Theorem ccatswrd 11362

Description: Joining two adjacent subwords makes a longer subword. (Contributed by Stefan O'Rear, 20-Aug-2015.)

Assertion

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

Proof of Theorem ccatswrd

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑆 ∈ Word 𝐴)
2		elfzelz 10359	. . . . . . . 8 ⊢ (𝑋 ∈ (0...𝑌) → 𝑋 ∈ ℤ)
3	2	3ad2ant1 1045	. . . . . . 7 ⊢ ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → 𝑋 ∈ ℤ)
4	3	adantl 277	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ ℤ)
5		elfzel2 10357	. . . . . . . 8 ⊢ (𝑋 ∈ (0...𝑌) → 𝑌 ∈ ℤ)
6	5	3ad2ant1 1045	. . . . . . 7 ⊢ ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → 𝑌 ∈ ℤ)
7	6	adantl 277	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℤ)
8		swrdclg 11342	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
9	1, 4, 7, 8	syl3anc 1274	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
10		elfzel2 10357	. . . . . . . 8 ⊢ (𝑌 ∈ (0...𝑍) → 𝑍 ∈ ℤ)
11	10	3ad2ant2 1046	. . . . . . 7 ⊢ ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → 𝑍 ∈ ℤ)
12	11	adantl 277	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ ℤ)
13		swrdclg 11342	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
14	1, 7, 12, 13	syl3anc 1274	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
15		ccatcl 11281	. . . . 5 ⊢ (((𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
16	9, 14, 15	syl2anc 411	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
17		wrdfn 11239	. . . 4 ⊢ (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴 → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))))
18	16, 17	syl 14	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))))
19		ccatlen 11283	. . . . . . 7 ⊢ (((𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴) → (♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))))
20	9, 14, 19	syl2anc 411	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))))
21		simpr1 1030	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ (0...𝑌))
22		simpr2 1031	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ (0...𝑍))
23		simpr3 1032	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ (0...(♯‘𝑆)))
24		fzass4 10396	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑍 ∈ (𝑌...(♯‘𝑆))) ↔ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))))
25	24	biimpri 133	. . . . . . . . . 10 ⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑍 ∈ (𝑌...(♯‘𝑆))))
26	25	simpld 112	. . . . . . . . 9 ⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → 𝑌 ∈ (0...(♯‘𝑆)))
27	22, 23, 26	syl2anc 411	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ (0...(♯‘𝑆)))
28		swrdlen 11344	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑌 − 𝑋))
29	1, 21, 27, 28	syl3anc 1274	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑌 − 𝑋))
30		swrdlen 11344	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍 − 𝑌))
31	30	3adant3r1 1239	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍 − 𝑌))
32	29, 31	oveq12d 6068	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = ((𝑌 − 𝑋) + (𝑍 − 𝑌)))
33	22	elfzelzd 10360	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℤ)
34	33	zcnd 9701	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℂ)
35	21	elfzelzd 10360	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ ℤ)
36	35	zcnd 9701	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ ℂ)
37	23	elfzelzd 10360	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ ℤ)
38	37	zcnd 9701	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ ℂ)
39	34, 36, 38	npncan3d 8620	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑌 − 𝑋) + (𝑍 − 𝑌)) = (𝑍 − 𝑋))
40	20, 32, 39	3eqtrd 2269	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑍 − 𝑋))
41	40	oveq2d 6066	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))) = (0..^(𝑍 − 𝑋)))
42	41	fneq2d 5447	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)))) ↔ ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(𝑍 − 𝑋))))
43	18, 42	mpbid 147	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) Fn (0..^(𝑍 − 𝑋)))
44		swrdclg 11342	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ ℤ ∧ 𝑍 ∈ ℤ) → (𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴)
45	1, 4, 12, 44	syl3anc 1274	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴)
46		wrdfn 11239	. . . 4 ⊢ ((𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴 → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑍⟩))))
