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Theorem revccat 13846

Description: Antiautomorphic property of the reversal operation. (Contributed by Stefan O'Rear, 27-Aug-2015.)

**Assertion**
Ref	Expression
revccat	⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (reverse‘(𝑆 ++ 𝑇)) = ((reverse‘𝑇) ++ (reverse‘𝑆)))

Proof of Theorem revccat Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ccatcl 13594	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑆 ++ 𝑇) ∈ Word 𝐴)
2		revcl 13841	. . . 4 ⊢ ((𝑆 ++ 𝑇) ∈ Word 𝐴 → (reverse‘(𝑆 ++ 𝑇)) ∈ Word 𝐴)
3		wrdf 13539	. . . 4 ⊢ ((reverse‘(𝑆 ++ 𝑇)) ∈ Word 𝐴 → (reverse‘(𝑆 ++ 𝑇)):(0..^(♯‘(reverse‘(𝑆 ++ 𝑇))))⟶𝐴)
4		ffn 6256	. . . 4 ⊢ ((reverse‘(𝑆 ++ 𝑇)):(0..^(♯‘(reverse‘(𝑆 ++ 𝑇))))⟶𝐴 → (reverse‘(𝑆 ++ 𝑇)) Fn (0..^(♯‘(reverse‘(𝑆 ++ 𝑇)))))
5	1, 2, 3, 4	4syl 19	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (reverse‘(𝑆 ++ 𝑇)) Fn (0..^(♯‘(reverse‘(𝑆 ++ 𝑇)))))
6		revlen 13842	. . . . . . 7 ⊢ ((𝑆 ++ 𝑇) ∈ Word 𝐴 → (♯‘(reverse‘(𝑆 ++ 𝑇))) = (♯‘(𝑆 ++ 𝑇)))
7	1, 6	syl 17	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(reverse‘(𝑆 ++ 𝑇))) = (♯‘(𝑆 ++ 𝑇)))
8		ccatlen 13595	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑆) + (♯‘𝑇)))
9		lencl 13553	. . . . . . . . 9 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℕ₀)
10	9	nn0cnd 11642	. . . . . . . 8 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℂ)
11		lencl 13553	. . . . . . . . 9 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℕ₀)
12	11	nn0cnd 11642	. . . . . . . 8 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℂ)
13		addcom 10512	. . . . . . . 8 ⊢ (((♯‘𝑆) ∈ ℂ ∧ (♯‘𝑇) ∈ ℂ) → ((♯‘𝑆) + (♯‘𝑇)) = ((♯‘𝑇) + (♯‘𝑆)))
14	10, 12, 13	syl2an 590	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑆) + (♯‘𝑇)) = ((♯‘𝑇) + (♯‘𝑆)))
15	8, 14	eqtrd 2833	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑇) + (♯‘𝑆)))
16	7, 15	eqtrd 2833	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(reverse‘(𝑆 ++ 𝑇))) = ((♯‘𝑇) + (♯‘𝑆)))
17	16	oveq2d 6894	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘(reverse‘(𝑆 ++ 𝑇)))) = (0..^((♯‘𝑇) + (♯‘𝑆))))
18	17	fneq2d 6193	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((reverse‘(𝑆 ++ 𝑇)) Fn (0..^(♯‘(reverse‘(𝑆 ++ 𝑇)))) ↔ (reverse‘(𝑆 ++ 𝑇)) Fn (0..^((♯‘𝑇) + (♯‘𝑆)))))
19	5, 18	mpbid 224	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (reverse‘(𝑆 ++ 𝑇)) Fn (0..^((♯‘𝑇) + (♯‘𝑆))))
20		revcl 13841	. . . . 5 ⊢ (𝑇 ∈ Word 𝐴 → (reverse‘𝑇) ∈ Word 𝐴)
21		revcl 13841	. . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → (reverse‘𝑆) ∈ Word 𝐴)
22		ccatcl 13594	. . . . 5 ⊢ (((reverse‘𝑇) ∈ Word 𝐴 ∧ (reverse‘𝑆) ∈ Word 𝐴) → ((reverse‘𝑇) ++ (reverse‘𝑆)) ∈ Word 𝐴)
23	20, 21, 22	syl2anr 591	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((reverse‘𝑇) ++ (reverse‘𝑆)) ∈ Word 𝐴)
24		wrdf 13539	. . . 4 ⊢ (((reverse‘𝑇) ++ (reverse‘𝑆)) ∈ Word 𝐴 → ((reverse‘𝑇) ++ (reverse‘𝑆)):(0..^(♯‘((reverse‘𝑇) ++ (reverse‘𝑆))))⟶𝐴)
25		ffn 6256	. . . 4 ⊢ (((reverse‘𝑇) ++ (reverse‘𝑆)):(0..^(♯‘((reverse‘𝑇) ++ (reverse‘𝑆))))⟶𝐴 → ((reverse‘𝑇) ++ (reverse‘𝑆)) Fn (0..^(♯‘((reverse‘𝑇) ++ (reverse‘𝑆)))))
