Theorem ccatswrd 14716

Description: Joining two adjacent subwords makes a longer subword. (Contributed by Stefan O'Rear, 20-Aug-2015.)

Assertion

Ref	Expression
ccatswrd	⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 substr ⟨𝑋, 𝑍⟩))

Proof of Theorem ccatswrd

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		swrdcl 14693	. . . . . 6 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
2	1	adantr 480	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
3		swrdcl 14693	. . . . . 6 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
4	3	adantr 480	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
5		ccatcl 14622	. . . . 5 ⊢ (((𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
6	2, 4, 5	syl2anc 583	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
7		wrdfn 14576	. . . 4 ⊢ (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴 → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))))
8	6, 7	syl 17	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))))
9		ccatlen 14623	. . . . . . 7 ⊢ (((𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴) → (♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))))
10	2, 4, 9	syl2anc 583	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))))
11		simpl 482	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑆 ∈ Word 𝐴)
12		simpr1 1194	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ (0...𝑌))
13		simpr2 1195	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ (0...𝑍))
14		simpr3 1196	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ (0...(♯‘𝑆)))
15		fzass4 13622	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑍 ∈ (𝑌...(♯‘𝑆))) ↔ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))))
16	15	biimpri 228	. . . . . . . . . 10 ⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑍 ∈ (𝑌...(♯‘𝑆))))
17	16	simpld 494	. . . . . . . . 9 ⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → 𝑌 ∈ (0...(♯‘𝑆)))
18	13, 14, 17	syl2anc 583	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ (0...(♯‘𝑆)))
19		swrdlen 14695	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑌 − 𝑋))
20	11, 12, 18, 19	syl3anc 1371	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑌 − 𝑋))
21		swrdlen 14695	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍 − 𝑌))
22	21	3adant3r1 1182	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍 − 𝑌))
23	20, 22	oveq12d 7466	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = ((𝑌 − 𝑋) + (𝑍 − 𝑌)))
24	13	elfzelzd 13585	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℤ)
25	24	zcnd 12748	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℂ)
26	12	elfzelzd 13585	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ ℤ)
27	26	zcnd 12748	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ ℂ)
28	14	elfzelzd 13585	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ ℤ)
29	28	zcnd 12748	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ ℂ)
30	25, 27, 29	npncan3d 11683	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑌 − 𝑋) + (𝑍 − 𝑌)) = (𝑍 − 𝑋))
31	10, 23, 30	3eqtrd 2784	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑍 − 𝑋))
32	31	oveq2d 7464	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))) = (0..^(𝑍 − 𝑋)))
33	32	fneq2d 6673	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))) ↔ ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(𝑍 − 𝑋))))
34	8, 33	mpbid 232	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(𝑍 − 𝑋)))
35		swrdcl 14693	. . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴)
36	35	adantr 480	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴)
37		wrdfn 14576	. . . 4 ⊢ ((𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴 → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑍⟩))))
38	36, 37	syl 17	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑍⟩))))
39		fzass4 13622	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0...𝑍) ∧ 𝑌 ∈ (𝑋...𝑍)) ↔ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)))
40	39	biimpri 228	. . . . . . . 8 ⊢ ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)) → (𝑋 ∈ (0...𝑍) ∧ 𝑌 ∈ (𝑋...𝑍)))
41	40	simpld 494	. . . . . . 7 ⊢ ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)) → 𝑋 ∈ (0...𝑍))
42	12, 13, 41	syl2anc 583	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ (0...𝑍))
43		swrdlen 14695	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨𝑋, 𝑍⟩)) = (𝑍 − 𝑋))
44	11, 42, 14, 43	syl3anc 1371	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 substr ⟨𝑋, 𝑍⟩)) = (𝑍 − 𝑋))
45	44	oveq2d 7464	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑍⟩))) = (0..^(𝑍 − 𝑋)))
46	45	fneq2d 6673	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑍⟩))) ↔ (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(𝑍 − 𝑋))))
47	38, 46	mpbid 232	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(𝑍 − 𝑋)))
48	24, 26	zsubcld 12752	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 − 𝑋) ∈ ℤ)
49	48	anim1ci 615	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → (𝑥 ∈ (0..^(𝑍 − 𝑋)) ∧ (𝑌 − 𝑋) ∈ ℤ))
50		fzospliti 13748	. . . . 5 ⊢ ((𝑥 ∈ (0..^(𝑍 − 𝑋)) ∧ (𝑌 − 𝑋) ∈ ℤ) → (𝑥 ∈ (0..^(𝑌 − 𝑋)) ∨ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))))
51	49, 50	syl 17	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → (𝑥 ∈ (0..^(𝑌 − 𝑋)) ∨ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))))
52	1	ad2antrr 725	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
53	3	ad2antrr 725	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
54	20	oveq2d 7464	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) = (0..^(𝑌 − 𝑋)))
55	54	eleq2d 2830	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) ↔ 𝑥 ∈ (0..^(𝑌 − 𝑋))))
56	55	biimpar 477	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑥 ∈ (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))))
57		ccatval1 14625	. . . . . . 7 ⊢ (((𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥))
58	52, 53, 56, 57	syl3anc 1371	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥))
59		simpll 766	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑆 ∈ Word 𝐴)
60		simplr1 1215	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑋 ∈ (0...𝑌))
61	18	adantr 480	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑌 ∈ (0...(♯‘𝑆)))
