Definition df-chn 18697

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	⊢ ( < Chain 𝐴) = {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 18696	. 2 class ( < Chain 𝐴)
4		vn	. . . . . . . 8 setvar 𝑛
5	4	cv 1569	. . . . . . 7 class 𝑛
6		c1 11128	. . . . . . 7 class 1
7		cmin 11468	. . . . . . 7 class −
8	5, 6, 7	co 7414	. . . . . 6 class (𝑛 − 1)
9		vc	. . . . . . 7 setvar 𝑐
10	9	cv 1569	. . . . . 6 class 𝑐
11	8, 10	cfv 6533	. . . . 5 class (𝑐‘(𝑛 − 1))
12	5, 10	cfv 6533	. . . . 5 class (𝑐‘𝑛)
13	11, 12, 2	wbr 5103	. . . 4 wff (𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)
14	10	cdm 5655	. . . . 5 class dom 𝑐
15		cc0 11127	. . . . . 6 class 0
16	15	csn 4584	. . . . 5 class {0}
17	14, 16	cdif 3896	. . . 4 class (dom 𝑐 ∖ {0})
18	13, 4, 17	wral 3076	. . 3 wff ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)
19	1	cword 14581	. . 3 class Word 𝐴
20	18, 9, 19	crab 3412	. 2 class {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)}
21	3, 20	wceq 1570	1 wff ( < Chain 𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)}

This definition is used by:  ischn  18698  nfchnd  18702  chneq1  18703  chneq2  18704