26	23, 24, 25	3syl 18	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((reverse‘𝑇) ++ (reverse‘𝑆)) Fn (0..^(♯‘((reverse‘𝑇) ++ (reverse‘𝑆)))))
27		ccatlen 13595	. . . . . . 7 ⊢ (((reverse‘𝑇) ∈ Word 𝐴 ∧ (reverse‘𝑆) ∈ Word 𝐴) → (♯‘((reverse‘𝑇) ++ (reverse‘𝑆))) = ((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆))))
28	20, 21, 27	syl2anr 591	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘((reverse‘𝑇) ++ (reverse‘𝑆))) = ((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆))))
29		revlen 13842	. . . . . . 7 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘(reverse‘𝑇)) = (♯‘𝑇))
30		revlen 13842	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘(reverse‘𝑆)) = (♯‘𝑆))
31	29, 30	oveqan12rd 6898	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆))) = ((♯‘𝑇) + (♯‘𝑆)))
32	28, 31	eqtrd 2833	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘((reverse‘𝑇) ++ (reverse‘𝑆))) = ((♯‘𝑇) + (♯‘𝑆)))
33	32	oveq2d 6894	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘((reverse‘𝑇) ++ (reverse‘𝑆)))) = (0..^((♯‘𝑇) + (♯‘𝑆))))
34	33	fneq2d 6193	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((reverse‘𝑇) ++ (reverse‘𝑆)) Fn (0..^(♯‘((reverse‘𝑇) ++ (reverse‘𝑆)))) ↔ ((reverse‘𝑇) ++ (reverse‘𝑆)) Fn (0..^((♯‘𝑇) + (♯‘𝑆)))))
35	26, 34	mpbid 224	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((reverse‘𝑇) ++ (reverse‘𝑆)) Fn (0..^((♯‘𝑇) + (♯‘𝑆))))
36		id 22	. . . 4 ⊢ (𝑥 ∈ (0..^((♯‘𝑇) + (♯‘𝑆))) → 𝑥 ∈ (0..^((♯‘𝑇) + (♯‘𝑆))))
37	11	nn0zd 11770	. . . . 5 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℤ)
38	37	adantl 474	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘𝑇) ∈ ℤ)
39		fzospliti 12755	. . . 4 ⊢ ((𝑥 ∈ (0..^((♯‘𝑇) + (♯‘𝑆))) ∧ (♯‘𝑇) ∈ ℤ) → (𝑥 ∈ (0..^(♯‘𝑇)) ∨ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))))
40	36, 38, 39	syl2anr 591	. . 3 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^((♯‘𝑇) + (♯‘𝑆)))) → (𝑥 ∈ (0..^(♯‘𝑇)) ∨ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))))
41		simpll 784	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑆 ∈ Word 𝐴)
42		simplr 786	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑇 ∈ Word 𝐴)
43		fzoval 12726	. . . . . . . . . . . 12 ⊢ ((♯‘𝑇) ∈ ℤ → (0..^(♯‘𝑇)) = (0...((♯‘𝑇) − 1)))
44	38, 43	syl 17	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘𝑇)) = (0...((♯‘𝑇) − 1)))
45	44	eleq2d 2864	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ (0..^(♯‘𝑇)) ↔ 𝑥 ∈ (0...((♯‘𝑇) − 1))))
46	45	biimpa 469	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑥 ∈ (0...((♯‘𝑇) − 1)))
47		fznn0sub2 12701	. . . . . . . . 9 ⊢ (𝑥 ∈ (0...((♯‘𝑇) − 1)) → (((♯‘𝑇) − 1) − 𝑥) ∈ (0...((♯‘𝑇) − 1)))
48	46, 47	syl 17	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (((♯‘𝑇) − 1) − 𝑥) ∈ (0...((♯‘𝑇) − 1)))
49	44	adantr 473	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (0..^(♯‘𝑇)) = (0...((♯‘𝑇) − 1)))
50	48, 49	eleqtrrd 2881	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (((♯‘𝑇) − 1) − 𝑥) ∈ (0..^(♯‘𝑇)))
