Theorem ccatpfx 14736

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 14712	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 prefix 𝑌) ∈ Word 𝐴)
2		swrdcl 14680	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
3		ccatcl 14609	. . . . . . 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 584	. . . . . 6 ⊢ (𝑆 ∈ Word 𝐴 → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
5		wrdfn 14563	. . . . . 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 14610	. . . . . . . . 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 584	. . . . . . . 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 13599	. . . . . . . . . . 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 14718	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 prefix 𝑌)) = 𝑌)
15	13, 14	sylan2 593	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 prefix 𝑌)) = 𝑌)
16		swrdlen 14682	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍 − 𝑌))
17	16	3expb 1119	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍 − 𝑌))
18	15, 17	oveq12d 7449	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑌 + (𝑍 − 𝑌)))
19		elfzelz 13561	. . . . . . . . . 10 ⊢ (𝑌 ∈ (0...𝑍) → 𝑌 ∈ ℤ)
20	19	zcnd 12721	. . . . . . . . 9 ⊢ (𝑌 ∈ (0...𝑍) → 𝑌 ∈ ℂ)
21		elfzelz 13561	. . . . . . . . . 10 ⊢ (𝑍 ∈ (0...(♯‘𝑆)) → 𝑍 ∈ ℤ)
22	21	zcnd 12721	. . . . . . . . 9 ⊢ (𝑍 ∈ (0...(♯‘𝑆)) → 𝑍 ∈ ℂ)
23		pncan3 11514	. . . . . . . . 9 ⊢ ((𝑌 ∈ ℂ ∧ 𝑍 ∈ ℂ) → (𝑌 + (𝑍 − 𝑌)) = 𝑍)
24	20, 22, 23	syl2an 596	. . . . . . . 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 2779	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = 𝑍)
27	26	oveq2d 7447	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))) = (0..^𝑍))
28	27	fneq2d 6663	. . . 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 14716	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (𝑆 prefix 𝑍) Fn (0..^𝑍))
31	30	adantrl 716	. . 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 728	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℤ)
34		fzospliti 13728	. . . . . 6 ⊢ ((𝑥 ∈ (0..^𝑍) ∧ 𝑌 ∈ ℤ) → (𝑥 ∈ (0..^𝑌) ∨ 𝑥 ∈ (𝑌..^𝑍)))
35	32, 33, 34	syl2anr 597	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑍)) → (𝑥 ∈ (0..^𝑌) ∨ 𝑥 ∈ (𝑌..^𝑍)))
36	1	ad2antrr 726	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑌)) → (𝑆 prefix 𝑌) ∈ Word 𝐴)
37	2	ad2antrr 726	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑌)) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
38	15	oveq2d 7447	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘(𝑆 prefix 𝑌))) = (0..^𝑌))
39	38	eleq2d 2825	. . . . . . . . 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 14612	. . . . . . . 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 1370	. . . . . . 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 14717	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑥 ∈ (0..^𝑌)) → ((𝑆 prefix 𝑌)‘𝑥) = (𝑆‘𝑥))
47	43, 44, 45, 46	syl2an3an 1421	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑌)) → ((𝑆 prefix 𝑌)‘𝑥) = (𝑆‘𝑥))
48	42, 47	eqtrd 2775	. . . . . 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 726	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑆 prefix 𝑌) ∈ Word 𝐴)
50	2	ad2antrr 726	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
51	18, 25	eqtrd 2775	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 prefix 𝑌)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = 𝑍)
52	15, 51	oveq12d 7449	. . . . . . . . . 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 2825	. . . . . . . . 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 14613	. . . . . . . 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 1370	. . . . . . 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 1119	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))))
59	15	oveq2d 7447	. . . . . . . . . 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 13760	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝑌..^𝑍) ∧ 𝑌 ∈ ℤ) → (𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)))
63	61, 33, 62	syl2anr 597	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)))
64	20	subidd 11606	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∈ (0...𝑍) → (𝑌 − 𝑌) = 0)
65	64	oveq1d 7446	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (0...𝑍) → ((𝑌 − 𝑌)..^(𝑍 − 𝑌)) = (0..^(𝑍 − 𝑌)))
66	65	eleq2d 2825	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (0...𝑍) → ((𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)) ↔ (𝑥 − 𝑌) ∈ (0..^(𝑍 − 𝑌))))
67	66	ad2antrl 728	. . . . . . . . . . 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 2839	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − (♯‘(𝑆 prefix 𝑌))) ∈ (0..^(𝑍 − 𝑌)))
71		swrdfv 14683	. . . . . . . 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 685	. . . . . . 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 7446	. . . . . . . . . 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 13696	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑌..^𝑍) → 𝑥 ∈ ℤ)
76	75	zcnd 12721	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑌..^𝑍) → 𝑥 ∈ ℂ)
77	20	ad2antrl 728	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℂ)
78		npcan 11515	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ 𝑌 ∈ ℂ) → ((𝑥 − 𝑌) + 𝑌) = 𝑥)
79	76, 77, 78	syl2anr 597	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − 𝑌) + 𝑌) = 𝑥)
80	74, 79	eqtrd 2775	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − (♯‘(𝑆 prefix 𝑌))) + 𝑌) = 𝑥)
81	80	fveq2d 6911	. . . . . . 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 2779	. . . . . 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 959	. . . . 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 591	. . . 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 14717	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑍 ∈ (0...(♯‘𝑆)) ∧ 𝑥 ∈ (0..^𝑍)) → ((𝑆 prefix 𝑍)‘𝑥) = (𝑆‘𝑥))
86	85	3expa 1117	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑍 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → ((𝑆 prefix 𝑍)‘𝑥) = (𝑆‘𝑥))
87	86	adantlrl 720	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^𝑍)) → ((𝑆 prefix 𝑍)‘𝑥) = (𝑆‘𝑥))
88	84, 87	eqtr4d 2778	. . 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 7054	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 prefix 𝑍))
90	89	3impb 1114	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 847   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ⟨cop 4637   Fn wfn 6558  ‘cfv 6563  (class class class)co 7431  ℂcc 11151  0cc0 11153   + caddc 11156   − cmin 11490  ℤcz 12611  ...cfz 13544  ..^cfzo 13691  ♯chash 14366  Word cword 14549   ++ cconcat 14605   substr csubstr 14675   prefix cpfx 14705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-concat 14606  df-substr 14676  df-pfx 14706

This theorem is referenced by:  pfxcctswrd  14745  wrdeqs1cat  14755  splid  14788  splval2  14792  efgredleme  19776  efgredlemc  19778  efgcpbllemb  19788  frgpuplem  19805  wrdsplex  32905