Theorem ccatpfx 11495

Description: Concatenating a prefix with an adjacent subword makes a longer prefix. (Contributed by AV, 7-May-2020.)

Assertion

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

Proof of Theorem ccatpfx

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfznn0 10538	. . . . . . . 8 ⊢ (𝑌 ∈ (0...𝑍) → 𝑌 ∈ ℕ0)
2	1	ad2antrl 494	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℕ0)
3		pfxclg 11472	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ ℕ0) → (𝑆 prefix 𝑌) ∈ Word 𝐴)
4	2, 3	syldan 282	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 prefix 𝑌) ∈ Word 𝐴)
5		simpl 109	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑆 ∈ Word 𝐴)
6	2	nn0zd 9777	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℤ)
7		elfzelz 10445	. . . . . . . . 9 ⊢ (𝑍 ∈ (0...(♯‘𝑆)) → 𝑍 ∈ ℤ)
8	7	adantl 277	. . . . . . . 8 ⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → 𝑍 ∈ ℤ)
9	8	adantl 277	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ ℤ)
10		swrdclg 11444	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
11	5, 6, 9, 10	syl3anc 1278	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
12		ccatcl 11383	. . . . . 6 ⊢ (((𝑆 prefix 𝑌) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
13	4, 11, 12	syl2anc 415	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
14		wrdfn 11341	. . . . 5 ⊢ (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴 → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(♯‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))))
15	13, 14	syl 14	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(♯‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))))
16		ccatlen 11385	. . . . . . . 8 ⊢ (((𝑆 prefix 𝑌) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴) → (♯‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = ((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))))
17	4, 11, 16	syl2anc 415	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = ((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))))
18		fzass4 10485	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑍 ∈ (𝑌...(♯‘𝑆))) ↔ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))))
19	18	biimpri 133	. . . . . . . . . 10 ⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑍 ∈ (𝑌...(♯‘𝑆))))
20	19	simpld 112	. . . . . . . . 9 ⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → 𝑌 ∈ (0...(♯‘𝑆)))
21		pfxlen 11479	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 prefix 𝑌)) = 𝑌)
22	20, 21	sylan2 286	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 prefix 𝑌)) = 𝑌)
23		swrdlen 11446	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍 − 𝑌))
24	23	3expb 1235	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍 − 𝑌))
25	22, 24	oveq12d 6106	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑌 + (𝑍 − 𝑌)))
26		elfzelz 10445	. . . . . . . . . 10 ⊢ (𝑌 ∈ (0...𝑍) → 𝑌 ∈ ℤ)
27	26	zcnd 9780	. . . . . . . . 9 ⊢ (𝑌 ∈ (0...𝑍) → 𝑌 ∈ ℂ)
28	7	zcnd 9780	. . . . . . . . 9 ⊢ (𝑍 ∈ (0...(♯‘𝑆)) → 𝑍 ∈ ℂ)
29		pncan3 8541	. . . . . . . . 9 ⊢ ((𝑌 ∈ ℂ ∧ 𝑍 ∈ ℂ) → (𝑌 + (𝑍 − 𝑌)) = 𝑍)
30	27, 28, 29	syl2an 289	. . . . . . . 8 ⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (𝑌 + (𝑍 − 𝑌)) = 𝑍)
31	30	adantl 277	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 + (𝑍 − 𝑌)) = 𝑍)
32	17, 25, 31	3eqtrd 2275	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = 𝑍)
33	32	oveq2d 6104	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))) = (0..^𝑍))
34	33	fneq2d 5475	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(♯‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))) ↔ ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^𝑍)))
35	15, 34	mpbid 147	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^𝑍))
