Theorem ccatpfx 14663

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 14640	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 prefix 𝑌) ∈ Word 𝐴)
2		swrdcl 14608	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
3		ccatcl 14536	. . . . . . 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 585	. . . . . 6 ⊢ (𝑆 ∈ Word 𝐴 → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
5		wrdfn 14490	. . . . . 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 14537	. . . . . . . . 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 585	. . . . . . . 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 13516	. . . . . . . . . . 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 14646	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 prefix 𝑌)) = 𝑌)
15	13, 14	sylan2 594	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 prefix 𝑌)) = 𝑌)
16		swrdlen 14610	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍 − 𝑌))
17	16	3expb 1121	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍 − 𝑌))
18	15, 17	oveq12d 7385	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑌 + (𝑍 − 𝑌)))
19		elfzelz 13478	. . . . . . . . . 10 ⊢ (𝑌 ∈ (0...𝑍) → 𝑌 ∈ ℤ)
20	19	zcnd 12634	. . . . . . . . 9 ⊢ (𝑌 ∈ (0...𝑍) → 𝑌 ∈ ℂ)
21		elfzelz 13478	. . . . . . . . . 10 ⊢ (𝑍 ∈ (0...(♯‘𝑆)) → 𝑍 ∈ ℤ)
22	21	zcnd 12634	. . . . . . . . 9 ⊢ (𝑍 ∈ (0...(♯‘𝑆)) → 𝑍 ∈ ℂ)
23		pncan3 11401	. . . . . . . . 9 ⊢ ((𝑌 ∈ ℂ ∧ 𝑍 ∈ ℂ) → (𝑌 + (𝑍 − 𝑌)) = 𝑍)
24	20, 22, 23	syl2an 597	. . . . . . . 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 2775	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = 𝑍)
27	26	oveq2d 7383	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))) = (0..^𝑍))
28	27	fneq2d 6592	. . . 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 14644	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (𝑆 prefix 𝑍) Fn (0..^𝑍))
31	30	adantrl 717	. . 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 729	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℤ)
34		fzospliti 13646	. . . . . 6 ⊢ ((𝑥 ∈ (0..^𝑍) ∧ 𝑌 ∈ ℤ) → (𝑥 ∈ (0..^𝑌) ∨ 𝑥 ∈ (𝑌..^𝑍)))
35	32, 33, 34	syl2anr 598	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑍)) → (𝑥 ∈ (0..^𝑌) ∨ 𝑥 ∈ (𝑌..^𝑍)))
36	1	ad2antrr 727	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑌)) → (𝑆 prefix 𝑌) ∈ Word 𝐴)
37	2	ad2antrr 727	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑌)) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
38	15	oveq2d 7383	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘(𝑆 prefix 𝑌))) = (0..^𝑌))
39	38	eleq2d 2822	. . . . . . . . 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 14539	. . . . . . . 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 1374	. . . . . . 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 14645	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑥 ∈ (0..^𝑌)) → ((𝑆 prefix 𝑌)‘𝑥) = (𝑆‘𝑥))
47	43, 44, 45, 46	syl2an3an 1425	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑌)) → ((𝑆 prefix 𝑌)‘𝑥) = (𝑆‘𝑥))
48	42, 47	eqtrd 2771	. . . . . 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 727	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑆 prefix 𝑌) ∈ Word 𝐴)
50	2	ad2antrr 727	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
51	18, 25	eqtrd 2771	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = 𝑍)
52	15, 51	oveq12d 7385	. . . . . . . . . 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 2822	. . . . . . . . 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 14540	. . . . . . . 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 1374	. . . . . . 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 1121	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))))
59	15	oveq2d 7383	. . . . . . . . . 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 13679	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝑌..^𝑍) ∧ 𝑌 ∈ ℤ) → (𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)))
63	61, 33, 62	syl2anr 598	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)))
64	20	subidd 11493	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∈ (0...𝑍) → (𝑌 − 𝑌) = 0)
65	64	oveq1d 7382	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (0...𝑍) → ((𝑌 − 𝑌)..^(𝑍 − 𝑌)) = (0..^(𝑍 − 𝑌)))
66	65	eleq2d 2822	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (0...𝑍) → ((𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)) ↔ (𝑥 − 𝑌) ∈ (0..^(𝑍 − 𝑌))))
67	66	ad2antrl 729	. . . . . . . . . . 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 2836	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − (♯‘(𝑆 prefix 𝑌))) ∈ (0..^(𝑍 − 𝑌)))
71		swrdfv 14611	. . . . . . . 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 686	. . . . . . 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 7382	. . . . . . . . . 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 13613	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑌..^𝑍) → 𝑥 ∈ ℤ)
76	75	zcnd 12634	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑌..^𝑍) → 𝑥 ∈ ℂ)
77	20	ad2antrl 729	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℂ)
78		npcan 11402	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑌 ∈ ℂ) → ((𝑥 − 𝑌) + 𝑌) = 𝑥)
79	76, 77, 78	syl2anr 598	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − 𝑌) + 𝑌) = 𝑥)
80	74, 79	eqtrd 2771	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − (♯‘(𝑆 prefix 𝑌))) + 𝑌) = 𝑥)
81	80	fveq2d 6844	. . . . . . 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 2775	. . . . . 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 960	. . . . 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 592	. . . 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 14645	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑍 ∈ (0...(♯‘𝑆)) ∧ 𝑥 ∈ (0..^𝑍)) → ((𝑆 prefix 𝑍)‘𝑥) = (𝑆‘𝑥))
86	85	3expa 1119	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑍 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → ((𝑆 prefix 𝑍)‘𝑥) = (𝑆‘𝑥))
87	86	adantlrl 721	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑍)) → ((𝑆 prefix 𝑍)‘𝑥) = (𝑆‘𝑥))
88	84, 87	eqtr4d 2774	. . 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 6986	. 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 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ⟨cop 4573   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367  ℂcc 11036  0cc0 11038   + caddc 11041   − cmin 11377  ℤcz 12524  ...cfz 13461  ..^cfzo 13608  ♯chash 14292  Word cword 14475   ++ cconcat 14532   substr csubstr 14603   prefix cpfx 14633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-n0 12438  df-z 12525  df-uz 12789  df-fz 13462  df-fzo 13609  df-hash 14293  df-word 14476  df-concat 14533  df-substr 14604  df-pfx 14634

This theorem is referenced by:  pfxcctswrd  14672  wrdeqs1cat  14682  splid  14715  splval2  14719  efgredleme  19718  efgredlemc  19720  efgcpbllemb  19730  frgpuplem  19747  wrdsplex  32996