51		ccatval3 13599	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ (((♯‘𝑇) − 1) − 𝑥) ∈ (0..^(♯‘𝑇))) → ((𝑆 ++ 𝑇)‘((((♯‘𝑇) − 1) − 𝑥) + (♯‘𝑆))) = (𝑇‘(((♯‘𝑇) − 1) − 𝑥)))
52	41, 42, 50, 51	syl3anc 1491	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((𝑆 ++ 𝑇)‘((((♯‘𝑇) − 1) − 𝑥) + (♯‘𝑆))) = (𝑇‘(((♯‘𝑇) − 1) − 𝑥)))
53	15	oveq1d 6893	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(𝑆 ++ 𝑇)) − 1) = (((♯‘𝑇) + (♯‘𝑆)) − 1))
54	12	adantl 474	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘𝑇) ∈ ℂ)
55	10	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘𝑆) ∈ ℂ)
56		1cnd 10323	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → 1 ∈ ℂ)
57	54, 55, 56	addsubd 10705	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘𝑇) + (♯‘𝑆)) − 1) = (((♯‘𝑇) − 1) + (♯‘𝑆)))
58	53, 57	eqtrd 2833	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(𝑆 ++ 𝑇)) − 1) = (((♯‘𝑇) − 1) + (♯‘𝑆)))
59	58	oveq1d 6893	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑇) − 1) + (♯‘𝑆)) − 𝑥))
60	59	adantr 473	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑇) − 1) + (♯‘𝑆)) − 𝑥))
61		peano2zm 11710	. . . . . . . . . . . 12 ⊢ ((♯‘𝑇) ∈ ℤ → ((♯‘𝑇) − 1) ∈ ℤ)
62	37, 61	syl 17	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → ((♯‘𝑇) − 1) ∈ ℤ)
63	62	zcnd 11773	. . . . . . . . . 10 ⊢ (𝑇 ∈ Word 𝐴 → ((♯‘𝑇) − 1) ∈ ℂ)
64	63	ad2antlr 719	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((♯‘𝑇) − 1) ∈ ℂ)
65	10	ad2antrr 718	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (♯‘𝑆) ∈ ℂ)
66		elfzoelz 12725	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0..^(♯‘𝑇)) → 𝑥 ∈ ℤ)
67	66	zcnd 11773	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0..^(♯‘𝑇)) → 𝑥 ∈ ℂ)
68	67	adantl 474	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑥 ∈ ℂ)
69	64, 65, 68	addsubd 10705	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((((♯‘𝑇) − 1) + (♯‘𝑆)) − 𝑥) = ((((♯‘𝑇) − 1) − 𝑥) + (♯‘𝑆)))
70	60, 69	eqtrd 2833	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑇) − 1) − 𝑥) + (♯‘𝑆)))
71	70	fveq2d 6415	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = ((𝑆 ++ 𝑇)‘((((♯‘𝑇) − 1) − 𝑥) + (♯‘𝑆))))
72		revfv 13843	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((reverse‘𝑇)‘𝑥) = (𝑇‘(((♯‘𝑇) − 1) − 𝑥)))
73	72	adantll 706	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((reverse‘𝑇)‘𝑥) = (𝑇‘(((♯‘𝑇) − 1) − 𝑥)))
74	52, 71, 73	3eqtr4d 2843	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = ((reverse‘𝑇)‘𝑥))
75	1	adantr 473	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (𝑆 ++ 𝑇) ∈ Word 𝐴)
76		uzid 11945	. . . . . . . . . . 11 ⊢ ((♯‘𝑇) ∈ ℤ → (♯‘𝑇) ∈ (ℤ_≥‘(♯‘𝑇)))
77	38, 76	syl 17	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘𝑇) ∈ (ℤ_≥‘(♯‘𝑇)))
78	9	adantr 473	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘𝑆) ∈ ℕ₀)
79		uzaddcl 11988	. . . . . . . . . 10 ⊢ (((♯‘𝑇) ∈ (ℤ_≥‘(♯‘𝑇)) ∧ (♯‘𝑆) ∈ ℕ₀) → ((♯‘𝑇) + (♯‘𝑆)) ∈ (ℤ_≥‘(♯‘𝑇)))
80	77, 78, 79	syl2anc 580	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇) + (♯‘𝑆)) ∈ (ℤ_≥‘(♯‘𝑇)))
81	15, 80	eqeltrd 2878	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(𝑆 ++ 𝑇)) ∈ (ℤ_≥‘(♯‘𝑇)))
