Theorem dihvalb 41613

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 2737	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2737	. . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾)
5		eqid 2737	. . . 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 2737	. . . 4 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊)
10		eqid 2737	. . . 4 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
11		eqid 2737	. . . 4 ⊢ (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊))
12		eqid 2737	. . . 4 ⊢ (LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	dihval 41608	. . 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 4487	. . 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 2792	. 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 1542   ∈ wcel 2114  ∀wral 3052  ifcif 4481   class class class wbr 5100  ‘cfv 6500  ℩crio 7324  (class class class)co 7368  Basecbs 17148  lecple 17196  joincjn 18246  meetcmee 18247  LSSumclsm 19575  LSubSpclss 20894  Atomscatm 39639  LHypclh 40360  DVecHcdvh 41454  DIsoBcdib 41514  DIsoCcdic 41548  DIsoHcdih 41604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-dih 41605

This theorem is referenced by:  dihopelvalbN  41614  dih1dimb  41616  dih2dimb  41620  dih2dimbALTN  41621  dihvalcq2  41623  dihlss  41626  dihord6apre  41632  dihord3  41633  dihord5b  41635  dihord5apre  41638  dih0  41656  dihwN  41665  dihglblem3N  41671  dihmeetlem2N  41675  dih1dimatlem  41705  dihjatcclem4  41797