47	45, 46	syl 14	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑍⟩))))
48		fzass4 10396	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0...𝑍) ∧ 𝑌 ∈ (𝑋...𝑍)) ↔ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)))
49	48	biimpri 133	. . . . . . . 8 ⊢ ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)) → (𝑋 ∈ (0...𝑍) ∧ 𝑌 ∈ (𝑋...𝑍)))
50	49	simpld 112	. . . . . . 7 ⊢ ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)) → 𝑋 ∈ (0...𝑍))
51	21, 22, 50	syl2anc 411	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ (0...𝑍))
52		swrdlen 11344	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨𝑋, 𝑍⟩)) = (𝑍 − 𝑋))
53	1, 51, 23, 52	syl3anc 1274	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 substr ⟨𝑋, 𝑍⟩)) = (𝑍 − 𝑋))
54	53	oveq2d 6066	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑍⟩))) = (0..^(𝑍 − 𝑋)))
55	54	fneq2d 5447	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑍⟩))) ↔ (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(𝑍 − 𝑋))))
56	47, 55	mpbid 147	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(𝑍 − 𝑋)))
57	33, 35	zsubcld 9705	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 − 𝑋) ∈ ℤ)
58	57	anim1ci 341	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → (𝑥 ∈ (0..^(𝑍 − 𝑋)) ∧ (𝑌 − 𝑋) ∈ ℤ))
59		fzospliti 10512	. . . . 5 ⊢ ((𝑥 ∈ (0..^(𝑍 − 𝑋)) ∧ (𝑌 − 𝑋) ∈ ℤ) → (𝑥 ∈ (0..^(𝑌 − 𝑋)) ∨ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))))
60	58, 59	syl 14	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → (𝑥 ∈ (0..^(𝑌 − 𝑋)) ∨ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))))
61	9	adantr 276	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
62	14	adantr 276	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
63	29	oveq2d 6066	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) = (0..^(𝑌 − 𝑋)))
64	63	eleq2d 2302	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) ↔ 𝑥 ∈ (0..^(𝑌 − 𝑋))))
65	64	biimpar 297	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑥 ∈ (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))))
66		ccatval1 11285	. . . . . . 7 ⊢ (((𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥))
67	61, 62, 65, 66	syl3anc 1274	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥))
68		simpll 527	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑆 ∈ Word 𝐴)
69		simplr1 1066	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑋 ∈ (0...𝑌))
70	27	adantr 276	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑌 ∈ (0...(♯‘𝑆)))
71		simpr 110	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑥 ∈ (0..^(𝑌 − 𝑋)))
72		swrdfv 11345	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
73	68, 69, 70, 71, 72	syl31anc 1277	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
74	67, 73	eqtrd 2265	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
75	9	adantr 276	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
76	14	adantr 276	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
77	32, 39	eqtrd 2265	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑍 − 𝑋))
78	29, 77	oveq12d 6068	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))..^((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)))) = ((𝑌 − 𝑋)..^(𝑍 − 𝑋)))
79	78	eleq2d 2302	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))..^((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)))) ↔ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))))
80	79	biimpar 297	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑥 ∈ ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))..^((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)))))
81		ccatval2 11286	. . . . . . 7 ⊢ (((𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴 ∧ 𝑥 ∈ ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))..^((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)))))
82	75, 76, 80, 81	syl3anc 1274	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)))))
83		simpll 527	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑆 ∈ Word 𝐴)
84		simplr2 1067	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑌 ∈ (0...𝑍))
85		simplr3 1068	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑍 ∈ (0...(♯‘𝑆)))
86	29	oveq2d 6066	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) = (𝑥 − (𝑌 − 𝑋)))
87	86	adantr 276	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) = (𝑥 − (𝑌 − 𝑋)))
88	39	oveq2d 6066	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌))) = ((𝑌 − 𝑋)..^(𝑍 − 𝑋)))