82		fzoss2 12751	. . . . . . . 8 ⊢ ((♯‘(𝑆 ++ 𝑇)) ∈ (ℤ_≥‘(♯‘𝑇)) → (0..^(♯‘𝑇)) ⊆ (0..^(♯‘(𝑆 ++ 𝑇))))
83	81, 82	syl 17	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘𝑇)) ⊆ (0..^(♯‘(𝑆 ++ 𝑇))))
84	83	sselda 3798	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑥 ∈ (0..^(♯‘(𝑆 ++ 𝑇))))
85		revfv 13843	. . . . . 6 ⊢ (((𝑆 ++ 𝑇) ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘(𝑆 ++ 𝑇)))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
86	75, 84, 85	syl2anc 580	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
87	20	ad2antlr 719	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (reverse‘𝑇) ∈ Word 𝐴)
88	21	ad2antrr 718	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (reverse‘𝑆) ∈ Word 𝐴)
89	29	adantl 474	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(reverse‘𝑇)) = (♯‘𝑇))
90	89	oveq2d 6894	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘(reverse‘𝑇))) = (0..^(♯‘𝑇)))
91	90	eleq2d 2864	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ (0..^(♯‘(reverse‘𝑇))) ↔ 𝑥 ∈ (0..^(♯‘𝑇))))
92	91	biimpar 470	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑥 ∈ (0..^(♯‘(reverse‘𝑇))))
93		ccatval1 13597	. . . . . 6 ⊢ (((reverse‘𝑇) ∈ Word 𝐴 ∧ (reverse‘𝑆) ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘(reverse‘𝑇)))) → (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥) = ((reverse‘𝑇)‘𝑥))
94	87, 88, 92, 93	syl3anc 1491	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥) = ((reverse‘𝑇)‘𝑥))
95	74, 86, 94	3eqtr4d 2843	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥))
96	8	oveq1d 6893	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(𝑆 ++ 𝑇)) − 1) = (((♯‘𝑆) + (♯‘𝑇)) − 1))
97	55, 54, 56	addsubd 10705	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘𝑆) + (♯‘𝑇)) − 1) = (((♯‘𝑆) − 1) + (♯‘𝑇)))
98	96, 97	eqtrd 2833	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(𝑆 ++ 𝑇)) − 1) = (((♯‘𝑆) − 1) + (♯‘𝑇)))
99	98	oveq1d 6893	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑆) − 1) + (♯‘𝑇)) − 𝑥))
100	99	adantr 473	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑆) − 1) + (♯‘𝑇)) − 𝑥))
101	9	nn0zd 11770	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℤ)
102		peano2zm 11710	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑆) ∈ ℤ → ((♯‘𝑆) − 1) ∈ ℤ)
103	101, 102	syl 17	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Word 𝐴 → ((♯‘𝑆) − 1) ∈ ℤ)
104	103	zcnd 11773	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝐴 → ((♯‘𝑆) − 1) ∈ ℂ)
105	104	ad2antrr 718	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((♯‘𝑆) − 1) ∈ ℂ)
106		elfzoelz 12725	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) → 𝑥 ∈ ℤ)
107	106	zcnd 11773	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) → 𝑥 ∈ ℂ)
108	107	adantl 474	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑥 ∈ ℂ)
109	12	ad2antlr 719	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (♯‘𝑇) ∈ ℂ)
110	105, 108, 109	subsub3d 10714	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘𝑆) − 1) − (𝑥 − (♯‘𝑇))) = ((((♯‘𝑆) − 1) + (♯‘𝑇)) − 𝑥))
111	100, 110	eqtr4d 2836	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = (((♯‘𝑆) − 1) − (𝑥 − (♯‘𝑇))))
