Definition df-chn 18563

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 18562	. 2 class ( < Chain 𝐴)
4		vn	. . . . . . . 8 setvar 𝑛
5	4	cv 1541	. . . . . . 7 class 𝑛
6		c1 11030	. . . . . . 7 class 1
7		cmin 11368	. . . . . . 7 class −
8	5, 6, 7	co 7360	. . . . . 6 class (𝑛 − 1)
9		vc	. . . . . . 7 setvar 𝑐
10	9	cv 1541	. . . . . 6 class 𝑐
11	8, 10	cfv 6492	. . . . 5 class (𝑐‘(𝑛 − 1))
12	5, 10	cfv 6492	. . . . 5 class (𝑐‘𝑛)
13	11, 12, 2	wbr 5086	. . . 4 wff (𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)
14	10	cdm 5624	. . . . 5 class dom 𝑐
15		cc0 11029	. . . . . 6 class 0
16	15	csn 4568	. . . . 5 class {0}
17	14, 16	cdif 3887	. . . 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 14466	. . 3 class Word 𝐴
20	18, 9, 19	crab 3390	. 2 class {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)}
21	3, 20	wceq 1542	1 wff ( < Chain 𝐴) = {𝑐 ∈ Word 𝐴 ∣ ∀𝑛 ∈ (dom 𝑐 ∖ {0})(𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)}

This definition is referenced by:  ischn  18564  nfchnd  18568  chneq1  18569  chneq2  18570