Theorem ccatswrd 13456

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 13419	. . . . . 6 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
2	1	adantr 481	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
3		swrdcl 13419	. . . . . 6 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
4	3	adantr 481	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
5		ccatcl 13359	. . . . 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 693	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
7		wrdf 13310	. . . 4 ⊢ (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴 → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)):(0..^(#‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))))⟶𝐴)
8		ffn 6045	. . . 4 ⊢ (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)):(0..^(#‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))))⟶𝐴 → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(#‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))))
9	6, 7, 8	3syl 18	. . 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⟩)))))
10		ccatlen 13360	. . . . . . 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⟩))))
11	2, 4, 10	syl2anc 693	. . . . . 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⟩))))
12		simpl 473	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑆 ∈ Word 𝐴)
13		simpr1 1067	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑋 ∈ (0...𝑌))
14		simpr2 1068	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ (0...𝑍))
15		simpr3 1069	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑍 ∈ (0...(#‘𝑆)))
16		fzass4 12379	. . . . . . . . . . . 12 ⊢ ((𝑌 ∈ (0...(#‘𝑆)) ∧ 𝑍 ∈ (𝑌...(#‘𝑆))) ↔ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))))
17	16	biimpri 218	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑌 ∈ (0...(#‘𝑆)) ∧ 𝑍 ∈ (𝑌...(#‘𝑆))))
18	17	simpld 475	. . . . . . . . . 10 ⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → 𝑌 ∈ (0...(#‘𝑆)))
19	14, 15, 18	syl2anc 693	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ (0...(#‘𝑆)))
20		swrdlen 13423	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆))) → (#‘(𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑌 − 𝑋))
21	12, 13, 19, 20	syl3anc 1326	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (#‘(𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑌 − 𝑋))
22		swrdlen 13423	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (#‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍 − 𝑌))
23	12, 14, 15, 22	syl3anc 1326	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (#‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍 − 𝑌))
24	21, 23	oveq12d 6668	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((#‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = ((𝑌 − 𝑋) + (𝑍 − 𝑌)))
25		elfzelz 12342	. . . . . . . . . 10 ⊢ (𝑌 ∈ (0...𝑍) → 𝑌 ∈ ℤ)
26	14, 25	syl 17	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ ℤ)
27	26	zcnd 11483	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ ℂ)
28		elfzelz 12342	. . . . . . . . . 10 ⊢ (𝑋 ∈ (0...𝑌) → 𝑋 ∈ ℤ)
29	13, 28	syl 17	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑋 ∈ ℤ)
30	29	zcnd 11483	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑋 ∈ ℂ)
31		elfzelz 12342	. . . . . . . . . 10 ⊢ (𝑍 ∈ (0...(#‘𝑆)) → 𝑍 ∈ ℤ)
32	15, 31	syl 17	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑍 ∈ ℤ)
33	32	zcnd 11483	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑍 ∈ ℂ)
34	27, 30, 33	npncan3d 10428	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((𝑌 − 𝑋) + (𝑍 − 𝑌)) = (𝑍 − 𝑋))
35	24, 34	eqtrd 2656	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((#‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑍 − 𝑋))
36	11, 35	eqtrd 2656	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (#‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑍 − 𝑋))
37	36	oveq2d 6666	. . . 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)))
38	37	fneq2d 5982	. . 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))))
39	9, 38	mpbid 222	. 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)))
40		swrdcl 13419	. . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴)
41	40	adantr 481	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴)
42		wrdf 13310	. . . 4 ⊢ ((𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴 → (𝑆 substr ⟨𝑋, 𝑍⟩):(0..^(#‘(𝑆 substr ⟨𝑋, 𝑍⟩)))⟶𝐴)
43		ffn 6045	. . . 4 ⊢ ((𝑆 substr ⟨𝑋, 𝑍⟩):(0..^(#‘(𝑆 substr ⟨𝑋, 𝑍⟩)))⟶𝐴 → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(#‘(𝑆 substr ⟨𝑋, 𝑍⟩))))
44	41, 42, 43	3syl 18	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(#‘(𝑆 substr ⟨𝑋, 𝑍⟩))))
45		fzass4 12379	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0...𝑍) ∧ 𝑌 ∈ (𝑋...𝑍)) ↔ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)))
46	45	biimpri 218	. . . . . . . 8 ⊢ ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)) → (𝑋 ∈ (0...𝑍) ∧ 𝑌 ∈ (𝑋...𝑍)))
47	46	simpld 475	. . . . . . 7 ⊢ ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)) → 𝑋 ∈ (0...𝑍))
48	13, 14, 47	syl2anc 693	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑋 ∈ (0...𝑍))
49		swrdlen 13423	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (#‘(𝑆 substr ⟨𝑋, 𝑍⟩)) = (𝑍 − 𝑋))
50	12, 48, 15, 49	syl3anc 1326	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (#‘(𝑆 substr ⟨𝑋, 𝑍⟩)) = (𝑍 − 𝑋))
51	50	oveq2d 6666	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (0..^(#‘(𝑆 substr ⟨𝑋, 𝑍⟩))) = (0..^(𝑍 − 𝑋)))
52	51	fneq2d 5982	. . 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))))
53	44, 52	mpbid 222	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(𝑍 − 𝑋)))
54		simpr 477	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑥 ∈ (0..^(𝑍 − 𝑋)))
55	26, 29	zsubcld 11487	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑌 − 𝑋) ∈ ℤ)
56	55	adantr 481	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → (𝑌 − 𝑋) ∈ ℤ)
57		fzospliti 12500	. . . . 5 ⊢ ((𝑥 ∈ (0..^(𝑍 − 𝑋)) ∧ (𝑌 − 𝑋) ∈ ℤ) → (𝑥 ∈ (0..^(𝑌 − 𝑋)) ∨ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))))
58	54, 56, 57	syl2anc 693	. . . 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))))
59	2	adantr 481	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