112	89	oveq2d 6894	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 − (♯‘(reverse‘𝑇))) = (𝑥 − (♯‘𝑇)))
113	112	oveq2d 6894	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘𝑆) − 1) − (𝑥 − (♯‘(reverse‘𝑇)))) = (((♯‘𝑆) − 1) − (𝑥 − (♯‘𝑇))))
114	113	adantr 473	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘𝑆) − 1) − (𝑥 − (♯‘(reverse‘𝑇)))) = (((♯‘𝑆) − 1) − (𝑥 − (♯‘𝑇))))
115	111, 114	eqtr4d 2836	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = (((♯‘𝑆) − 1) − (𝑥 − (♯‘(reverse‘𝑇)))))
116	115	fveq2d 6415	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (𝑆‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = (𝑆‘(((♯‘𝑆) − 1) − (𝑥 − (♯‘(reverse‘𝑇))))))
117		simpll 784	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑆 ∈ Word 𝐴)
118		simplr 786	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑇 ∈ Word 𝐴)
119		zaddcl 11707	. . . . . . . . . . . 12 ⊢ (((♯‘𝑇) ∈ ℤ ∧ (♯‘𝑆) ∈ ℤ) → ((♯‘𝑇) + (♯‘𝑆)) ∈ ℤ)
120	37, 101, 119	syl2anr 591	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇) + (♯‘𝑆)) ∈ ℤ)
121		peano2zm 11710	. . . . . . . . . . 11 ⊢ (((♯‘𝑇) + (♯‘𝑆)) ∈ ℤ → (((♯‘𝑇) + (♯‘𝑆)) − 1) ∈ ℤ)
122	120, 121	syl 17	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘𝑇) + (♯‘𝑆)) − 1) ∈ ℤ)
123	122	adantr 473	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘𝑇) + (♯‘𝑆)) − 1) ∈ ℤ)
124		fzoval 12726	. . . . . . . . . . . 12 ⊢ (((♯‘𝑇) + (♯‘𝑆)) ∈ ℤ → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) = ((♯‘𝑇)...(((♯‘𝑇) + (♯‘𝑆)) − 1)))
125	120, 124	syl 17	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) = ((♯‘𝑇)...(((♯‘𝑇) + (♯‘𝑆)) − 1)))
126	125	eleq2d 2864	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ↔ 𝑥 ∈ ((♯‘𝑇)...(((♯‘𝑇) + (♯‘𝑆)) − 1))))
127	126	biimpa 469	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑥 ∈ ((♯‘𝑇)...(((♯‘𝑇) + (♯‘𝑆)) − 1)))
128		fzrev2i 12659	. . . . . . . . 9 ⊢ (((((♯‘𝑇) + (♯‘𝑆)) − 1) ∈ ℤ ∧ 𝑥 ∈ ((♯‘𝑇)...(((♯‘𝑇) + (♯‘𝑆)) − 1))) → ((((♯‘𝑇) + (♯‘𝑆)) − 1) − 𝑥) ∈ (((((♯‘𝑇) + (♯‘𝑆)) − 1) − (((♯‘𝑇) + (♯‘𝑆)) − 1))...((((♯‘𝑇) + (♯‘𝑆)) − 1) − (♯‘𝑇))))
129	123, 127, 128	syl2anc 580	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((((♯‘𝑇) + (♯‘𝑆)) − 1) − 𝑥) ∈ (((((♯‘𝑇) + (♯‘𝑆)) − 1) − (((♯‘𝑇) + (♯‘𝑆)) − 1))...((((♯‘𝑇) + (♯‘𝑆)) − 1) − (♯‘𝑇))))
130	53	oveq1d 6893	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑇) + (♯‘𝑆)) − 1) − 𝑥))
131	130	adantr 473	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑇) + (♯‘𝑆)) − 1) − 𝑥))
132	101	adantr 473	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘𝑆) ∈ ℤ)
133		fzoval 12726	. . . . . . . . . . 11 ⊢ ((♯‘𝑆) ∈ ℤ → (0..^(♯‘𝑆)) = (0...((♯‘𝑆) − 1)))
134	132, 133	syl 17	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘𝑆)) = (0...((♯‘𝑆) − 1)))
135	122	zcnd 11773	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘𝑇) + (♯‘𝑆)) − 1) ∈ ℂ)
136	135	subidd 10672	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((♯‘𝑇) + (♯‘𝑆)) − 1) − (((♯‘𝑇) + (♯‘𝑆)) − 1)) = 0)
137		addcl 10306	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑆) ∈ ℂ) → ((♯‘𝑇) + (♯‘𝑆)) ∈ ℂ)
