Definition df-chn 32938

Description: Define the class of (finite) chains. A chain is defined to be a sequence of objects, where each object is less than the next one in the sequence. The term "chain" is usually used in order theory. In the context of algebra, chains are often called "towers", for example for fields, or "series", for example for subgroup or subnormal series. (Contributed by Thierry Arnoux, 19-Jun-2025.)

Assertion

Ref	Expression
df-chn	⊢ ( < Chain𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)}

Distinct variable groups:   𝐴,𝑐,𝑛   < ,𝑐,𝑛

Detailed syntax breakdown of Definition df-chn

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		c.lt	. . 3 class <
3	1, 2	cchn 32937	. 2 class ( < Chain𝐴)
4		vn	. . . . . . . 8 setvar 𝑛
5	4	cv 1539	. . . . . . 7 class 𝑛
6		c1 11076	. . . . . . 7 class 1
7		cmin 11412	. . . . . . 7 class −
8	5, 6, 7	co 7390	. . . . . 6 class (𝑛 − 1)
9		vc	. . . . . . 7 setvar 𝑐
10	9	cv 1539	. . . . . 6 class 𝑐
11	8, 10	cfv 6514	. . . . 5 class (𝑐‘(𝑛 − 1))
12	5, 10	cfv 6514	. . . . 5 class (𝑐‘𝑛)
13	11, 12, 2	wbr 5110	. . . 4 wff (𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)
14	10	cdm 5641	. . . . 5 class dom 𝑐
15		cc0 11075	. . . . . 6 class 0
16	15	csn 4592	. . . . 5 class {0}
17	14, 16	cdif 3914	. . . 4 class (dom 𝑐 ∖ {0})
18	13, 4, 17	wral 3045	. . 3 wff ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)
19	1	cword 14485	. . 3 class Word 𝐴
20	18, 9, 19	crab 3408	. 2 class {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)}
21	3, 20	wceq 1540	1 wff ( < Chain𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)}

This definition is referenced by:  ischn  32939