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Theorem dihval 38355
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 38354 . . 3 ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))))))
1413fveq1d 6665 . 2 ((𝐾𝑉𝑊𝐻) → (𝐼𝑋) = ((𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))))))‘𝑋))
15 breq1 5060 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑊𝑋 𝑊))
16 fveq2 6663 . . . 4 (𝑥 = 𝑋 → (𝐷𝑥) = (𝐷𝑋))
17 oveq1 7155 . . . . . . . . . 10 (𝑥 = 𝑋 → (𝑥 𝑊) = (𝑋 𝑊))
1817oveq2d 7164 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑞 (𝑥 𝑊)) = (𝑞 (𝑋 𝑊)))
19 id 22 . . . . . . . . 9 (𝑥 = 𝑋𝑥 = 𝑋)
2018, 19eqeq12d 2835 . . . . . . . 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 7171 . . . . . . . . 9 (𝑥 = 𝑋 → (𝐷‘(𝑥 𝑊)) = (𝐷‘(𝑋 𝑊)))
2322oveq2d 7164 . . . . . . . 8 (𝑥 = 𝑋 → ((𝐶𝑞) (𝐷‘(𝑥 𝑊))) = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))
2423eqeq2d 2830 . . . . . . 7 (𝑥 = 𝑋 → (𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))) ↔ 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))
2521, 24imbi12d 347 . . . . . 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 3195 . . . . 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 7108 . . . 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 2819 . . 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 6676 . . . 4 (𝐷𝑋) ∈ V
31 riotaex 7110 . . . 4 (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))) ∈ V
3230, 31ifex 4513 . . 3 if(𝑋 𝑊, (𝐷𝑋), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))) ∈ V
3328, 29, 32fvmpt 6761 . 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 2874 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 398   = wceq 1530  wcel 2107  wral 3136  ifcif 4465   class class class wbr 5057  cmpt 5137  cfv 6348  crio 7105  (class class class)co 7148  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  LSSumclsm 18751  LSubSpclss 19695  Atomscatm 36386  LHypclh 37107  DVecHcdvh 38201  DIsoBcdib 38261  DIsoCcdic 38295  DIsoHcdih 38351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  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 7106  df-ov 7151  df-dih 38352
This theorem is referenced by:  dihvalc  38356  dihvalb  38360
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