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Theorem dihfval 41233
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 41232 . . . 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 6908 . . 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, 9eqtrid 2789 . 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 5147 . . . . 5 (𝑤 = 𝑊 → (𝑥 𝑤𝑥 𝑊))
12 fveq2 6906 . . . . . . 7 (𝑤 = 𝑊 → ((DIsoB‘𝐾)‘𝑤) = ((DIsoB‘𝐾)‘𝑊))
13 dihval.d . . . . . . 7 𝐷 = ((DIsoB‘𝐾)‘𝑊)
1412, 13eqtr4di 2795 . . . . . 6 (𝑤 = 𝑊 → ((DIsoB‘𝐾)‘𝑤) = 𝐷)
1514fveq1d 6908 . . . . 5 (𝑤 = 𝑊 → (((DIsoB‘𝐾)‘𝑤)‘𝑥) = (𝐷𝑥))
16 fveq2 6906 . . . . . . . . 9 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
17 dihval.u . . . . . . . . 9 𝑈 = ((DVecH‘𝐾)‘𝑊)
1816, 17eqtr4di 2795 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
1918fveq2d 6910 . . . . . . 7 (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = (LSubSp‘𝑈))
20 dihval.s . . . . . . 7 𝑆 = (LSubSp‘𝑈)
2119, 20eqtr4di 2795 . . . . . 6 (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = 𝑆)
22 breq2 5147 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑞 𝑤𝑞 𝑊))
2322notbid 318 . . . . . . . . 9 (𝑤 = 𝑊 → (¬ 𝑞 𝑤 ↔ ¬ 𝑞 𝑊))
24 oveq2 7439 . . . . . . . . . . 11 (𝑤 = 𝑊 → (𝑥 𝑤) = (𝑥 𝑊))
2524oveq2d 7447 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑞 (𝑥 𝑤)) = (𝑞 (𝑥 𝑊)))
2625eqeq1d 2739 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑞 (𝑥 𝑤)) = 𝑥 ↔ (𝑞 (𝑥 𝑊)) = 𝑥))
2723, 26anbi12d 632 . . . . . . . 8 (𝑤 = 𝑊 → ((¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥) ↔ (¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥)))
2818fveq2d 6910 . . . . . . . . . . 11 (𝑤 = 𝑊 → (LSSum‘((DVecH‘𝐾)‘𝑤)) = (LSSum‘𝑈))
29 dihval.p . . . . . . . . . . 11 = (LSSum‘𝑈)
3028, 29eqtr4di 2795 . . . . . . . . . 10 (𝑤 = 𝑊 → (LSSum‘((DVecH‘𝐾)‘𝑤)) = )
31 fveq2 6906 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ((DIsoC‘𝐾)‘𝑤) = ((DIsoC‘𝐾)‘𝑊))
32 dihval.c . . . . . . . . . . . 12 𝐶 = ((DIsoC‘𝐾)‘𝑊)
3331, 32eqtr4di 2795 . . . . . . . . . . 11 (𝑤 = 𝑊 → ((DIsoC‘𝐾)‘𝑤) = 𝐶)
3433fveq1d 6908 . . . . . . . . . 10 (𝑤 = 𝑊 → (((DIsoC‘𝐾)‘𝑤)‘𝑞) = (𝐶𝑞))
3514, 24fveq12d 6913 . . . . . . . . . 10 (𝑤 = 𝑊 → (((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤)) = (𝐷‘(𝑥 𝑊)))
3630, 34, 35oveq123d 7452 . . . . . . . . 9 (𝑤 = 𝑊 → ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤))) = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))
3736eqeq2d 2748 . . . . . . . 8 (𝑤 = 𝑊 → (𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤))) ↔ 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))))
3827, 37imbi12d 344 . . . . . . 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 3178 . . . . . 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 7391 . . . . 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 4554 . . . 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 5244 . . 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 2737 . . 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))))))))
4442, 43, 2mptfvmpt 7248 . 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))))))))
4510, 44sylan9eq 2797 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 395   = wceq 1540  wcel 2108  wral 3061  ifcif 4525   class class class wbr 5143  cmpt 5225  cfv 6561  crio 7387  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  meetcmee 18358  LSSumclsm 19652  LSubSpclss 20929  Atomscatm 39264  LHypclh 39986  DVecHcdvh 41080  DIsoBcdib 41140  DIsoCcdic 41174  DIsoHcdih 41230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-dih 41231
This theorem is referenced by:  dihval  41234  dihf11lem  41268
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