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Theorem dihval 39225
Description: Value of isomorphism H for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 3-Feb-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
dihval (((𝐾𝑉𝑊𝐻) ∧ 𝑋𝐵) → (𝐼𝑋) = if(𝑋 𝑊, (𝐷𝑋), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))))
Distinct variable groups:   𝐴,𝑞   𝑢,𝑞,𝐾   𝑢,𝑆   𝑊,𝑞,𝑢   𝑋,𝑞,𝑢
Allowed substitution hints:   𝐴(𝑢)   𝐵(𝑢,𝑞)   𝐶(𝑢,𝑞)   𝐷(𝑢,𝑞)   (𝑢,𝑞)   𝑆(𝑞)   𝑈(𝑢,𝑞)   𝐻(𝑢,𝑞)   𝐼(𝑢,𝑞)   (𝑢,𝑞)   (𝑢,𝑞)   (𝑢,𝑞)   𝑉(𝑢,𝑞)

Proof of Theorem dihval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dihval.b . . . 4 𝐵 = (Base‘𝐾)
2 dihval.l . . . 4 = (le‘𝐾)
3 dihval.j . . . 4 = (join‘𝐾)
4 dihval.m . . . 4 = (meet‘𝐾)
5 dihval.a . . . 4 𝐴 = (Atoms‘𝐾)
6 dihval.h . . . 4 𝐻 = (LHyp‘𝐾)
7 dihval.i . . . 4 𝐼 = ((DIsoH‘𝐾)‘𝑊)
8 dihval.d . . . 4 𝐷 = ((DIsoB‘𝐾)‘𝑊)
9 dihval.c . . . 4 𝐶 = ((DIsoC‘𝐾)‘𝑊)
10 dihval.u . . . 4 𝑈 = ((DVecH‘𝐾)‘𝑊)
11 dihval.s . . . 4 𝑆 = (LSubSp‘𝑈)
12 dihval.p . . . 4 = (LSSum‘𝑈)
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12dihfval 39224 . . 3 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))))))
1413fveq1d 6770 . 2 ((𝐾𝑉𝑊𝐻) → (𝐼𝑋) = ((𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))))))‘𝑋))
15 breq1 5081 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑊𝑋 𝑊))
16 fveq2 6768 . . . 4 (𝑥 = 𝑋 → (𝐷𝑥) = (𝐷𝑋))
17 oveq1 7275 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥 𝑊) = (𝑋 𝑊))
1817oveq2d 7284 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑞 (𝑥 𝑊)) = (𝑞 (𝑋 𝑊)))
19 id 22 . . . . . . . . 9 (𝑥 = 𝑋𝑥 = 𝑋)
2018, 19eqeq12d 2755 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑞 (𝑥 𝑊)) = 𝑥 ↔ (𝑞 (𝑋 𝑊)) = 𝑋))
2120anbi2d 628 . . . . . . 7 (𝑥 = 𝑋 → ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) ↔ (¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋)))
22 fvoveq1 7291 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐷‘(𝑥 𝑊)) = (𝐷‘(𝑋 𝑊)))
2322oveq2d 7284 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐶𝑞) (𝐷‘(𝑥 𝑊))) = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))
2423eqeq2d 2750 . . . . . . 7 (𝑥 = 𝑋 → (𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))) ↔ 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))
2521, 24imbi12d 344 . . . . . 6 (𝑥 = 𝑋 → (((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))) ↔ ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))))
2625ralbidv 3122 . . . . 5 (𝑥 = 𝑋 → (∀𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))) ↔ ∀𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))))
2726riotabidv 7227 . . . 4 (𝑥 = 𝑋 → (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))) = (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))))
2815, 16, 27ifbieq12d 4492 . . 3 (𝑥 = 𝑋 → if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))))) = if(𝑋 𝑊, (𝐷𝑋), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))))
29 eqid 2739 . . 3 (𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))))) = (𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))))))
30 fvex 6781 . . . 4 (𝐷𝑋) ∈ V
31 riotaex 7229 . . . 4 (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))) ∈ V
3230, 31ifex 4514 . . 3 if(𝑋 𝑊, (𝐷𝑋), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))) ∈ V
3328, 29, 32fvmpt 6869 . 2 (𝑋𝐵 → ((𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))))))‘𝑋) = if(𝑋 𝑊, (𝐷𝑋), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))))
3414, 33sylan9eq 2799 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑋𝐵) → (𝐼𝑋) = 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 395   = wceq 1541  wcel 2109  wral 3065  ifcif 4464   class class class wbr 5078  cmpt 5161  cfv 6430  crio 7224  (class class class)co 7268  Basecbs 16893  lecple 16950  joincjn 18010  meetcmee 18011  LSSumclsm 19220  LSubSpclss 20174  Atomscatm 37256  LHypclh 37977  DVecHcdvh 39071  DIsoBcdib 39131  DIsoCcdic 39165  DIsoHcdih 39221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pr 5355
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-ov 7271  df-dih 39222
This theorem is referenced by:  dihvalc  39226  dihvalb  39230
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