Definition df-substr 11099

Description: Define an operation which extracts portions (called subwords or substrings) of words. Definition in Section 9.1 of [AhoHopUll] p. 318. (Contributed by Stefan O'Rear, 15-Aug-2015.)

Assertion

Ref	Expression
df-substr	⊢ substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅))

Distinct variable group:   𝑠,𝑏,𝑥

Detailed syntax breakdown of Definition df-substr

Step	Hyp	Ref	Expression
1		csubstr 11098	. 2 class substr
2		vs	. . 3 setvar 𝑠
3		vb	. . 3 setvar 𝑏
4		cvv 2772	. . 3 class V
5		cz 9372	. . . 4 class ℤ
6	5, 5	cxp 4673	. . 3 class (ℤ × ℤ)
7	3	cv 1372	. . . . . . 7 class 𝑏
8		c1st 6224	. . . . . . 7 class 1st
9	7, 8	cfv 5271	. . . . . 6 class (1st ‘𝑏)
10		c2nd 6225	. . . . . . 7 class 2nd
11	7, 10	cfv 5271	. . . . . 6 class (2nd ‘𝑏)
12		cfzo 10264	. . . . . 6 class ..^
13	9, 11, 12	co 5944	. . . . 5 class ((1st ‘𝑏)..^(2nd ‘𝑏))
14	2	cv 1372	. . . . . 6 class 𝑠
15	14	cdm 4675	. . . . 5 class dom 𝑠
16	13, 15	wss 3166	. . . 4 wff ((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠
17		vx	. . . . 5 setvar 𝑥
18		cc0 7925	. . . . . 6 class 0
19		cmin 8243	. . . . . . 7 class −
20	11, 9, 19	co 5944	. . . . . 6 class ((2nd ‘𝑏) − (1st ‘𝑏))
21	18, 20, 12	co 5944	. . . . 5 class (0..^((2nd ‘𝑏) − (1st ‘𝑏)))
22	17	cv 1372	. . . . . . 7 class 𝑥
23		caddc 7928	. . . . . . 7 class +
24	22, 9, 23	co 5944	. . . . . 6 class (𝑥 + (1st ‘𝑏))
25	24, 14	cfv 5271	. . . . 5 class (𝑠‘(𝑥 + (1st ‘𝑏)))
26	17, 21, 25	cmpt 4105	. . . 4 class (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏))))
27		c0 3460	. . . 4 class ∅
28	16, 26, 27	cif 3571	. . 3 class if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅)
29	2, 3, 4, 6, 28	cmpo 5946	. 2 class (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅))
30	1, 29	wceq 1373	1 wff substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1st ‘𝑏)..^(2nd ‘𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd ‘𝑏) − (1st ‘𝑏))) ↦ (𝑠‘(𝑥 + (1st ‘𝑏)))), ∅))

This definition is referenced by:  swrdval  11101