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Theorem dihval 41215
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 41214 . . 3 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))))))
1413fveq1d 6909 . 2 ((𝐾𝑉𝑊𝐻) → (𝐼𝑋) = ((𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))))))‘𝑋))
15 breq1 5151 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑊𝑋 𝑊))
16 fveq2 6907 . . . 4 (𝑥 = 𝑋 → (𝐷𝑥) = (𝐷𝑋))
17 oveq1 7438 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥 𝑊) = (𝑋 𝑊))
1817oveq2d 7447 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑞 (𝑥 𝑊)) = (𝑞 (𝑋 𝑊)))
19 id 22 . . . . . . . . 9 (𝑥 = 𝑋𝑥 = 𝑋)
2018, 19eqeq12d 2751 . . . . . . . 8 (𝑥 = 𝑋 → ((𝑞 (𝑥 𝑊)) = 𝑥 ↔ (𝑞 (𝑋 𝑊)) = 𝑋))
2120anbi2d 630 . . . . . . 7 (𝑥 = 𝑋 → ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) ↔ (¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋)))
22 fvoveq1 7454 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐷‘(𝑥 𝑊)) = (𝐷‘(𝑋 𝑊)))
2322oveq2d 7447 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐶𝑞) (𝐷‘(𝑥 𝑊))) = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))
2423eqeq2d 2746 . . . . . . 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 3176 . . . . 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 7390 . . . 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 4559 . . 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 2735 . . 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 6920 . . . 4 (𝐷𝑋) ∈ V
31 riotaex 7392 . . . 4 (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))) ∈ V
3230, 31ifex 4581 . . 3 if(𝑋 𝑊, (𝐷𝑋), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))) ∈ V
3328, 29, 32fvmpt 7016 . 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 2795 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 1537  wcel 2106  wral 3059  ifcif 4531   class class class wbr 5148  cmpt 5231  cfv 6563  crio 7387  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  LSSumclsm 19667  LSubSpclss 20947  Atomscatm 39245  LHypclh 39967  DVecHcdvh 41061  DIsoBcdib 41121  DIsoCcdic 41155  DIsoHcdih 41211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-dih 41212
This theorem is referenced by:  dihvalc  41216  dihvalb  41220
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