Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihval Structured version   Visualization version   GIF version

Theorem dihval 36838
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 36837 . . 3 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))))))
1413fveq1d 6231 . 2 ((𝐾𝑉𝑊𝐻) → (𝐼𝑋) = ((𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))))))‘𝑋))
15 breq1 4688 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑊𝑋 𝑊))
16 fveq2 6229 . . . 4 (𝑥 = 𝑋 → (𝐷𝑥) = (𝐷𝑋))
17 oveq1 6697 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥 𝑊) = (𝑋 𝑊))
1817oveq2d 6706 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑞 (𝑥 𝑊)) = (𝑞 (𝑋 𝑊)))
19 id 22 . . . . . . . . 9 (𝑥 = 𝑋𝑥 = 𝑋)
2018, 19eqeq12d 2666 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑞 (𝑥 𝑊)) = 𝑥 ↔ (𝑞 (𝑋 𝑊)) = 𝑋))
2120anbi2d 740 . . . . . . 7 (𝑥 = 𝑋 → ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) ↔ (¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋)))
2217fveq2d 6233 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐷‘(𝑥 𝑊)) = (𝐷‘(𝑋 𝑊)))
2322oveq2d 6706 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐶𝑞) (𝐷‘(𝑥 𝑊))) = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))
2423eqeq2d 2661 . . . . . . 7 (𝑥 = 𝑋 → (𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))) ↔ 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))
2521, 24imbi12d 333 . . . . . 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 3015 . . . . 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 6653 . . . 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 4146 . . 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 2651 . . 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 6239 . . . 4 (𝐷𝑋) ∈ V
31 riotaex 6655 . . . 4 (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))) ∈ V
3230, 31ifex 4189 . . 3 if(𝑋 𝑊, (𝐷𝑋), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))) ∈ V
3328, 29, 32fvmpt 6321 . 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 2705 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 383   = wceq 1523  wcel 2030  wral 2941  ifcif 4119   class class class wbr 4685  cmpt 4762  cfv 5926  crio 6650  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  meetcmee 16992  LSSumclsm 18095  LSubSpclss 18980  Atomscatm 34868  LHypclh 35588  DVecHcdvh 36684  DIsoBcdib 36744  DIsoCcdic 36778  DIsoHcdih 36834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-dih 36835
This theorem is referenced by:  dihvalc  36839  dihvalb  36843
  Copyright terms: Public domain W3C validator