Theorem dihvalc 41215

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

Hypotheses

Ref	Expression
dihval.b	⊢ 𝐵 = (Base‘𝐾)
dihval.l	⊢ ≤ = (le‘𝐾)
dihval.j	⊢ ∨ = (join‘𝐾)
dihval.m	⊢ ∧ = (meet‘𝐾)
dihval.a	⊢ 𝐴 = (Atoms‘𝐾)
dihval.h	⊢ 𝐻 = (LHyp‘𝐾)
dihval.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihval.d	⊢ 𝐷 = ((DIsoB‘𝐾)‘𝑊)
dihval.c	⊢ 𝐶 = ((DIsoC‘𝐾)‘𝑊)
dihval.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihval.s	⊢ 𝑆 = (LSubSp‘𝑈)
dihval.p	⊢ ⊕ = (LSSum‘𝑈)

Assertion

Ref	Expression
dihvalc	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))))

Distinct variable groups:   𝐴,𝑞   𝑢,𝑞,𝐾   𝑢,𝑆   𝑊,𝑞,𝑢   𝑋,𝑞,𝑢

Allowed substitution hints:   𝐴(𝑢)   𝐵(𝑢,𝑞)   𝐶(𝑢,𝑞)   𝐷(𝑢,𝑞)   ⊕ (𝑢,𝑞)   𝑆(𝑞)   𝑈(𝑢,𝑞)   𝐻(𝑢,𝑞)   𝐼(𝑢,𝑞)   ∨ (𝑢,𝑞)   ≤ (𝑢,𝑞)   ∧ (𝑢,𝑞)   𝑉(𝑢,𝑞)

Proof of Theorem dihvalc

Step	Hyp	Ref	Expression
1		dihval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		dihval.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		dihval.j	. . . 4 ⊢ ∨ = (join‘𝐾)
4		dihval.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
5		dihval.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
6		dihval.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
7		dihval.i	. . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
8		dihval.d	. . . 4 ⊢ 𝐷 = ((DIsoB‘𝐾)‘𝑊)
9		dihval.c	. . . 4 ⊢ 𝐶 = ((DIsoC‘𝐾)‘𝑊)
10		dihval.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
11		dihval.s	. . . 4 ⊢ 𝑆 = (LSubSp‘𝑈)
12		dihval.p	. . . 4 ⊢ ⊕ = (LSSum‘𝑈)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	dihval 41214	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) = if(𝑋 ≤ 𝑊, (𝐷‘𝑋), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))))))
14		iffalse 4539	. . 3 ⊢ (¬ 𝑋 ≤ 𝑊 → if(𝑋 ≤ 𝑊, (𝐷‘𝑋), (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))))) = (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))))
15	13, 14	sylan9eq 2794	. 2 ⊢ ((((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) ∧ ¬ 𝑋 ≤ 𝑊) → (𝐼‘𝑋) = (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))))
16	15	anasss 466	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ifcif 4530   class class class wbr 5147  ‘cfv 6562  ℩crio 7386  (class class class)co 7430  Basecbs 17244  lecple 17304  joincjn 18368  meetcmee 18369  LSSumclsm 19666  LSubSpclss 20946  Atomscatm 39244  LHypclh 39966  DVecHcdvh 41060  DIsoBcdib 41120  DIsoCcdic 41154  DIsoHcdih 41210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-dih 41211

This theorem is referenced by:  dihlsscpre  41216  dihvalcqpre  41217