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Theorem dihfval 35337
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 35336 . . . 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 6086 . . 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 2651 . 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 4577 . . . . 5 (𝑤 = 𝑊 → (𝑥 𝑤𝑥 𝑊))
12 fveq2 6084 . . . . . . 7 (𝑤 = 𝑊 → ((DIsoB‘𝐾)‘𝑤) = ((DIsoB‘𝐾)‘𝑊))
13 dihval.d . . . . . . 7 𝐷 = ((DIsoB‘𝐾)‘𝑊)
1412, 13syl6eqr 2657 . . . . . 6 (𝑤 = 𝑊 → ((DIsoB‘𝐾)‘𝑤) = 𝐷)
1514fveq1d 6086 . . . . 5 (𝑤 = 𝑊 → (((DIsoB‘𝐾)‘𝑤)‘𝑥) = (𝐷𝑥))
16 fveq2 6084 . . . . . . . . 9 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
17 dihval.u . . . . . . . . 9 𝑈 = ((DVecH‘𝐾)‘𝑊)
1816, 17syl6eqr 2657 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
1918fveq2d 6088 . . . . . . 7 (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = (LSubSp‘𝑈))
20 dihval.s . . . . . . 7 𝑆 = (LSubSp‘𝑈)
2119, 20syl6eqr 2657 . . . . . 6 (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = 𝑆)
22 breq2 4577 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑞 𝑤𝑞 𝑊))
2322notbid 306 . . . . . . . . 9 (𝑤 = 𝑊 → (¬ 𝑞 𝑤 ↔ ¬ 𝑞 𝑊))
24 oveq2 6531 . . . . . . . . . . 11 (𝑤 = 𝑊 → (𝑥 𝑤) = (𝑥 𝑊))
2524oveq2d 6539 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑞 (𝑥 𝑤)) = (𝑞 (𝑥 𝑊)))
2625eqeq1d 2607 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑞 (𝑥 𝑤)) = 𝑥 ↔ (𝑞 (𝑥 𝑊)) = 𝑥))
2723, 26anbi12d 742 . . . . . . . 8 (𝑤 = 𝑊 → ((¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥) ↔ (¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥)))
2818fveq2d 6088 . . . . . . . . . . 11 (𝑤 = 𝑊 → (LSSum‘((DVecH‘𝐾)‘𝑤)) = (LSSum‘𝑈))
29 dihval.p . . . . . . . . . . 11 = (LSSum‘𝑈)
3028, 29syl6eqr 2657 . . . . . . . . . 10 (𝑤 = 𝑊 → (LSSum‘((DVecH‘𝐾)‘𝑤)) = )
31 fveq2 6084 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ((DIsoC‘𝐾)‘𝑤) = ((DIsoC‘𝐾)‘𝑊))
32 dihval.c . . . . . . . . . . . 12 𝐶 = ((DIsoC‘𝐾)‘𝑊)
3331, 32syl6eqr 2657 . . . . . . . . . . 11 (𝑤 = 𝑊 → ((DIsoC‘𝐾)‘𝑤) = 𝐶)
3433fveq1d 6086 . . . . . . . . . 10 (𝑤 = 𝑊 → (((DIsoC‘𝐾)‘𝑤)‘𝑞) = (𝐶𝑞))
3514, 24fveq12d 6090 . . . . . . . . . 10 (𝑤 = 𝑊 → (((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤)) = (𝐷‘(𝑥 𝑊)))
3630, 34, 35oveq123d 6544 . . . . . . . . 9 (𝑤 = 𝑊 → ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤))) = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))
3736eqeq2d 2615 . . . . . . . 8 (𝑤 = 𝑊 → (𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤))) ↔ 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))))
3827, 37imbi12d 332 . . . . . . 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 2964 . . . . . 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 6488 . . . . 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 4058 . . . 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 4663 . . 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 2605 . . 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))))))))
44 fvex 6094 . . . . 5 (Base‘𝐾) ∈ V
452, 44eqeltri 2679 . . . 4 𝐵 ∈ V
4645mptex 6364 . . 3 (𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))))) ∈ V
4742, 43, 46fvmpt 6172 . 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))))))))
4810, 47sylan9eq 2659 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 382   = wceq 1474  wcel 1975  wral 2891  Vcvv 3168  ifcif 4031   class class class wbr 4573  cmpt 4633  cfv 5786  crio 6484  (class class class)co 6523  Basecbs 15637  lecple 15717  joincjn 16709  meetcmee 16710  LSSumclsm 17814  LSubSpclss 18695  Atomscatm 33367  LHypclh 34087  DVecHcdvh 35184  DIsoBcdib 35244  DIsoCcdic 35278  DIsoHcdih 35334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-dih 35335
This theorem is referenced by:  dihval  35338  dihf11lem  35372
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