Theorem ccatswrd 14381

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		swrdcl 14358	. . . . . 6 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
2	1	adantr 481	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
3		swrdcl 14358	. . . . . 6 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
4	3	adantr 481	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
5		ccatcl 14277	. . . . 5 ⊢ (((𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴 ∧ (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
6	2, 4, 5	syl2anc 584	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩)) ∈ Word 𝐴)
7		wrdfn 14231	. . . 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⟩)))))
8	6, 7	syl 17	. . 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⟩)))))
9		ccatlen 14278	. . . . . . 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⟩))))
10	2, 4, 9	syl2anc 584	. . . . . 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⟩))))
11		simpl 483	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑆 ∈ Word 𝐴)
12		simpr1 1193	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ (0...𝑌))
13		simpr2 1194	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ (0...𝑍))
14		simpr3 1195	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ (0...(♯‘𝑆)))
15		fzass4 13294	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑍 ∈ (𝑌...(♯‘𝑆))) ↔ (𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))))
16	15	biimpri 227	. . . . . . . . . 10 ⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (𝑌 ∈ (0...(♯‘𝑆)) ∧ 𝑍 ∈ (𝑌...(♯‘𝑆))))
17	16	simpld 495	. . . . . . . . 9 ⊢ ((𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → 𝑌 ∈ (0...(♯‘𝑆)))
18	13, 14, 17	syl2anc 584	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ (0...(♯‘𝑆)))
19		swrdlen 14360	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑌 − 𝑋))
20	11, 12, 18, 19	syl3anc 1370	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) = (𝑌 − 𝑋))
21		swrdlen 14360	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍 − 𝑌))
22	21	3adant3r1 1181	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩)) = (𝑍 − 𝑌))
23	20, 22	oveq12d 7293	. . . . . 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)))
24	13	elfzelzd 13257	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℤ)
25	24	zcnd 12427	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑌 ∈ ℂ)
26	12	elfzelzd 13257	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ ℤ)
27	26	zcnd 12427	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ ℂ)
28	14	elfzelzd 13257	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ ℤ)
29	28	zcnd 12427	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑍 ∈ ℂ)
30	25, 27, 29	npncan3d 11368	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑌 − 𝑋) + (𝑍 − 𝑌)) = (𝑍 − 𝑋))
31	10, 23, 30	3eqtrd 2782	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘((𝑆 substr ⟨𝑋, 𝑌⟩) ++ (𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑍 − 𝑋))
32	31	oveq2d 7291	. . . 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)))
33	32	fneq2d 6527	. . 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))))
34	8, 33	mpbid 231	. 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)))
35		swrdcl 14358	. . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → (𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴)
36	35	adantr 481	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴)
37		wrdfn 14231	. . . 4 ⊢ ((𝑆 substr ⟨𝑋, 𝑍⟩) ∈ Word 𝐴 → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑍⟩))))
38	36, 37	syl 17	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑍⟩))))
39		fzass4 13294	. . . . . . . . 9 ⊢ ((𝑋 ∈ (0...𝑍) ∧ 𝑌 ∈ (𝑋...𝑍)) ↔ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)))
40	39	biimpri 227	. . . . . . . 8 ⊢ ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)) → (𝑋 ∈ (0...𝑍) ∧ 𝑌 ∈ (𝑋...𝑍)))
41	40	simpld 495	. . . . . . 7 ⊢ ((𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍)) → 𝑋 ∈ (0...𝑍))
42	12, 13, 41	syl2anc 584	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → 𝑋 ∈ (0...𝑍))
43		swrdlen 14360	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) → (♯‘(𝑆 substr ⟨𝑋, 𝑍⟩)) = (𝑍 − 𝑋))
44	11, 42, 14, 43	syl3anc 1370	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (♯‘(𝑆 substr ⟨𝑋, 𝑍⟩)) = (𝑍 − 𝑋))
45	44	oveq2d 7291	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑍⟩))) = (0..^(𝑍 − 𝑋)))
46	45	fneq2d 6527	. . 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))))
47	38, 46	mpbid 231	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑆 substr ⟨𝑋, 𝑍⟩) Fn (0..^(𝑍 − 𝑋)))
48	24, 26	zsubcld 12431	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 − 𝑋) ∈ ℤ)
49	48	anim1ci 616	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → (𝑥 ∈ (0..^(𝑍 − 𝑋)) ∧ (𝑌 − 𝑋) ∈ ℤ))
50		fzospliti 13419	. . . . 5 ⊢ ((𝑥 ∈ (0..^(𝑍 − 𝑋)) ∧ (𝑌 − 𝑋) ∈ ℤ) → (𝑥 ∈ (0..^(𝑌 − 𝑋)) ∨ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))))
51	49, 50	syl 17	. . . 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))))
52	1	ad2antrr 723	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
53	3	ad2antrr 723	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
54	20	oveq2d 7291	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) = (0..^(𝑌 − 𝑋)))
55	54	eleq2d 2824	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 ∈ (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) ↔ 𝑥 ∈ (0..^(𝑌 − 𝑋))))
56	55	biimpar 478	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑥 ∈ (0..^(♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))))
57		ccatval1 14281	. . . . . . 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))
58	52, 53, 56, 57	syl3anc 1370	. . . . . 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))
59		simpll 764	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑆 ∈ Word 𝐴)
