Theorem ccatpfx 14749

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		pfxcl 14725	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 prefix 𝑌) ∈ Word 𝐴)
2		swrdcl 14693	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
3		ccatcl 14622	. . . . . . 7 ⊢ (((𝑆 prefix 𝑌) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
4	1, 2, 3	syl2anc 583	. . . . . 6 ⊢ (𝑆 ∈ Word 𝐴 → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
5		wrdfn 14576	. . . . . 6 ⊢ (((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⟩)))))
6	4, 5	syl 17	. . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(♯‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))))
7	6	adantr 480	. . . 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⟩)))))
8		ccatlen 14623	. . . . . . . . 9 ⊢ (((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⟩))))
9	1, 2, 8	syl2anc 583	. . . . . . . 8 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = ((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))))
10	9	adantr 480	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = ((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))))
11		fzass4 13622	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑍 ∈ (𝑌...(♯‘𝑆))) ↔ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))))
12	11	biimpri 228	. . . . . . . . . 10 ⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑍 ∈ (𝑌...(♯‘𝑆))))
13	12	simpld 494	. . . . . . . . 9 ⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → 𝑌 ∈ (0...(♯‘𝑆)))
14		pfxlen 14731	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 prefix 𝑌)) = 𝑌)
15	13, 14	sylan2 592	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 prefix 𝑌)) = 𝑌)
16		swrdlen 14695	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍 − 𝑌))
17	16	3expb 1120	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍 − 𝑌))
18	15, 17	oveq12d 7466	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑌 + (𝑍 − 𝑌)))
19		elfzelz 13584	. . . . . . . . . 10 ⊢ (𝑌 ∈ (0...𝑍) → 𝑌 ∈ ℤ)
20	19	zcnd 12748	. . . . . . . . 9 ⊢ (𝑌 ∈ (0...𝑍) → 𝑌 ∈ ℂ)
21		elfzelz 13584	. . . . . . . . . 10 ⊢ (𝑍 ∈ (0...(♯‘𝑆)) → 𝑍 ∈ ℤ)
22	21	zcnd 12748	. . . . . . . . 9 ⊢ (𝑍 ∈ (0...(♯‘𝑆)) → 𝑍 ∈ ℂ)
23		pncan3 11544	. . . . . . . . 9 ⊢ ((𝑌 ∈ ℂ ∧ 𝑍 ∈ ℂ) → (𝑌 + (𝑍 − 𝑌)) = 𝑍)
24	20, 22, 23	syl2an 595	. . . . . . . 8 ⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (𝑌 + (𝑍 − 𝑌)) = 𝑍)
25	24	adantl 481	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 + (𝑍 − 𝑌)) = 𝑍)
26	10, 18, 25	3eqtrd 2784	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = 𝑍)
27	26	oveq2d 7464	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))) = (0..^𝑍))
28	27	fneq2d 6673	. . . 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)))
29	7, 28	mpbid 232	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^𝑍))
30		pfxfn 14729	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (𝑆 prefix 𝑍) Fn (0..^𝑍))
31	30	adantrl 715	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 prefix 𝑍) Fn (0..^𝑍))
32		id 22	. . . . . 6 ⊢ (𝑥 ∈ (0..^𝑍) → 𝑥 ∈ (0..^𝑍))
33	19	ad2antrl 727	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℤ)
34		fzospliti 13748	. . . . . 6 ⊢ ((𝑥 ∈ (0..^𝑍) ∧ 𝑌 ∈ ℤ) → (𝑥 ∈ (0..^𝑌) ∨ 𝑥 ∈ (𝑌..^𝑍)))
35	32, 33, 34	syl2anr 596	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑍)) → (𝑥 ∈ (0..^𝑌) ∨ 𝑥 ∈ (𝑌..^𝑍)))
36	1	ad2antrr 725	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑌)) → (𝑆 prefix 𝑌) ∈ Word 𝐴)
37	2	ad2antrr 725	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑌)) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
38	15	oveq2d 7464	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘(𝑆 prefix 𝑌))) = (0..^𝑌))
39	38	eleq2d 2830	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ (0..^(♯‘(𝑆 prefix 𝑌))) ↔ 𝑥 ∈ (0..^𝑌)))
40	39	biimpar 477	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑌)) → 𝑥 ∈ (0..^(♯‘(𝑆 prefix 𝑌))))
41		ccatval1 14625	. . . . . . . 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))
42	36, 37, 40, 41	syl3anc 1371	. . . . . . 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))
43		simpl 482	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑆 ∈ Word 𝐴)
44	13	adantl 481	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ (0...(♯‘𝑆)))
45		id 22	. . . . . . . 8 ⊢ (𝑥 ∈ (0..^𝑌) → 𝑥 ∈ (0..^𝑌))
46		pfxfv 14730	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑥 ∈ (0..^𝑌)) → ((𝑆 prefix 𝑌)‘𝑥) = (𝑆‘𝑥))
47	43, 44, 45, 46	syl2an3an 1422	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑌)) → ((𝑆 prefix 𝑌)‘𝑥) = (𝑆‘𝑥))
48	42, 47	eqtrd 2780	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑌)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘𝑥))
49	1	ad2antrr 725	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑆 prefix 𝑌) ∈ Word 𝐴)