60	4	adantr 481	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
61	21	oveq2d 6666	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (0..^(#‘(𝑆 substr ⟨𝑋, 𝑌⟩))) = (0..^(𝑌 − 𝑋)))
62	61	eleq2d 2687	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑥 ∈ (0..^(#‘(𝑆 substr ⟨𝑋, 𝑌⟩))) ↔ 𝑥 ∈ (0..^(𝑌 − 𝑋))))
63	62	biimpar 502	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑥 ∈ (0..^(#‘(𝑆 substr ⟨𝑋, 𝑌⟩))))
64		ccatval1 13361	. . . . . . 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))
65	59, 60, 63, 64	syl3anc 1326	. . . . . 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))
66		simpll 790	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑆 ∈ Word 𝐴)
67		simplr1 1103	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑋 ∈ (0...𝑌))
68	19	adantr 481	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑌 ∈ (0...(#‘𝑆)))
69		simpr 477	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑥 ∈ (0..^(𝑌 − 𝑋)))
70		swrdfv 13424	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
71	66, 67, 68, 69, 70	syl31anc 1329	. . . . . 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)))
72	65, 71	eqtrd 2656	. . . . 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)))
73	2	adantr 481	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
74	4	adantr 481	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
75	21, 35	oveq12d 6668	. . . . . . . . 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)))
76	75	eleq2d 2687	. . . . . . . 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))))
77	76	biimpar 502	. . . . . . 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⟩)))))
78		ccatval2 13362	. . . . . . 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⟩)))))
79	73, 74, 77, 78	syl3anc 1326	. . . . . 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⟩)))))
80		simpll 790	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑆 ∈ Word 𝐴)
81		simplr2 1104	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑌 ∈ (0...𝑍))
82		simplr3 1105	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑍 ∈ (0...(#‘𝑆)))
83	21	oveq2d 6666	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑥 − (#‘(𝑆 substr ⟨𝑋, 𝑌⟩))) = (𝑥 − (𝑌 − 𝑋)))
84	83	adantr 481	. . . . . . . 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)))
85	34	oveq2d 6666	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌))) = ((𝑌 − 𝑋)..^(𝑍 − 𝑋)))
86	85	eleq2d 2687	. . . . . . . . . 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))))
87	86	biimpar 502	. . . . . . . . 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))))
88	32, 26	zsubcld 11487	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑍 − 𝑌) ∈ ℤ)
89	88	adantr 481	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑍 − 𝑌) ∈ ℤ)
90		fzosubel3 12528	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌))) ∧ (𝑍 − 𝑌) ∈ ℤ) → (𝑥 − (𝑌 − 𝑋)) ∈ (0..^(𝑍 − 𝑌)))
91	87, 89, 90	syl2anc 693	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑥 − (𝑌 − 𝑋)) ∈ (0..^(𝑍 − 𝑌)))
92	84, 91	eqeltrd 2701	. . . . . . 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)))
93		swrdfv 13424	. . . . . . 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)))
94	80, 81, 82, 92, 93	syl31anc 1329	. . . . . 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)))
95	83	oveq1d 6665	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → ((𝑥 − (#‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌) = ((𝑥 − (𝑌 − 𝑋)) + 𝑌))
96	95	adantr 481	. . . . . . . 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))
97		elfzoelz 12470	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)) → 𝑥 ∈ ℤ)
98	97	zcnd 11483	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)) → 𝑥 ∈ ℂ)
99	98	adantl 482	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑥 ∈ ℂ)
100	27, 30	subcld 10392	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑌 − 𝑋) ∈ ℂ)
101	100	adantr 481	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑌 − 𝑋) ∈ ℂ)
102	27	adantr 481	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑌 ∈ ℂ)
103	99, 101, 102	subadd23d 10414	. . . . . . . 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))))
104	27, 30	nncand 10397	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑌 − (𝑌 − 𝑋)) = 𝑋)
105	104	oveq2d 6666	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → (𝑥 + (𝑌 − (𝑌 − 𝑋))) = (𝑥 + 𝑋))
106	105	adantr 481	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑥 + (𝑌 − (𝑌 − 𝑋))) = (𝑥 + 𝑋))
107	96, 103, 106	3eqtrd 2660	. . . . . . 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))
108	107	fveq2d 6195	. . . . . 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)))
109	79, 94, 108	3eqtrd 2660	. . . . 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)))
110	72, 109	jaodan 826	. . . 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)))
111	58, 110	syldan 487	. . 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)))
112		simpll 790	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑆 ∈ Word 𝐴)
113	48	adantr 481	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑋 ∈ (0...𝑍))
114		simplr3 1105	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑍 ∈ (0...(#‘𝑆)))
115		swrdfv 13424	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → ((𝑆 substr ⟨𝑋, 𝑍⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
116	112, 113, 114, 54, 115	syl31anc 1329	. . 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)))
117	111, 116	eqtr4d 2659	. 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))
118	39, 53, 117	eqfnfvd 6314	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   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ⟨cop 4183   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ℂcc 9934  0cc0 9936   + caddc 9939   − cmin 10266  ℤcz 11377  ...cfz 12326  ..^cfzo 12465  #chash 13117  Word cword 13291   ++ cconcat 13293   substr csubstr 13295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303

This theorem is referenced by:  wrdcctswrd  13465  swrdccatwrd  13468  wrdeqs1cat  13474  splid  13504  splval2  13508  swrds2  13685  efgredleme  18156  efgredlemc  18158  efgcpbllemb  18168  frgpuplem  18185  wrdsplex  30618