138	12, 10, 137	syl2anr 591	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇) + (♯‘𝑆)) ∈ ℂ)
139	138, 56, 54	sub32d 10716	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((♯‘𝑇) + (♯‘𝑆)) − 1) − (♯‘𝑇)) = ((((♯‘𝑇) + (♯‘𝑆)) − (♯‘𝑇)) − 1))
140		pncan2 10579	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑆) ∈ ℂ) → (((♯‘𝑇) + (♯‘𝑆)) − (♯‘𝑇)) = (♯‘𝑆))
141	12, 10, 140	syl2anr 591	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘𝑇) + (♯‘𝑆)) − (♯‘𝑇)) = (♯‘𝑆))
142	141	oveq1d 6893	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((♯‘𝑇) + (♯‘𝑆)) − (♯‘𝑇)) − 1) = ((♯‘𝑆) − 1))
143	139, 142	eqtrd 2833	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((♯‘𝑇) + (♯‘𝑆)) − 1) − (♯‘𝑇)) = ((♯‘𝑆) − 1))
144	136, 143	oveq12d 6896	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((((♯‘𝑇) + (♯‘𝑆)) − 1) − (((♯‘𝑇) + (♯‘𝑆)) − 1))...((((♯‘𝑇) + (♯‘𝑆)) − 1) − (♯‘𝑇))) = (0...((♯‘𝑆) − 1)))
145	134, 144	eqtr4d 2836	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘𝑆)) = (((((♯‘𝑇) + (♯‘𝑆)) − 1) − (((♯‘𝑇) + (♯‘𝑆)) − 1))...((((♯‘𝑇) + (♯‘𝑆)) − 1) − (♯‘𝑇))))
146	145	adantr 473	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (0..^(♯‘𝑆)) = (((((♯‘𝑇) + (♯‘𝑆)) − 1) − (((♯‘𝑇) + (♯‘𝑆)) − 1))...((((♯‘𝑇) + (♯‘𝑆)) − 1) − (♯‘𝑇))))
147	129, 131, 146	3eltr4d 2893	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) ∈ (0..^(♯‘𝑆)))
148		ccatval1 13597	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = (𝑆‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
149	117, 118, 147, 148	syl3anc 1491	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = (𝑆‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
150	29	ad2antlr 719	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (♯‘(reverse‘𝑇)) = (♯‘𝑇))
151	150	oveq2d 6894	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (𝑥 − (♯‘(reverse‘𝑇))) = (𝑥 − (♯‘𝑇)))
152		id 22	. . . . . . . . 9 ⊢ (𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) → 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))))
153		fzosubel3 12784	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ∧ (♯‘𝑆) ∈ ℤ) → (𝑥 − (♯‘𝑇)) ∈ (0..^(♯‘𝑆)))
154	152, 132, 153	syl2anr 591	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (𝑥 − (♯‘𝑇)) ∈ (0..^(♯‘𝑆)))
155	151, 154	eqeltrd 2878	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (𝑥 − (♯‘(reverse‘𝑇))) ∈ (0..^(♯‘𝑆)))
156		revfv 13843	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑥 − (♯‘(reverse‘𝑇))) ∈ (0..^(♯‘𝑆))) → ((reverse‘𝑆)‘(𝑥 − (♯‘(reverse‘𝑇)))) = (𝑆‘(((♯‘𝑆) − 1) − (𝑥 − (♯‘(reverse‘𝑇))))))
157	117, 155, 156	syl2anc 580	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((reverse‘𝑆)‘(𝑥 − (♯‘(reverse‘𝑇)))) = (𝑆‘(((♯‘𝑆) − 1) − (𝑥 − (♯‘(reverse‘𝑇))))))
158	116, 149, 157	3eqtr4d 2843	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = ((reverse‘𝑆)‘(𝑥 − (♯‘(reverse‘𝑇)))))
159	1	adantr 473	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (𝑆 ++ 𝑇) ∈ Word 𝐴)
160	11	adantl 474	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘𝑇) ∈ ℕ₀)
161		fzoss1 12750	. . . . . . . . . 10 ⊢ ((♯‘𝑇) ∈ (ℤ_≥‘0) → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ⊆ (0..^((♯‘𝑇) + (♯‘𝑆))))
