((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))))))))‘𝑊))
11 breq2 5061 . . . . 5 (𝑤 = 𝑊 → (𝑥 𝑤𝑥 𝑊))
12 fveq2 6663 . . . . . . 7 (𝑤 = 𝑊 → ((DIsoB‘𝐾)‘𝑤) = ((DIsoB‘𝐾)‘𝑊))
13 dihval.d . . . . . . 7 𝐷 = ((DIsoB‘𝐾)‘𝑊)
1412, 13syl6eqr 2871 . . . . . 6 (𝑤 = 𝑊 → ((DIsoB‘𝐾)‘𝑤) = 𝐷)
1514fveq1d 6665 . . . . 5 (𝑤 = 𝑊 → (((DIsoB‘𝐾)‘𝑤)‘𝑥) = (𝐷𝑥))
16 fveq2 6663 . . . . . . . . 9 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
17 dihval.u . . . . . . . . 9 𝑈 = ((DVecH‘𝐾)‘𝑊)
1816, 17syl6eqr 2871 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
1918fveq2d 6667 . . . . . . 7 (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = (LSubSp‘𝑈))
20 dihval.s . . . . . . 7 𝑆 = (LSubSp‘𝑈)
2119, 20syl6eqr 2871 . . . . . 6 (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = 𝑆)
22 breq2 5061 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑞 𝑤𝑞 𝑊))
2322notbid 319 . . . . . . . . 9 (𝑤 = 𝑊 → (¬ 𝑞 𝑤 ↔ ¬ 𝑞 𝑊))
24 oveq2 7153 . . . . . . . . . . 11 (𝑤 = 𝑊 → (𝑥 𝑤) = (𝑥 𝑊))
2524oveq2d 7161 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑞 (𝑥 𝑤)) = (𝑞 (𝑥 𝑊)))
2625eqeq1d 2820 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑞 (𝑥 𝑤)) = 𝑥 ↔ (𝑞 (𝑥 𝑊)) = 𝑥))
2723, 26anbi12d 630 . . . . . . . 8 (𝑤 = 𝑊 → ((¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥) ↔ (¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥)))
2818fveq2d 6667 . . . . . . . . . . 11 (𝑤 = 𝑊 → (LSSum‘((DVecH‘𝐾)‘𝑤)) = (LSSum‘𝑈))
29 dihval.p . . . . . . . . . . 11 = (LSSum‘𝑈)
3028, 29syl6eqr 2871 . . . . . . . . . 10 (𝑤 = 𝑊 → (LSSum‘((DVecH‘𝐾)‘𝑤)) = )
31 fveq2 6663 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ((DIsoC‘𝐾)‘𝑤) = ((DIsoC‘𝐾)‘𝑊))
32 dihval.c . . . . . . . . . . . 12 𝐶 = ((DIsoC‘𝐾)‘𝑊)
3331, 32syl6eqr 2871 . . . . . . . . . . 11 (𝑤 = 𝑊 → ((DIsoC‘𝐾)‘𝑤) = 𝐶)
3433fveq1d 6665 . . . . . . . . . 10 (𝑤 = 𝑊 → (((DIsoC‘𝐾)‘𝑤)‘𝑞) = (𝐶𝑞))
3514, 24fveq12d 6670 . . . . . . . . . 10 (𝑤 = 𝑊 → (((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤)) = (𝐷‘(𝑥 𝑊)))
3630, 34, 35oveq123d 7166 . . . . . . . . 9 (𝑤 = 𝑊 → ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤))) = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))
3736eqeq2d 2829 . . . . . . . 8 (𝑤 = 𝑊 → (𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤))) ↔ 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))))
3827, 37imbi12d 346 . . . . . . 7 (𝑤 = 𝑊 → (((¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤)))) ↔ ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))))
3938ralbidv 3194 . . . . . 6 (𝑤 = 𝑊 → (∀𝑞𝐴 ((¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤)))) ↔ ∀𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))))
4021, 39riotaeqbidv 7106 . . . . 5 (𝑤 = 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))))) = (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))))
4111, 15, 40ifbieq12d 4490 . . . 4 (𝑤 = 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)))))) = if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))))))
4241mpteq2dv 5153 . . 3 (𝑤 = 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))))))) = (𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))))))
43 eqid 2818 . . 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)))))))) = (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))))))))
4442, 43, 2mptfvmpt 6981 . 2 (𝑊𝐻 → ((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))))))))‘𝑊) = (𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))))))
4510, 44sylan9eq 2873 1 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  ifcif 4463   class class class wbr 5057  cmpt 5137  cfv 6348  crio 7102  (class class class)co 7145  Basecbs 16471  lecple 16560  joincjn 17542  meetcmee 17543  LSSumclsm 18688  LSubSpclss 19632  Atomscatm 36279  LHypclh 37000  DVecHcdvh 38094  DIsoBcdib 38154  DIsoCcdic 38188  DIsoHcdih 38244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-dih 38245
This theorem is referenced by:  dihval  38248  dihf11lem  38282
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