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Definition df-tgrp 37915
Description: Define the class of all translation groups. 𝑘 is normally a member of HL. Each base set is the set of all lattice translations with respect to a hyperplane 𝑤, and the operation is function composition. Similar to definition of G in [Crawley] p. 116, third paragraph (which defines this for geomodular lattices). (Contributed by NM, 5-Jun-2013.)
Assertion
Ref Expression
df-tgrp TGrp = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓𝑔))⟩}))
Distinct variable group:   𝑤,𝑘,𝑓,𝑔

Detailed syntax breakdown of Definition df-tgrp
StepHypRef Expression
1 ctgrp 37914 . 2 class TGrp
2 vk . . 3 setvar 𝑘
3 cvv 3473 . . 3 class V
4 vw . . . 4 setvar 𝑤
52cv 1536 . . . . 5 class 𝑘
6 clh 37156 . . . . 5 class LHyp
75, 6cfv 6331 . . . 4 class (LHyp‘𝑘)
8 cnx 16459 . . . . . . 7 class ndx
9 cbs 16462 . . . . . . 7 class Base
108, 9cfv 6331 . . . . . 6 class (Base‘ndx)
114cv 1536 . . . . . . 7 class 𝑤
12 cltrn 37273 . . . . . . . 8 class LTrn
135, 12cfv 6331 . . . . . . 7 class (LTrn‘𝑘)
1411, 13cfv 6331 . . . . . 6 class ((LTrn‘𝑘)‘𝑤)
1510, 14cop 4549 . . . . 5 class ⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩
16 cplusg 16544 . . . . . . 7 class +g
178, 16cfv 6331 . . . . . 6 class (+g‘ndx)
18 vf . . . . . . 7 setvar 𝑓
19 vg . . . . . . 7 setvar 𝑔
2018cv 1536 . . . . . . . 8 class 𝑓
2119cv 1536 . . . . . . . 8 class 𝑔
2220, 21ccom 5535 . . . . . . 7 class (𝑓𝑔)
2318, 19, 14, 14, 22cmpo 7135 . . . . . 6 class (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓𝑔))
2417, 23cop 4549 . . . . 5 class ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓𝑔))⟩
2515, 24cpr 4545 . . . 4 class {⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓𝑔))⟩}
264, 7, 25cmpt 5122 . . 3 class (𝑤 ∈ (LHyp‘𝑘) ↦ {⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓𝑔))⟩})
272, 3, 26cmpt 5122 . 2 class (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓𝑔))⟩}))
281, 27wceq 1537 1 wff TGrp = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓𝑔))⟩}))
Colors of variables: wff setvar class
This definition is referenced by:  tgrpfset  37916
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