50	2	ad2antrr 725	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
51	18, 25	eqtrd 2780	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = 𝑍)
52	15, 51	oveq12d 7466	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 prefix 𝑌))..^((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)))) = (𝑌..^𝑍))
53	52	eleq2d 2830	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ ((♯‘(𝑆 prefix 𝑌))..^((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)))) ↔ 𝑥 ∈ (𝑌..^𝑍)))
54	53	biimpar 477	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → 𝑥 ∈ ((♯‘(𝑆 prefix 𝑌))..^((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)))))
55		ccatval2 14626	. . . . . . . 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)))))
56	49, 50, 54, 55	syl3anc 1371	. . . . . . 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)))))
57		id 22	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))))
58	57	3expb 1120	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))))
59	15	oveq2d 7464	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 − (♯‘(𝑆 prefix 𝑌))) = (𝑥 − 𝑌))
60	59	adantr 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − (♯‘(𝑆 prefix 𝑌))) = (𝑥 − 𝑌))
61		id 22	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑌..^𝑍) → 𝑥 ∈ (𝑌..^𝑍))
62		fzosubel 13775	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝑌..^𝑍) ∧ 𝑌 ∈ ℤ) → (𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)))
63	61, 33, 62	syl2anr 596	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)))
64	20	subidd 11635	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∈ (0...𝑍) → (𝑌 − 𝑌) = 0)
65	64	oveq1d 7463	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (0...𝑍) → ((𝑌 − 𝑌)..^(𝑍 − 𝑌)) = (0..^(𝑍 − 𝑌)))
66	65	eleq2d 2830	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (0...𝑍) → ((𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)) ↔ (𝑥 − 𝑌) ∈ (0..^(𝑍 − 𝑌))))
67	66	ad2antrl 727	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)) ↔ (𝑥 − 𝑌) ∈ (0..^(𝑍 − 𝑌))))
68	67	adantr 480	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)) ↔ (𝑥 − 𝑌) ∈ (0..^(𝑍 − 𝑌))))
69	63, 68	mpbid 232	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − 𝑌) ∈ (0..^(𝑍 − 𝑌)))
70	60, 69	eqeltrd 2844	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − (♯‘(𝑆 prefix 𝑌))) ∈ (0..^(𝑍 − 𝑌)))
71		swrdfv 14696	. . . . . . . 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)))
72	58, 70, 71	syl2an2r 684	. . . . . . 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)))
73	59	oveq1d 7463	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑥 − (♯‘(𝑆 prefix 𝑌))) + 𝑌) = ((𝑥 − 𝑌) + 𝑌))
74	73	adantr 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − (♯‘(𝑆 prefix 𝑌))) + 𝑌) = ((𝑥 − 𝑌) + 𝑌))
75		elfzoelz 13716	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑌..^𝑍) → 𝑥 ∈ ℤ)
76	75	zcnd 12748	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑌..^𝑍) → 𝑥 ∈ ℂ)
77	20	ad2antrl 727	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℂ)
78		npcan 11545	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑌 ∈ ℂ) → ((𝑥 − 𝑌) + 𝑌) = 𝑥)
79	76, 77, 78	syl2anr 596	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − 𝑌) + 𝑌) = 𝑥)
80	74, 79	eqtrd 2780	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − (♯‘(𝑆 prefix 𝑌))) + 𝑌) = 𝑥)
81	80	fveq2d 6924	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑆‘((𝑥 − (♯‘(𝑆 prefix 𝑌))) + 𝑌)) = (𝑆‘𝑥))
82	56, 72, 81	3eqtrd 2784	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘𝑥))
83	48, 82	jaodan 958	. . . . 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))
84	35, 83	syldan 590	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑍)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘𝑥))
85		pfxfv 14730	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑍 ∈ (0...(♯‘𝑆)) ∧ 𝑥 ∈ (0..^𝑍)) → ((𝑆 prefix 𝑍)‘𝑥) = (𝑆‘𝑥))
86	85	3expa 1118	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑍 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → ((𝑆 prefix 𝑍)‘𝑥) = (𝑆‘𝑥))
87	86	adantlrl 719	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑍)) → ((𝑆 prefix 𝑍)‘𝑥) = (𝑆‘𝑥))
88	84, 87	eqtr4d 2783	. . 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))
89	29, 31, 88	eqfnfvd 7067	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 prefix 𝑍))
90	89	3impb 1115	1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 prefix 𝑍))

Syntax hints:   → wi 4   ↔ wb 206   ∧ 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   prefix cpfx 14718

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  df-pfx 14719

This theorem is referenced by:  pfxcctswrd  14758  wrdeqs1cat  14768  splid  14801  splval2  14805  efgredleme  19785  efgredlemc  19787  efgcpbllemb  19797  frgpuplem  19814  wrdsplex  32902