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Definition df-tgrp 39614
Description: Define the class of all translation groups. π‘˜ is normally a member of HL. Each base set is the set of all lattice translations with respect to a hyperplane 𝑀, 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 = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {⟨(Baseβ€˜ndx), ((LTrnβ€˜π‘˜)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩}))
Distinct variable group:   𝑀,π‘˜,𝑓,𝑔

Detailed syntax breakdown of Definition df-tgrp
StepHypRef Expression
1 ctgrp 39613 . 2 class TGrp
2 vk . . 3 setvar π‘˜
3 cvv 3475 . . 3 class V
4 vw . . . 4 setvar 𝑀
52cv 1541 . . . . 5 class π‘˜
6 clh 38855 . . . . 5 class LHyp
75, 6cfv 6544 . . . 4 class (LHypβ€˜π‘˜)
8 cnx 17126 . . . . . . 7 class ndx
9 cbs 17144 . . . . . . 7 class Base
108, 9cfv 6544 . . . . . 6 class (Baseβ€˜ndx)
114cv 1541 . . . . . . 7 class 𝑀
12 cltrn 38972 . . . . . . . 8 class LTrn
135, 12cfv 6544 . . . . . . 7 class (LTrnβ€˜π‘˜)
1411, 13cfv 6544 . . . . . 6 class ((LTrnβ€˜π‘˜)β€˜π‘€)
1510, 14cop 4635 . . . . 5 class ⟨(Baseβ€˜ndx), ((LTrnβ€˜π‘˜)β€˜π‘€)⟩
16 cplusg 17197 . . . . . . 7 class +g
178, 16cfv 6544 . . . . . 6 class (+gβ€˜ndx)
18 vf . . . . . . 7 setvar 𝑓
19 vg . . . . . . 7 setvar 𝑔
2018cv 1541 . . . . . . . 8 class 𝑓
2119cv 1541 . . . . . . . 8 class 𝑔
2220, 21ccom 5681 . . . . . . 7 class (𝑓 ∘ 𝑔)
2318, 19, 14, 14, 22cmpo 7411 . . . . . 6 class (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))
2417, 23cop 4635 . . . . 5 class ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩
2515, 24cpr 4631 . . . 4 class {⟨(Baseβ€˜ndx), ((LTrnβ€˜π‘˜)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩}
264, 7, 25cmpt 5232 . . 3 class (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {⟨(Baseβ€˜ndx), ((LTrnβ€˜π‘˜)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩})
272, 3, 26cmpt 5232 . 2 class (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {⟨(Baseβ€˜ndx), ((LTrnβ€˜π‘˜)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩}))
281, 27wceq 1542 1 wff TGrp = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {⟨(Baseβ€˜ndx), ((LTrnβ€˜π‘˜)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩}))
Colors of variables: wff setvar class
This definition is referenced by:  tgrpfset  39615
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