Metamath Proof Explorer

Definition df-chn

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)

Ref Expression
Assertion df-chn ⊢ Chain A < ˙ = c ∈ Word A | ∀ n ∈ dom ⁡ c ∖ 0 c ⁡ n − 1 < ˙ c ⁡ n

Detailed syntax breakdown

Step Hyp Ref Expression
0 c.lt class < ˙
1 cA class A
2 1 0 cchn class Chain A < ˙
3 vc setvar c
4 1 cword class Word A
5 vn setvar n
6 3 cv setvar c
7 6 cdm class dom ⁡ c
8 cc0 class 0
9 8 csn class 0
10 7 9 cdif class dom ⁡ c ∖ 0
11 5 cv setvar n
12 cmin class −
13 c1 class 1
14 11 13 12 co class n − 1
15 14 6 cfv class c ⁡ n − 1
16 11 6 cfv class c ⁡ n
17 15 16 0 wbr wff c ⁡ n − 1 < ˙ c ⁡ n
18 17 5 10 wral wff ∀ n ∈ dom ⁡ c ∖ 0 c ⁡ n − 1 < ˙ c ⁡ n
19 18 3 4 crab class c ∈ Word A | ∀ n ∈ dom ⁡ c ∖ 0 c ⁡ n − 1 < ˙ c ⁡ n
20 2 19 wceq wff Chain A < ˙ = c ∈ Word A | ∀ n ∈ dom ⁡ c ∖ 0 c ⁡ n − 1 < ˙ c ⁡ n