Definition df-chn 18541

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 18540	. 2 class ( < Chain 𝐴)
4		vn	. . . . . . . 8 setvar 𝑛
5	4	cv 1541	. . . . . . 7 class 𝑛
6		c1 11039	. . . . . . 7 class 1
7		cmin 11376	. . . . . . 7 class −
8	5, 6, 7	co 7368	. . . . . 6 class (𝑛 − 1)
9		vc	. . . . . . 7 setvar 𝑐
10	9	cv 1541	. . . . . 6 class 𝑐
11	8, 10	cfv 6500	. . . . 5 class (𝑐‘(𝑛 − 1))
12	5, 10	cfv 6500	. . . . 5 class (𝑐‘𝑛)
13	11, 12, 2	wbr 5100	. . . 4 wff (𝑐‘(𝑛 − 1)) < (𝑐‘𝑛)
14	10	cdm 5632	. . . . 5 class dom 𝑐
15		cc0 11038	. . . . . 6 class 0
16	15	csn 4582	. . . . 5 class {0}
17	14, 16	cdif 3900	. . . 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 14448	. . 3 class Word 𝐴
20	18, 9, 19	crab 3401	. 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  18542  nfchnd  18546  chneq1  18547  chneq2  18548