Definition df-chn 32990

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 32989	. 2 class ( < Chain𝐴)
4		vn	. . . . . . . 8 setvar 𝑛
5	4	cv 1539	. . . . . . 7 class 𝑛
6		c1 11135	. . . . . . 7 class 1
7		cmin 11471	. . . . . . 7 class −
8	5, 6, 7	co 7410	. . . . . 6 class (𝑛 − 1)
9		vc	. . . . . . 7 setvar 𝑐
10	9	cv 1539	. . . . . 6 class 𝑐
11	8, 10	cfv 6536	. . . . 5 class (𝑐‘(𝑛 − 1))
12	5, 10	cfv 6536	. . . . 5 class (𝑐‘𝑛)
13	11, 12, 2	wbr 5124	. . . 4 wff (𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)
14	10	cdm 5659	. . . . 5 class dom 𝑐
15		cc0 11134	. . . . . 6 class 0
16	15	csn 4606	. . . . 5 class {0}
17	14, 16	cdif 3928	. . . 4 class (dom 𝑐 ∖ {0})
18	13, 4, 17	wral 3052	. . 3 wff ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)
19	1	cword 14536	. . 3 class Word 𝐴
20	18, 9, 19	crab 3420	. 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  32991