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Theorem dihfval 37119
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 37118 . . . 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 6376 . . 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, 9syl5eq 2810 . 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 4812 . . . . 5 (𝑤 = 𝑊 → (𝑥 𝑤𝑥 𝑊))
12 fveq2 6374 . . . . . . 7 (𝑤 = 𝑊 → ((DIsoB‘𝐾)‘𝑤) = ((DIsoB‘𝐾)‘𝑊))
13 dihval.d . . . . . . 7 𝐷 = ((DIsoB‘𝐾)‘𝑊)
1412, 13syl6eqr 2816 . . . . . 6 (𝑤 = 𝑊 → ((DIsoB‘𝐾)‘𝑤) = 𝐷)
1514fveq1d 6376 . . . . 5 (𝑤 = 𝑊 → (((DIsoB‘𝐾)‘𝑤)‘𝑥) = (𝐷𝑥))
16 fveq2 6374 . . . . . . . . 9 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
17 dihval.u . . . . . . . . 9 𝑈 = ((DVecH‘𝐾)‘𝑊)
1816, 17syl6eqr 2816 . . . . . . . 8 (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
1918fveq2d 6378 . . . . . . 7 (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = (LSubSp‘𝑈))
20 dihval.s . . . . . . 7 𝑆 = (LSubSp‘𝑈)
2119, 20syl6eqr 2816 . . . . . 6 (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = 𝑆)
22 breq2 4812 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑞 𝑤𝑞 𝑊))
2322notbid 309 . . . . . . . . 9 (𝑤 = 𝑊 → (¬ 𝑞 𝑤 ↔ ¬ 𝑞 𝑊))
24 oveq2 6849 . . . . . . . . . . 11 (𝑤 = 𝑊 → (𝑥 𝑤) = (𝑥 𝑊))
2524oveq2d 6857 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑞 (𝑥 𝑤)) = (𝑞 (𝑥 𝑊)))
2625eqeq1d 2766 . . . . . . . . 9 (𝑤 = 𝑊 → ((𝑞 (𝑥 𝑤)) = 𝑥 ↔ (𝑞 (𝑥 𝑊)) = 𝑥))
2723, 26anbi12d 624 . . . . . . . 8 (𝑤 = 𝑊 → ((¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥) ↔ (¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥)))
2818fveq2d 6378 . . . . . . . . . . 11 (𝑤 = 𝑊 → (LSSum‘((DVecH‘𝐾)‘𝑤)) = (LSSum‘𝑈))
29 dihval.p . . . . . . . . . . 11 = (LSSum‘𝑈)
3028, 29syl6eqr 2816 . . . . . . . . . 10 (𝑤 = 𝑊 → (LSSum‘((DVecH‘𝐾)‘𝑤)) = )
31 fveq2 6374 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ((DIsoC‘𝐾)‘𝑤) = ((DIsoC‘𝐾)‘𝑊))
32 dihval.c . . . . . . . . . . . 12 𝐶 = ((DIsoC‘𝐾)‘𝑊)
3331, 32syl6eqr 2816 . . . . . . . . . . 11 (𝑤 = 𝑊 → ((DIsoC‘𝐾)‘𝑤) = 𝐶)
3433fveq1d 6376 . . . . . . . . . 10 (𝑤 = 𝑊 → (((DIsoC‘𝐾)‘𝑤)‘𝑞) = (𝐶𝑞))
3514, 24fveq12d 6381 . . . . . . . . . 10 (𝑤 = 𝑊 → (((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤)) = (𝐷‘(𝑥 𝑊)))
3630, 34, 35oveq123d 6862 . . . . . . . . 9 (𝑤 = 𝑊 → ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤))) = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))
3736eqeq2d 2774 . . . . . . . 8 (𝑤 = 𝑊 → (𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤))) ↔ 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊)))))
3827, 37imbi12d 335 . . . . . . 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 3132 . . . . . 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 6805 . . . . 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 4269 . . . 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 4903 . . 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 2764 . . 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 6682 . 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 2818 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 384   = wceq 1652  wcel 2155  wral 3054  ifcif 4242   class class class wbr 4808  cmpt 4887  cfv 6067  crio 6801  (class class class)co 6841  Basecbs 16131  lecple 16222  joincjn 17211  meetcmee 17212  LSSumclsm 18314  LSubSpclss 19200  Atomscatm 35151  LHypclh 35872  DVecHcdvh 36966  DIsoBcdib 37026  DIsoCcdic 37060  DIsoHcdih 37116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pr 5061
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-dih 37117
This theorem is referenced by:  dihval  37120  dihf11lem  37154
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