36		pfxfn 11477	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (𝑆 prefix 𝑍) Fn (0..^𝑍))
37	36	adantrl 482	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 prefix 𝑍) Fn (0..^𝑍))
38		id 19	. . . . . 6 ⊢ (𝑥 ∈ (0..^𝑍) → 𝑥 ∈ (0..^𝑍))
39	26	ad2antrl 494	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℤ)
40		fzospliti 10602	. . . . . 6 ⊢ ((𝑥 ∈ (0..^𝑍) ∧ 𝑌 ∈ ℤ) → (𝑥 ∈ (0..^𝑌) ∨ 𝑥 ∈ (𝑌..^𝑍)))
41	38, 39, 40	syl2anr 290	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑍)) → (𝑥 ∈ (0..^𝑌) ∨ 𝑥 ∈ (𝑌..^𝑍)))
42	4	adantr 276	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑌)) → (𝑆 prefix 𝑌) ∈ Word 𝐴)
43	11	adantr 276	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑌)) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
44	22	oveq2d 6104	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘(𝑆 prefix 𝑌))) = (0..^𝑌))
45	44	eleq2d 2308	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ (0..^(♯‘(𝑆 prefix 𝑌))) ↔ 𝑥 ∈ (0..^𝑌)))
46	45	biimpar 297	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑌)) → 𝑥 ∈ (0..^(♯‘(𝑆 prefix 𝑌))))
47		ccatval1 11387	. . . . . . . 8 ⊢ (((𝑆 prefix 𝑌) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘(𝑆 prefix 𝑌)))) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 prefix 𝑌)‘𝑥))
48	42, 43, 46, 47	syl3anc 1278	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑌)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 prefix 𝑌)‘𝑥))
49	20	adantl 277	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ (0...(♯‘𝑆)))
50		id 19	. . . . . . . 8 ⊢ (𝑥 ∈ (0..^𝑌) → 𝑥 ∈ (0..^𝑌))
51		pfxfv 11478	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑥 ∈ (0..^𝑌)) → ((𝑆 prefix 𝑌)‘𝑥) = (𝑆‘𝑥))
52	5, 49, 50, 51	syl2an3an 1339	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑌)) → ((𝑆 prefix 𝑌)‘𝑥) = (𝑆‘𝑥))
53	48, 52	eqtrd 2271	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑌)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘𝑥))
54	4	adantr 276	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑆 prefix 𝑌) ∈ Word 𝐴)
55	11	adantr 276	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
56	25, 31	eqtrd 2271	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = 𝑍)
57	22, 56	oveq12d 6106	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 prefix 𝑌))..^((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)))) = (𝑌..^𝑍))
58	57	eleq2d 2308	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ ((♯‘(𝑆 prefix 𝑌))..^((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)))) ↔ 𝑥 ∈ (𝑌..^𝑍)))
59	58	biimpar 297	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → 𝑥 ∈ ((♯‘(𝑆 prefix 𝑌))..^((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)))))
60		ccatval2 11388	. . . . . . . 8 ⊢ (((𝑆 prefix 𝑌) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴 ∧ 𝑥 ∈ ((♯‘(𝑆 prefix 𝑌))..^((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))))) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (♯‘(𝑆 prefix 𝑌)))))
61	54, 55, 59, 60	syl3anc 1278	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (♯‘(𝑆 prefix 𝑌)))))
62		id 19	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))))
63	62	3expb 1235	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))))
64	22	oveq2d 6104	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 − (♯‘(𝑆 prefix 𝑌))) = (𝑥 − 𝑌))
65	64	adantr 276	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − (♯‘(𝑆 prefix 𝑌))) = (𝑥 − 𝑌))
66		id 19	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑌..^𝑍) → 𝑥 ∈ (𝑌..^𝑍))
67		fzosubel 10629	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝑌..^𝑍) ∧ 𝑌 ∈ ℤ) → (𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)))
68	66, 39, 67	syl2anr 290	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)))
