Theorem dihvalb 38388

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 2821	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2821	. . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		eqid 2821	. . . 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 2821	. . . 4 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊)
10		eqid 2821	. . . 4 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
11		eqid 2821	. . . 4 ⊢ (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊))
12		eqid 2821	. . . 4 ⊢ (LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	dihval 38383	. . 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 4473	. . 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 2876	. 2 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 ≤ 𝑊) → (𝐼‘𝑋) = (𝐷‘𝑋))
16	15	anasss 469	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (𝐷‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ifcif 4467   class class class wbr 5066  ‘cfv 6355  ℩crio 7113  (class class class)co 7156  Basecbs 16483  lecple 16572  joincjn 17554  meetcmee 17555  LSSumclsm 18759  LSubSpclss 19703  Atomscatm 36414  LHypclh 37135  DVecHcdvh 38229  DIsoBcdib 38289  DIsoCcdic 38323  DIsoHcdih 38379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-dih 38380

This theorem is referenced by:  dihopelvalbN  38389  dih1dimb  38391  dih2dimb  38395  dih2dimbALTN  38396  dihvalcq2  38398  dihlss  38401  dihord6apre  38407  dihord3  38408  dihord5b  38410  dihord5apre  38413  dih0  38431  dihwN  38440  dihglblem3N  38446  dihmeetlem2N  38450  dih1dimatlem  38480  dihjatcclem4  38572