Theorem dihvalb 41220

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 2729	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2729	. . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		eqid 2729	. . . 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 2729	. . . 4 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊)
10		eqid 2729	. . . 4 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
11		eqid 2729	. . . 4 ⊢ (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊))
12		eqid 2729	. . . 4 ⊢ (LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	dihval 41215	. . 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 4482	. . 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 2784	. 2 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 ≤ 𝑊) → (𝐼‘𝑋) = (𝐷‘𝑋))
16	15	anasss 466	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (𝐷‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ifcif 4476   class class class wbr 5092  ‘cfv 6482  ℩crio 7305  (class class class)co 7349  Basecbs 17120  lecple 17168  joincjn 18217  meetcmee 18218  LSSumclsm 19513  LSubSpclss 20834  Atomscatm 39246  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 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-dih 41212

This theorem is referenced by:  dihopelvalbN  41221  dih1dimb  41223  dih2dimb  41227  dih2dimbALTN  41228  dihvalcq2  41230  dihlss  41233  dihord6apre  41239  dihord3  41240  dihord5b  41242  dihord5apre  41245  dih0  41263  dihwN  41272  dihglblem3N  41278  dihmeetlem2N  41282  dih1dimatlem  41312  dihjatcclem4  41404