69	27	subidd 8632	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∈ (0...𝑍) → (𝑌 − 𝑌) = 0)
70	69	oveq1d 6103	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (0...𝑍) → ((𝑌 − 𝑌)..^(𝑍 − 𝑌)) = (0..^(𝑍 − 𝑌)))
71	70	eleq2d 2308	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (0...𝑍) → ((𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)) ↔ (𝑥 − 𝑌) ∈ (0..^(𝑍 − 𝑌))))
72	71	ad2antrl 494	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)) ↔ (𝑥 − 𝑌) ∈ (0..^(𝑍 − 𝑌))))
73	72	adantr 276	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)) ↔ (𝑥 − 𝑌) ∈ (0..^(𝑍 − 𝑌))))
74	68, 73	mpbid 147	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − 𝑌) ∈ (0..^(𝑍 − 𝑌)))
75	65, 74	eqeltrd 2315	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − (♯‘(𝑆 prefix 𝑌))) ∈ (0..^(𝑍 − 𝑌)))
76		swrdfv 11447	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) ∧ (𝑥 − (♯‘(𝑆 prefix 𝑌))) ∈ (0..^(𝑍 − 𝑌))) → ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (♯‘(𝑆 prefix 𝑌)))) = (𝑆‘((𝑥 − (♯‘(𝑆 prefix 𝑌))) + 𝑌)))
77	63, 75, 76	syl2an2r 603	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (♯‘(𝑆 prefix 𝑌)))) = (𝑆‘((𝑥 − (♯‘(𝑆 prefix 𝑌))) + 𝑌)))
78	64	oveq1d 6103	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑥 − (♯‘(𝑆 prefix 𝑌))) + 𝑌) = ((𝑥 − 𝑌) + 𝑌))
79	78	adantr 276	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − (♯‘(𝑆 prefix 𝑌))) + 𝑌) = ((𝑥 − 𝑌) + 𝑌))
80		elfzoelz 10571	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑌..^𝑍) → 𝑥 ∈ ℤ)
81	80	zcnd 9780	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑌..^𝑍) → 𝑥 ∈ ℂ)
82	27	ad2antrl 494	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℂ)
83		npcan 8542	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑌 ∈ ℂ) → ((𝑥 − 𝑌) + 𝑌) = 𝑥)
84	81, 82, 83	syl2anr 290	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − 𝑌) + 𝑌) = 𝑥)
85	79, 84	eqtrd 2271	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − (♯‘(𝑆 prefix 𝑌))) + 𝑌) = 𝑥)
86	85	fveq2d 5702	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑆‘((𝑥 − (♯‘(𝑆 prefix 𝑌))) + 𝑌)) = (𝑆‘𝑥))
87	61, 77, 86	3eqtrd 2275	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘𝑥))
88	53, 87	jaodan 809	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ (𝑥 ∈ (0..^𝑌) ∨ 𝑥 ∈ (𝑌..^𝑍))) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘𝑥))
89	41, 88	syldan 282	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑍)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘𝑥))
90		pfxfv 11478	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑍 ∈ (0...(♯‘𝑆)) ∧ 𝑥 ∈ (0..^𝑍)) → ((𝑆 prefix 𝑍)‘𝑥) = (𝑆‘𝑥))
91	90	3expa 1234	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑍 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → ((𝑆 prefix 𝑍)‘𝑥) = (𝑆‘𝑥))
92	91	adantlrl 486	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑍)) → ((𝑆 prefix 𝑍)‘𝑥) = (𝑆‘𝑥))
93	89, 92	eqtr4d 2274	. . 3 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑍)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 prefix 𝑍)‘𝑥))
94	35, 37, 93	eqfnfvd 5812	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 prefix 𝑍))
95	94	3impb 1230	1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 prefix 𝑍))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ⟨cop 3712   Fn wfn 5375  ‘cfv 5380  (class class class)co 6088  ℂcc 8183  0cc0 8185   + caddc 8188   − cmin 8504  ℕ₀cn0 9574  ℤcz 9655  ...cfz 10428  ..^cfzo 10566  ♯chash 11236  Word cword 11326   ++ cconcat 11380   substr csubstr 11439   prefix cpfx 11466

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-fzo 10567  df-ihash 11237  df-word 11327  df-concat 11381  df-substr 11440  df-pfx 11467

This theorem is used by:  pfxcctswrd  11504  wrdeqs1cat  11514