Definition df-chn 32960

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 32959	. 2 class ( < Chain𝐴)
4		vn	. . . . . . . 8 setvar 𝑛
5	4	cv 1539	. . . . . . 7 class 𝑛
6		c1 11029	. . . . . . 7 class 1
7		cmin 11365	. . . . . . 7 class −
8	5, 6, 7	co 7353	. . . . . 6 class (𝑛 − 1)
9		vc	. . . . . . 7 setvar 𝑐
10	9	cv 1539	. . . . . 6 class 𝑐
11	8, 10	cfv 6486	. . . . 5 class (𝑐‘(𝑛 − 1))
12	5, 10	cfv 6486	. . . . 5 class (𝑐‘𝑛)
13	11, 12, 2	wbr 5095	. . . 4 wff (𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)
14	10	cdm 5623	. . . . 5 class dom 𝑐
15		cc0 11028	. . . . . 6 class 0
16	15	csn 4579	. . . . 5 class {0}
17	14, 16	cdif 3902	. . . 4 class (dom 𝑐 ∖ {0})
18	13, 4, 17	wral 3044	. . 3 wff ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)
19	1	cword 14438	. . 3 class Word 𝐴
20	18, 9, 19	crab 3396	. 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  32961