Theorem ccatpfx 40272

Description: Joining 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 40249	. . . . . 6 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 prefix 𝑌) ∈ Word 𝐴)
2	1	3ad2ant1 1075	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑆 prefix 𝑌) ∈ Word 𝐴)
3		swrdcl 13271	. . . . . 6 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
4	3	3ad2ant1 1075	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
5		ccatcl 13212	. . . . 5 ⊢ (((𝑆 prefix 𝑌) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
6	2, 4, 5	syl2anc 691	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
7		wrdf 13165	. . . 4 ⊢ (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴 → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)):(0..^(#‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))))⟶𝐴)
8		ffn 5958	. . . 4 ⊢ (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)):(0..^(#‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))))⟶𝐴 → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(#‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))))
9	6, 7, 8	3syl 18	. . 3 ⊢ ((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⟩)))))
10		ccatlen 13213	. . . . . . 7 ⊢ (((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⟩))))
11	2, 4, 10	syl2anc 691	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (#‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = ((#‘(𝑆 prefix 𝑌)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩))))
12		simp1 1054	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → 𝑆 ∈ Word 𝐴)
13		fzass4 12250	. . . . . . . . . . . 12 ⊢ ((𝑌 ∈ (0...(#‘𝑆)) ∧ 𝑍 ∈ (𝑌...(#‘𝑆))) ↔ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))))
14	13	biimpri 217	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑌 ∈ (0...(#‘𝑆)) ∧ 𝑍 ∈ (𝑌...(#‘𝑆))))
15	14	simpld 474	. . . . . . . . . 10 ⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → 𝑌 ∈ (0...(#‘𝑆)))
16	15	3adant1 1072	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → 𝑌 ∈ (0...(#‘𝑆)))
17		pfxlen 40254	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...(#‘𝑆))) → (#‘(𝑆 prefix 𝑌)) = 𝑌)
18	12, 16, 17	syl2anc 691	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (#‘(𝑆 prefix 𝑌)) = 𝑌)
19		swrdlen 13275	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (#‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍 − 𝑌))
20	18, 19	oveq12d 6567	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → ((#‘(𝑆 prefix 𝑌)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑌 + (𝑍 − 𝑌)))
21		elfzelz 12213	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (0...𝑍) → 𝑌 ∈ ℤ)
22	21	ad2antrl 760	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ ℤ)
23	22	zcnd 11359	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑌 ∈ ℂ)
24	23	3impb 1252	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → 𝑌 ∈ ℂ)
25		elfzelz 12213	. . . . . . . . . . 11 ⊢ (𝑍 ∈ (0...(#‘𝑆)) → 𝑍 ∈ ℤ)
26	25	ad2antll 761	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑍 ∈ ℤ)
27	26	zcnd 11359	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆)))) → 𝑍 ∈ ℂ)
28	27	3impb 1252	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → 𝑍 ∈ ℂ)
29	24, 28	pncan3d 10274	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑌 + (𝑍 − 𝑌)) = 𝑍)
30	20, 29	eqtrd 2644	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → ((#‘(𝑆 prefix 𝑌)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = 𝑍)
31	11, 30	eqtrd 2644	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (#‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = 𝑍)
32	31	oveq2d 6565	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (0..^(#‘((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))) = (0..^𝑍))
33	32	fneq2d 5896	. . 3 ⊢ ((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)))
34	9, 33	mpbid 221	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → ((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^𝑍))
35		pfxfn 40253	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑆 prefix 𝑍) Fn (0..^𝑍))
36	35	3adant2 1073	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑆 prefix 𝑍) Fn (0..^𝑍))
37		simpr 476	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → 𝑥 ∈ (0..^𝑍))
38	21	3ad2ant2 1076	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → 𝑌 ∈ ℤ)
39	38	adantr 480	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → 𝑌 ∈ ℤ)
40		fzospliti 12369	. . . . 5 ⊢ ((𝑥 ∈ (0..^𝑍) ∧ 𝑌 ∈ ℤ) → (𝑥 ∈ (0..^𝑌) ∨ 𝑥 ∈ (𝑌..^𝑍)))
41	37, 39, 40	syl2anc 691	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → (𝑥 ∈ (0..^𝑌) ∨ 𝑥 ∈ (𝑌..^𝑍)))
42	2	adantr 480	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑌)) → (𝑆 prefix 𝑌) ∈ Word 𝐴)
43	4	adantr 480	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑌)) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
44	18	oveq2d 6565	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (0..^(#‘(𝑆 prefix 𝑌))) = (0..^𝑌))
45	44	eleq2d 2673	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑥 ∈ (0..^(#‘(𝑆 prefix 𝑌))) ↔ 𝑥 ∈ (0..^𝑌)))
46	45	biimpar 501	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑌)) → 𝑥 ∈ (0..^(#‘(𝑆 prefix 𝑌))))