62		simpr 484	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑥 ∈ (0..^(𝑌 − 𝑋)))
63		swrdfv 14696	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
64	59, 60, 61, 62, 63	syl31anc 1373	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
65	58, 64	eqtrd 2780	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
66	1	ad2antrr 725	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
67	3	ad2antrr 725	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
68	23, 30	eqtrd 2780	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑍 − 𝑋))
69	20, 68	oveq12d 7466	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))..^((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)))) = ((𝑌 − 𝑋)..^(𝑍 − 𝑋)))
70	69	eleq2d 2830	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))..^((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)))) ↔ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))))
71	70	biimpar 477	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑥 ∈ ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))..^((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)))))
72		ccatval2 14626	. . . . . . 7 ⊢ (((𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴 ∧ 𝑥 ∈ ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))..^((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)))))
73	66, 67, 71, 72	syl3anc 1371	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)))))
74		simpll 766	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑆 ∈ Word 𝐴)
75		simplr2 1216	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑌 ∈ (0...𝑍))
76		simplr3 1217	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑍 ∈ (0...(♯‘𝑆)))
77	20	oveq2d 7464	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) = (𝑥 − (𝑌 − 𝑋)))
78	77	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) = (𝑥 − (𝑌 − 𝑋)))
79	30	oveq2d 7464	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌))) = ((𝑌 − 𝑋)..^(𝑍 − 𝑋)))
80	79	eleq2d 2830	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌))) ↔ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))))
81	80	biimpar 477	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑥 ∈ ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌))))
82	28, 24	zsubcld 12752	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑍 − 𝑌) ∈ ℤ)
83	82	adantr 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑍 − 𝑌) ∈ ℤ)
84		fzosubel3 13777	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌))) ∧ (𝑍 − 𝑌) ∈ ℤ) → (𝑥 − (𝑌 − 𝑋)) ∈ (0..^(𝑍 − 𝑌)))
85	81, 83, 84	syl2anc 583	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑥 − (𝑌 − 𝑋)) ∈ (0..^(𝑍 − 𝑌)))
86	78, 85	eqeltrd 2844	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) ∈ (0..^(𝑍 − 𝑌)))
87		swrdfv 14696	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) ∧ (𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) ∈ (0..^(𝑍 − 𝑌))) → ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)))) = (𝑆‘((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌)))
88	74, 75, 76, 86, 87	syl31anc 1373	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)))) = (𝑆‘((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌)))
89	77	oveq1d 7463	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌) = ((𝑥 − (𝑌 − 𝑋)) + 𝑌))
90	89	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → ((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌) = ((𝑥 − (𝑌 − 𝑋)) + 𝑌))
91		elfzoelz 13716	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)) → 𝑥 ∈ ℤ)
92	91	zcnd 12748	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)) → 𝑥 ∈ ℂ)
93	92	adantl 481	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑥 ∈ ℂ)
94	25, 27	subcld 11647	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 − 𝑋) ∈ ℂ)
95	94	adantr 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑌 − 𝑋) ∈ ℂ)
96	25	adantr 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑌 ∈ ℂ)
97	93, 95, 96	subadd23d 11669	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → ((𝑥 − (𝑌 − 𝑋)) + 𝑌) = (𝑥 + (𝑌 − (𝑌 − 𝑋))))
98	25, 27	nncand 11652	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 − (𝑌 − 𝑋)) = 𝑋)
99	98	oveq2d 7464	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 + (𝑌 − (𝑌 − 𝑋))) = (𝑥 + 𝑋))
100	99	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑥 + (𝑌 − (𝑌 − 𝑋))) = (𝑥 + 𝑋))
101	90, 97, 100	3eqtrd 2784	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → ((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌) = (𝑥 + 𝑋))
102	101	fveq2d 6924	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑆‘((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌)) = (𝑆‘(𝑥 + 𝑋)))
103	73, 88, 102	3eqtrd 2784	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
104	65, 103	jaodan 958	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ (𝑥 ∈ (0..^(𝑌 − 𝑋)) ∨ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
105	51, 104	syldan 590	. . 3 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
106		simpll 766	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑆 ∈ Word 𝐴)
107	42	adantr 480	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑋 ∈ (0...𝑍))
108		simplr3 1217	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑍 ∈ (0...(♯‘𝑆)))
109		simpr 484	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑥 ∈ (0..^(𝑍 − 𝑋)))
110		swrdfv 14696	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → ((𝑆 substr ⟨𝑋, 𝑍⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
111	106, 107, 108, 109, 110	syl31anc 1373	. . 3 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → ((𝑆 substr ⟨𝑋, 𝑍⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
112	105, 111	eqtr4d 2783	. 2 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑋, 𝑍⟩)‘𝑥))
113	34, 47, 112	eqfnfvd 7067	1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 substr ⟨𝑋, 𝑍⟩))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ⟨cop 4654   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448  ℂcc 11182  0cc0 11184   + caddc 11187   − cmin 11520  ℤcz 12639  ...cfz 13567  ..^cfzo 13711  ♯chash 14379  Word cword 14562   ++ cconcat 14618   substr csubstr 14688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380  df-word 14563  df-concat 14619  df-substr 14689

This theorem is referenced by:  swrds2  14989  efgredleme  19785