Theorem dihvalb 40860

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 2725	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2725	. . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		eqid 2725	. . . 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 2725	. . . 4 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊)
10		eqid 2725	. . . 4 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
11		eqid 2725	. . . 4 ⊢ (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊))
12		eqid 2725	. . . 4 ⊢ (LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	dihval 40855	. . 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 4536	. . 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 2785	. 2 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 ≤ 𝑊) → (𝐼‘𝑋) = (𝐷‘𝑋))
16	15	anasss 465	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (𝐷‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3050  ifcif 4530   class class class wbr 5149  ‘cfv 6549  ℩crio 7374  (class class class)co 7419  Basecbs 17199  lecple 17259  joincjn 18322  meetcmee 18323  LSSumclsm 19618  LSubSpclss 20844  Atomscatm 38885  LHypclh 39607  DVecHcdvh 40701  DIsoBcdib 40761  DIsoCcdic 40795  DIsoHcdih 40851

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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-dih 40852

This theorem is referenced by:  dihopelvalbN  40861  dih1dimb  40863  dih2dimb  40867  dih2dimbALTN  40868  dihvalcq2  40870  dihlss  40873  dihord6apre  40879  dihord3  40880  dihord5b  40882  dihord5apre  40885  dih0  40903  dihwN  40912  dihglblem3N  40918  dihmeetlem2N  40922  dih1dimatlem  40952  dihjatcclem4  41044