Definition df-frmd 18808

Description: Define a free monoid over a set 𝑖 of generators, defined as the set of finite strings on 𝐼 with the operation of concatenation. (Contributed by Mario Carneiro, 27-Sep-2015.)

Assertion

Ref	Expression
df-frmd	⊢ freeMnd = (𝑖 ∈ V ↦ {⟨(Base‘ndx), Word 𝑖⟩, ⟨(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))⟩})

Detailed syntax breakdown of Definition df-frmd

Step	Hyp	Ref	Expression
1		cfrmd 18806	. 2 class freeMnd
2		vi	. . 3 setvar 𝑖
3		cvv 3430	. . 3 class V
4		cnx 17154	. . . . . 6 class ndx
5		cbs 17170	. . . . . 6 class Base
6	4, 5	cfv 6492	. . . . 5 class (Base‘ndx)
7	2	cv 1541	. . . . . 6 class 𝑖
8	7	cword 14466	. . . . 5 class Word 𝑖
9	6, 8	cop 4574	. . . 4 class ⟨(Base‘ndx), Word 𝑖⟩
10		cplusg 17211	. . . . . 6 class +g
11	4, 10	cfv 6492	. . . . 5 class (+g‘ndx)
12		cconcat 14523	. . . . . 6 class ++
13	8, 8	cxp 5622	. . . . . 6 class (Word 𝑖 × Word 𝑖)
14	12, 13	cres 5626	. . . . 5 class ( ++ ↾ (Word 𝑖 × Word 𝑖))
15	11, 14	cop 4574	. . . 4 class ⟨(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))⟩
16	9, 15	cpr 4570	. . 3 class {⟨(Base‘ndx), Word 𝑖⟩, ⟨(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))⟩}
17	2, 3, 16	cmpt 5167	. 2 class (𝑖 ∈ V ↦ {⟨(Base‘ndx), Word 𝑖⟩, ⟨(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))⟩})
18	1, 17	wceq 1542	1 wff freeMnd = (𝑖 ∈ V ↦ {⟨(Base‘ndx), Word 𝑖⟩, ⟨(+g‘ndx), ( ++ ↾ (Word 𝑖 × Word 𝑖))⟩})

This definition is referenced by:  frmdval  18810