Theorem dihvalb 36526

Description: Value of isomorphism H for a lattice 𝐾 when 𝑋 ≤ 𝑊. (Contributed by NM, 4-Mar-2014.)

Hypotheses

Ref	Expression
dihvalb.b	⊢ 𝐵 = (Base‘𝐾)
dihvalb.l	⊢ ≤ = (le‘𝐾)
dihvalb.h	⊢ 𝐻 = (LHyp‘𝐾)
dihvalb.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihvalb.d	⊢ 𝐷 = ((DIsoB‘𝐾)‘𝑊)

Assertion

Ref	Expression
dihvalb	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (𝐷‘𝑋))

Proof of Theorem dihvalb

Dummy variables 𝑢 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dihvalb.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		dihvalb.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		eqid 2622	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2622	. . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		eqid 2622	. . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
6		dihvalb.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
7		dihvalb.i	. . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
8		dihvalb.d	. . . 4 ⊢ 𝐷 = ((DIsoB‘𝐾)‘𝑊)
9		eqid 2622	. . . 4 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊)
10		eqid 2622	. . . 4 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
11		eqid 2622	. . . 4 ⊢ (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊))
12		eqid 2622	. . . 4 ⊢ (LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	dihval 36521	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) = if(𝑋 ≤ 𝑊, (𝐷‘𝑋), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊))∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) → 𝑢 = ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(𝐷‘(𝑋(meet‘𝐾)𝑊)))))))
14		iftrue 4092	. . 3 ⊢ (𝑋 ≤ 𝑊 → if(𝑋 ≤ 𝑊, (𝐷‘𝑋), (℩𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊))∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑞 ≤ 𝑊 ∧ (𝑞(join‘𝐾)(𝑋(meet‘𝐾)𝑊)) = 𝑋) → 𝑢 = ((((DIsoC‘𝐾)‘𝑊)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑊))(𝐷‘(𝑋(meet‘𝐾)𝑊)))))) = (𝐷‘𝑋))
15	13, 14	sylan9eq 2676	. 2 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 ≤ 𝑊) → (𝐼‘𝑋) = (𝐷‘𝑋))
16	15	anasss 679	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (𝐷‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ifcif 4086   class class class wbr 4653  ‘cfv 5888  ℩crio 6610  (class class class)co 6650  Basecbs 15857  lecple 15948  joincjn 16944  meetcmee 16945  LSSumclsm 18049  LSubSpclss 18932  Atomscatm 34550  LHypclh 35270  DVecHcdvh 36367  DIsoBcdib 36427  DIsoCcdic 36461  DIsoHcdih 36517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-dih 36518

This theorem is referenced by:  dihopelvalbN  36527  dih1dimb  36529  dih2dimb  36533  dih2dimbALTN  36534  dihvalcq2  36536  dihlss  36539  dihord6apre  36545  dihord3  36546  dihord5b  36548  dihord5apre  36551  dih0  36569  dihwN  36578  dihglblem3N  36584  dihmeetlem2N  36588  dih1dimatlem  36618  dihjatcclem4  36710