47		ccatval1 13214	. . . . . . 7 ⊢ (((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 1318	. . . . . 6 ⊢ (((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	12	adantr 480	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑌)) → 𝑆 ∈ Word 𝐴)
50	16	adantr 480	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑌)) → 𝑌 ∈ (0...(#‘𝑆)))
51		simpr 476	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑌)) → 𝑥 ∈ (0..^𝑌))
52		pfxfv 40262	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...(#‘𝑆)) ∧ 𝑥 ∈ (0..^𝑌)) → ((𝑆 prefix 𝑌)‘𝑥) = (𝑆‘𝑥))
53	49, 50, 51, 52	syl3anc 1318	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑌)) → ((𝑆 prefix 𝑌)‘𝑥) = (𝑆‘𝑥))
54	48, 53	eqtrd 2644	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑌)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘𝑥))
55	2	adantr 480	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑆 prefix 𝑌) ∈ Word 𝐴)
56	4	adantr 480	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
57	18, 30	oveq12d 6567	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → ((#‘(𝑆 prefix 𝑌))..^((#‘(𝑆 prefix 𝑌)) + (#‘(𝑆 substr ⟨𝑌, 𝑍⟩)))) = (𝑌..^𝑍))
58	57	eleq2d 2673	. . . . . . . 8 ⊢ ((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 501	. . . . . . 7 ⊢ (((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 13215	. . . . . . 7 ⊢ (((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	55, 56, 59, 60	syl3anc 1318	. . . . . 6 ⊢ (((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	18	oveq2d 6565	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (𝑥 − (#‘(𝑆 prefix 𝑌))) = (𝑥 − 𝑌))
63	62	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − (#‘(𝑆 prefix 𝑌))) = (𝑥 − 𝑌))
64	38	anim1i 590	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑌 ∈ ℤ ∧ 𝑥 ∈ (𝑌..^𝑍)))
65	64	ancomd 466	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 ∈ (𝑌..^𝑍) ∧ 𝑌 ∈ ℤ))
66		fzosubel 12394	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑌..^𝑍) ∧ 𝑌 ∈ ℤ) → (𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)))
67	65, 66	syl 17	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌)))
68	21	zcnd 11359	. . . . . . . . . . . . . . 15 ⊢ (𝑌 ∈ (0...𝑍) → 𝑌 ∈ ℂ)
69	68	subidd 10259	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∈ (0...𝑍) → (𝑌 − 𝑌) = 0)
70	69	eqcomd 2616	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ (0...𝑍) → 0 = (𝑌 − 𝑌))
71	70	3ad2ant2 1076	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → 0 = (𝑌 − 𝑌))
72	71	oveq1d 6564	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → (0..^(𝑍 − 𝑌)) = ((𝑌 − 𝑌)..^(𝑍 − 𝑌)))
73	72	eleq2d 2673	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) → ((𝑥 − 𝑌) ∈ (0..^(𝑍 − 𝑌)) ↔ (𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌))))
74	73	adantr 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − 𝑌) ∈ (0..^(𝑍 − 𝑌)) ↔ (𝑥 − 𝑌) ∈ ((𝑌 − 𝑌)..^(𝑍 − 𝑌))))
75	67, 74	mpbird 246	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − 𝑌) ∈ (0..^(𝑍 − 𝑌)))
76	63, 75	eqeltrd 2688	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑥 − (#‘(𝑆 prefix 𝑌))) ∈ (0..^(𝑍 − 𝑌)))
77		swrdfv 13276	. . . . . . 7 ⊢ (((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)))
78	76, 77	syldan 486	. . . . . 6 ⊢ (((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)))
79	63	oveq1d 6564	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − (#‘(𝑆 prefix 𝑌))) + 𝑌) = ((𝑥 − 𝑌) + 𝑌))
80		elfzoelz 12339	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑌..^𝑍) → 𝑥 ∈ ℤ)
81	80	zcnd 11359	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑌..^𝑍) → 𝑥 ∈ ℂ)
82	81	adantl 481	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → 𝑥 ∈ ℂ)
83	24	adantr 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → 𝑌 ∈ ℂ)
84	82, 83	npcand 10275	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − 𝑌) + 𝑌) = 𝑥)
85	79, 84	eqtrd 2644	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → ((𝑥 − (#‘(𝑆 prefix 𝑌))) + 𝑌) = 𝑥)
86	85	fveq2d 6107	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (𝑆‘((𝑥 − (#‘(𝑆 prefix 𝑌))) + 𝑌)) = (𝑆‘𝑥))
87	61, 78, 86	3eqtrd 2648	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (𝑌..^𝑍)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘𝑥))
88	54, 87	jaodan 822	. . . 4 ⊢ (((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 486	. . 3 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘𝑥))
90	12	adantr 480	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → 𝑆 ∈ Word 𝐴)
91		simpl3 1059	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → 𝑍 ∈ (0...(#‘𝑆)))
92		pfxfv 40262	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑍 ∈ (0...(#‘𝑆)) ∧ 𝑥 ∈ (0..^𝑍)) → ((𝑆 prefix 𝑍)‘𝑥) = (𝑆‘𝑥))
93	90, 91, 37, 92	syl3anc 1318	. . 3 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → ((𝑆 prefix 𝑍)‘𝑥) = (𝑆‘𝑥))
94	89, 93	eqtr4d 2647	. 2 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(#‘𝑆))) ∧ 𝑥 ∈ (0..^𝑍)) → (((𝑆 prefix 𝑌) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 prefix 𝑍)‘𝑥))
95	34, 36, 94	eqfnfvd 6222	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 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  0cc0 9815   + caddc 9818   − cmin 10145  ℤcz 11254  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   ++ cconcat 13148   substr csubstr 13150   prefix cpfx 40244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158  df-pfx 40245

This theorem is referenced by:  pfxcctswrd  40280