60		simplr1 1214	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑋 ∈ (0...𝑌))
61	18	adantr 481	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑌 ∈ (0...(♯‘𝑆)))
62		simpr 485	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → 𝑥 ∈ (0..^(𝑌 − 𝑋)))
63		swrdfv 14361	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑌 − 𝑋))) → ((𝑆 substr ⟨𝑋, 𝑌⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
64	59, 60, 61, 62, 63	syl31anc 1372	. . . . . 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)))
65	58, 64	eqtrd 2778	. . . . 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)))
66	1	ad2antrr 723	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑆 substr ⟨𝑋, 𝑌⟩) ∈ Word 𝐴)
67	3	ad2antrr 723	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑆 substr ⟨𝑌, 𝑍⟩) ∈ Word 𝐴)
68	23, 30	eqtrd 2778	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((♯‘(𝑆 substr ⟨𝑋, 𝑌⟩)) + (♯‘(𝑆 substr ⟨𝑌, 𝑍⟩))) = (𝑍 − 𝑋))
69	20, 68	oveq12d 7293	. . . . . . . . 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)))
70	69	eleq2d 2824	. . . . . . . 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))))
71	70	biimpar 478	. . . . . . 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⟩)))))
72		ccatval2 14283	. . . . . . 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⟩)))))
73	66, 67, 71, 72	syl3anc 1370	. . . . . 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⟩)))))
74		simpll 764	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑆 ∈ Word 𝐴)
75		simplr2 1215	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑌 ∈ (0...𝑍))
76		simplr3 1216	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑍 ∈ (0...(♯‘𝑆)))
77	20	oveq2d 7291	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) = (𝑥 − (𝑌 − 𝑋)))
78	77	adantr 481	. . . . . . . 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)))
79	30	oveq2d 7291	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌))) = ((𝑌 − 𝑋)..^(𝑍 − 𝑋)))
80	79	eleq2d 2824	. . . . . . . . . 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))))
81	80	biimpar 478	. . . . . . . . 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))))
82	28, 24	zsubcld 12431	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑍 − 𝑌) ∈ ℤ)
83	82	adantr 481	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑍 − 𝑌) ∈ ℤ)
84		fzosubel3 13448	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((𝑌 − 𝑋)..^((𝑌 − 𝑋) + (𝑍 − 𝑌))) ∧ (𝑍 − 𝑌) ∈ ℤ) → (𝑥 − (𝑌 − 𝑋)) ∈ (0..^(𝑍 − 𝑌)))
85	81, 83, 84	syl2anc 584	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑥 − (𝑌 − 𝑋)) ∈ (0..^(𝑍 − 𝑌)))
86	78, 85	eqeltrd 2839	. . . . . . 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)))
87		swrdfv 14361	. . . . . . 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)))
88	74, 75, 76, 86, 87	syl31anc 1372	. . . . . 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)))
89	77	oveq1d 7290	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → ((𝑥 − (♯‘(𝑆 substr ⟨𝑋, 𝑌⟩))) + 𝑌) = ((𝑥 − (𝑌 − 𝑋)) + 𝑌))
90	89	adantr 481	. . . . . . . 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))
91		elfzoelz 13387	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)) → 𝑥 ∈ ℤ)
92	91	zcnd 12427	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋)) → 𝑥 ∈ ℂ)
93	92	adantl 482	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑥 ∈ ℂ)
94	25, 27	subcld 11332	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 − 𝑋) ∈ ℂ)
95	94	adantr 481	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑌 − 𝑋) ∈ ℂ)
96	25	adantr 481	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → 𝑌 ∈ ℂ)
97	93, 95, 96	subadd23d 11354	. . . . . . . 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))))
98	25, 27	nncand 11337	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑌 − (𝑌 − 𝑋)) = 𝑋)
99	98	oveq2d 7291	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) → (𝑥 + (𝑌 − (𝑌 − 𝑋))) = (𝑥 + 𝑋))
100	99	adantr 481	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ ((𝑌 − 𝑋)..^(𝑍 − 𝑋))) → (𝑥 + (𝑌 − (𝑌 − 𝑋))) = (𝑥 + 𝑋))
101	90, 97, 100	3eqtrd 2782	. . . . . . 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))
102	101	fveq2d 6778	. . . . . 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)))
103	73, 88, 102	3eqtrd 2782	. . . . 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)))
104	65, 103	jaodan 955	. . . 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)))
105	51, 104	syldan 591	. . 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)))
106		simpll 764	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑆 ∈ Word 𝐴)
107	42	adantr 481	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑋 ∈ (0...𝑍))
108		simplr3 1216	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑍 ∈ (0...(♯‘𝑆)))
109		simpr 485	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ (𝑋 ∈ (0...𝑌) ∧ 𝑌 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆)))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → 𝑥 ∈ (0..^(𝑍 − 𝑋)))
110		swrdfv 14361	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑋 ∈ (0...𝑍) ∧ 𝑍 ∈ (0...(♯‘𝑆))) ∧ 𝑥 ∈ (0..^(𝑍 − 𝑋))) → ((𝑆 substr ⟨𝑋, 𝑍⟩)‘𝑥) = (𝑆‘(𝑥 + 𝑋)))
111	106, 107, 108, 109, 110	syl31anc 1372	. . 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)))
112	105, 111	eqtr4d 2781	. 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))
113	34, 47, 112	eqfnfvd 6912	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 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ⟨cop 4567   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275  ℂcc 10869  0cc0 10871   + caddc 10874   − cmin 11205  ℤcz 12319  ...cfz 13239  ..^cfzo 13382  ♯chash 14044  Word cword 14217   ++ cconcat 14273   substr csubstr 14353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-hash 14045  df-word 14218  df-concat 14274  df-substr 14354

This theorem is referenced by:  swrds2  14653  efgredleme  19349