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Definition df-dih 35339
Description: Define isomorphism H. (Contributed by NM, 28-Jan-2014.)
Assertion
Ref Expression
df-dih DIsoH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))))))))
Distinct variable group:   𝑘,𝑞,𝑤,𝑢,𝑥

Detailed syntax breakdown of Definition df-dih
StepHypRef Expression
1 cdih 35338 . 2 class DIsoH
2 vk . . 3 setvar 𝑘
3 cvv 3172 . . 3 class V
4 vw . . . 4 setvar 𝑤
52cv 1473 . . . . 5 class 𝑘
6 clh 34091 . . . . 5 class LHyp
75, 6cfv 5790 . . . 4 class (LHyp‘𝑘)
8 vx . . . . 5 setvar 𝑥
9 cbs 15641 . . . . . 6 class Base
105, 9cfv 5790 . . . . 5 class (Base‘𝑘)
118cv 1473 . . . . . . 7 class 𝑥
124cv 1473 . . . . . . 7 class 𝑤
13 cple 15721 . . . . . . . 8 class le
145, 13cfv 5790 . . . . . . 7 class (le‘𝑘)
1511, 12, 14wbr 4577 . . . . . 6 wff 𝑥(le‘𝑘)𝑤
16 cdib 35248 . . . . . . . . 9 class DIsoB
175, 16cfv 5790 . . . . . . . 8 class (DIsoB‘𝑘)
1812, 17cfv 5790 . . . . . . 7 class ((DIsoB‘𝑘)‘𝑤)
1911, 18cfv 5790 . . . . . 6 class (((DIsoB‘𝑘)‘𝑤)‘𝑥)
20 vq . . . . . . . . . . . . 13 setvar 𝑞
2120cv 1473 . . . . . . . . . . . 12 class 𝑞
2221, 12, 14wbr 4577 . . . . . . . . . . 11 wff 𝑞(le‘𝑘)𝑤
2322wn 3 . . . . . . . . . 10 wff ¬ 𝑞(le‘𝑘)𝑤
24 cmee 16714 . . . . . . . . . . . . . 14 class meet
255, 24cfv 5790 . . . . . . . . . . . . 13 class (meet‘𝑘)
2611, 12, 25co 6527 . . . . . . . . . . . 12 class (𝑥(meet‘𝑘)𝑤)
27 cjn 16713 . . . . . . . . . . . . 13 class join
285, 27cfv 5790 . . . . . . . . . . . 12 class (join‘𝑘)
2921, 26, 28co 6527 . . . . . . . . . . 11 class (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤))
3029, 11wceq 1474 . . . . . . . . . 10 wff (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥
3123, 30wa 382 . . . . . . . . 9 wff 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥)
32 vu . . . . . . . . . . 11 setvar 𝑢
3332cv 1473 . . . . . . . . . 10 class 𝑢
34 cdic 35282 . . . . . . . . . . . . . 14 class DIsoC
355, 34cfv 5790 . . . . . . . . . . . . 13 class (DIsoC‘𝑘)
3612, 35cfv 5790 . . . . . . . . . . . 12 class ((DIsoC‘𝑘)‘𝑤)
3721, 36cfv 5790 . . . . . . . . . . 11 class (((DIsoC‘𝑘)‘𝑤)‘𝑞)
3826, 18cfv 5790 . . . . . . . . . . 11 class (((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))
39 cdvh 35188 . . . . . . . . . . . . . 14 class DVecH
405, 39cfv 5790 . . . . . . . . . . . . 13 class (DVecH‘𝑘)
4112, 40cfv 5790 . . . . . . . . . . . 12 class ((DVecH‘𝑘)‘𝑤)
42 clsm 17818 . . . . . . . . . . . 12 class LSSum
4341, 42cfv 5790 . . . . . . . . . . 11 class (LSSum‘((DVecH‘𝑘)‘𝑤))
4437, 38, 43co 6527 . . . . . . . . . 10 class ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))
4533, 44wceq 1474 . . . . . . . . 9 wff 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))
4631, 45wi 4 . . . . . . . 8 wff ((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))))
47 catm 33371 . . . . . . . . 9 class Atoms
485, 47cfv 5790 . . . . . . . 8 class (Atoms‘𝑘)
4946, 20, 48wral 2895 . . . . . . 7 wff 𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))))
50 clss 18699 . . . . . . . 8 class LSubSp
5141, 50cfv 5790 . . . . . . 7 class (LSubSp‘((DVecH‘𝑘)‘𝑤))
5249, 32, 51crio 6488 . . . . . 6 class (𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))))
5315, 19, 52cif 4035 . . . . 5 class if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))))))
548, 10, 53cmpt 4637 . . . 4 class (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))))))
554, 7, 54cmpt 4637 . . 3 class (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤))))))))
562, 3, 55cmpt 4637 . 2 class (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))))))))
571, 56wceq 1474 1 wff DIsoH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))))))))
Colors of variables: wff setvar class
This definition is referenced by:  dihffval  35340
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