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Definition df-ldil 34211
Description: Define set of all lattice dilations. Similar to definition of dilation in [Crawley] p. 111. (Contributed by NM, 11-May-2012.)
Assertion
Ref Expression
df-ldil LDil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥)}))
Distinct variable group:   𝑤,𝑘,𝑓,𝑥

Detailed syntax breakdown of Definition df-ldil
StepHypRef Expression
1 cldil 34207 . 2 class LDil
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 . . . . . . . . 9 setvar 𝑥
98cv 1473 . . . . . . . 8 class 𝑥
104cv 1473 . . . . . . . 8 class 𝑤
11 cple 15721 . . . . . . . . 9 class le
125, 11cfv 5790 . . . . . . . 8 class (le‘𝑘)
139, 10, 12wbr 4577 . . . . . . 7 wff 𝑥(le‘𝑘)𝑤
14 vf . . . . . . . . . 10 setvar 𝑓
1514cv 1473 . . . . . . . . 9 class 𝑓
169, 15cfv 5790 . . . . . . . 8 class (𝑓𝑥)
1716, 9wceq 1474 . . . . . . 7 wff (𝑓𝑥) = 𝑥
1813, 17wi 4 . . . . . 6 wff (𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥)
19 cbs 15641 . . . . . . 7 class Base
205, 19cfv 5790 . . . . . 6 class (Base‘𝑘)
2118, 8, 20wral 2895 . . . . 5 wff 𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥)
22 claut 34092 . . . . . 6 class LAut
235, 22cfv 5790 . . . . 5 class (LAut‘𝑘)
2421, 14, 23crab 2899 . . . 4 class {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥)}
254, 7, 24cmpt 4637 . . 3 class (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥)})
262, 3, 25cmpt 4637 . 2 class (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥)}))
271, 26wceq 1474 1 wff LDil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∈ (LAut‘𝑘) ∣ ∀𝑥 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑤 → (𝑓𝑥) = 𝑥)}))
Colors of variables: wff setvar class
This definition is referenced by:  ldilfset  34215
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