162		nn0uz 11966	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
163	161, 162	eleq2s 2896	. . . . . . . . 9 ⊢ ((♯‘𝑇) ∈ ℕ₀ → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ⊆ (0..^((♯‘𝑇) + (♯‘𝑆))))
164	160, 163	syl 17	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ⊆ (0..^((♯‘𝑇) + (♯‘𝑆))))
165	15	oveq2d 6894	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘(𝑆 ++ 𝑇))) = (0..^((♯‘𝑇) + (♯‘𝑆))))
166	164, 165	sseqtr4d 3838	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ⊆ (0..^(♯‘(𝑆 ++ 𝑇))))
167	166	sselda 3798	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑥 ∈ (0..^(♯‘(𝑆 ++ 𝑇))))
168	159, 167, 85	syl2anc 580	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
169	20	ad2antlr 719	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (reverse‘𝑇) ∈ Word 𝐴)
170	21	ad2antrr 718	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (reverse‘𝑆) ∈ Word 𝐴)
171	89, 31	oveq12d 6896	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(reverse‘𝑇))..^((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆)))) = ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))))
172	171	eleq2d 2864	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ ((♯‘(reverse‘𝑇))..^((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆)))) ↔ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))))
173	172	biimpar 470	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑥 ∈ ((♯‘(reverse‘𝑇))..^((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆)))))
174		ccatval2 13598	. . . . . 6 ⊢ (((reverse‘𝑇) ∈ Word 𝐴 ∧ (reverse‘𝑆) ∈ Word 𝐴 ∧ 𝑥 ∈ ((♯‘(reverse‘𝑇))..^((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆))))) → (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥) = ((reverse‘𝑆)‘(𝑥 − (♯‘(reverse‘𝑇)))))
175	169, 170, 173, 174	syl3anc 1491	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥) = ((reverse‘𝑆)‘(𝑥 − (♯‘(reverse‘𝑇)))))
176	158, 168, 175	3eqtr4d 2843	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥))
177	95, 176	jaodan 981	. . 3 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ (𝑥 ∈ (0..^(♯‘𝑇)) ∨ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥))
178	40, 177	syldan 586	. 2 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^((♯‘𝑇) + (♯‘𝑆)))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥))
179	19, 35, 178	eqfnfvd 6540	1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (reverse‘(𝑆 ++ 𝑇)) = ((reverse‘𝑇) ++ (reverse‘𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 385 ∨ wo 874 = wceq 1653 ∈ wcel 2157 ⊆ wss 3769 Fn wfn 6096 ⟶wf 6097 ‘cfv 6101 (class class class)co 6878 ℂcc 10222 0cc0 10224 1c1 10225 + caddc 10227 − cmin 10556 ℕ₀cn0 11580 ℤcz 11666 ℤ_≥cuz 11930 ...cfz 12580 ..^cfzo 12720 ♯chash 13370 Word cword 13534 ++ cconcat 13590 reversecreverse 13838

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-fzo 12721 df-hash 13371 df-word 13535 df-concat 13591 df-reverse 13839

This theorem is referenced by: gsumwrev 18108 efginvrel2 18453

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