89	88	eleq2d 2302	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌))) ↔ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))))
90	89	biimpar 297	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑥 ∈ ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌))))
91	37, 33	zsubcld 9705	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑍 − 𝑌) ∈ ℤ)
92	91	adantr 276	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑍 − 𝑌) ∈ ℤ)
93		fzosubel3 10541	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌))) ∧ (𝑍 − 𝑌) ∈ ℤ) → (𝑥 − (𝑌 − 𝑋)) ∈ (0..^(𝑍 − 𝑌)))
94	90, 92, 93	syl2anc 411	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑥 − (𝑌 − 𝑋)) ∈ (0..^(𝑍 − 𝑌)))
95	87, 94	eqeltrd 2309	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) ∈ (0..^(𝑍 − 𝑌)))
96		swrdfv 11345	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) ∧ (𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) ∈ (0..^(𝑍 − 𝑌))) → ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)))) = (𝑆‘((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌)))
97	83, 84, 85, 95, 96	syl31anc 1277	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → ((𝑆 substr ⟨𝑌, 𝑍⟩)‘(𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)))) = (𝑆‘((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌)))
98	86	oveq1d 6065	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌) = ((𝑥 − (𝑌 − 𝑋)) + 𝑌))
99	98	adantr 276	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → ((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌) = ((𝑥 − (𝑌 − 𝑋)) + 𝑌))
100		elfzoelz 10481	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)) → 𝑥 ∈ ℤ)
101	100	zcnd 9701	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)) → 𝑥 ∈ ℂ)
102	101	adantl 277	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑥 ∈ ℂ)
103	34, 36	subcld 8584	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 − 𝑋) ∈ ℂ)
104	103	adantr 276	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑌 − 𝑋) ∈ ℂ)
105	34	adantr 276	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑌 ∈ ℂ)
106	102, 104, 105	subadd23d 8606	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → ((𝑥 − (𝑌 − 𝑋)) + 𝑌) = (𝑥 + (𝑌 − (𝑌 − 𝑋))))
107	34, 36	nncand 8589	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 − (𝑌 − 𝑋)) = 𝑋)
108	107	oveq2d 6066	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 + (𝑌 − (𝑌 − 𝑋))) = (𝑥 + 𝑋))
109	108	adantr 276	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑥 + (𝑌 − (𝑌 − 𝑋))) = (𝑥 + 𝑋))
110	99, 106, 109	3eqtrd 2269	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → ((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌) = (𝑥 + 𝑋))
111	110	fveq2d 5674	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑆‘((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌)) = (𝑆‘(𝑥 + 𝑋)))
112	82, 97, 111	3eqtrd 2269	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
113	74, 112	jaodan 805	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ (𝑥 ∈ (0..^(𝑌 − 𝑋)) ∨ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
114	60, 113	syldan 282	. . 3 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
115		simpll 527	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑆 ∈ Word 𝐴)
116	51	adantr 276	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑋 ∈ (0...𝑍))
117		simplr3 1068	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑍 ∈ (0...(♯‘𝑆)))
118		simpr 110	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑥 ∈ (0..^(𝑍 − 𝑋)))
119		swrdfv 11345	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → ((𝑆 substr ⟨𝑋, 𝑍⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
120	115, 116, 117, 118, 119	syl31anc 1277	. . 3 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → ((𝑆 substr ⟨𝑋, 𝑍⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
121	114, 120	eqtr4d 2268	. 2 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → (((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))‘𝑥) = ((𝑆 substr ⟨𝑋, 𝑍⟩)‘𝑥))
122	43, 56, 121	eqfnfvd 5778	1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑆 substr ⟨𝑋, 𝑍⟩))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 716   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  ⟨cop 3692   Fn wfn 5347  ‘cfv 5352  (class class class)co 6050  ℂcc 8125  0cc0 8127   + caddc 8130   − cmin 8444  ℤcz 9577  ...cfz 10342  ..^cfzo 10476  ♯chash 11138  Word cword 11224   ++ cconcat 11278   substr csubstr 11337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-concat 11279  df-substr 11338

This theorem is referenced by: (None)