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Theorem revccat 13366

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 13212	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑆 ++ 𝑇) ∈ Word 𝐴)
2		revcl 13361	. . . 4 ⊢ ((𝑆 ++ 𝑇) ∈ Word 𝐴 → (reverse‘(𝑆 ++ 𝑇)) ∈ Word 𝐴)
3		wrdf 13165	. . . 4 ⊢ ((reverse‘(𝑆 ++ 𝑇)) ∈ Word 𝐴 → (reverse‘(𝑆 ++ 𝑇)):(0..^(#‘(reverse‘(𝑆 ++ 𝑇))))⟶𝐴)
4		ffn 5958	. . . 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 13362	. . . . . . 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 13213	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘(𝑆 ++ 𝑇)) = ((#‘𝑆) + (#‘𝑇)))
9		lencl 13179	. . . . . . . . 9 ⊢ (𝑆 ∈ Word 𝐴 → (#‘𝑆) ∈ ℕ₀)
10	9	nn0cnd 11230	. . . . . . . 8 ⊢ (𝑆 ∈ Word 𝐴 → (#‘𝑆) ∈ ℂ)
11		lencl 13179	. . . . . . . . 9 ⊢ (𝑇 ∈ Word 𝐴 → (#‘𝑇) ∈ ℕ₀)
12	11	nn0cnd 11230	. . . . . . . 8 ⊢ (𝑇 ∈ Word 𝐴 → (#‘𝑇) ∈ ℂ)
13		addcom 10101	. . . . . . . 8 ⊢ (((#‘𝑆) ∈ ℂ ∧ (#‘𝑇) ∈ ℂ) → ((#‘𝑆) + (#‘𝑇)) = ((#‘𝑇) + (#‘𝑆)))
14	10, 12, 13	syl2an 493	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘𝑆) + (#‘𝑇)) = ((#‘𝑇) + (#‘𝑆)))
15	8, 14	eqtrd 2644	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘(𝑆 ++ 𝑇)) = ((#‘𝑇) + (#‘𝑆)))
16	7, 15	eqtrd 2644	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘(reverse‘(𝑆 ++ 𝑇))) = ((#‘𝑇) + (#‘𝑆)))
17	16	oveq2d 6565	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(#‘(reverse‘(𝑆 ++ 𝑇)))) = (0..^((#‘𝑇) + (#‘𝑆))))
18	17	fneq2d 5896	. . 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 221	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (reverse‘(𝑆 ++ 𝑇)) Fn (0..^((#‘𝑇) + (#‘𝑆))))
20		revcl 13361	. . . . 5 ⊢ (𝑇 ∈ Word 𝐴 → (reverse‘𝑇) ∈ Word 𝐴)
21		revcl 13361	. . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → (reverse‘𝑆) ∈ Word 𝐴)
22		ccatcl 13212	. . . . 5 ⊢ (((reverse‘𝑇) ∈ Word 𝐴 ∧ (reverse‘𝑆) ∈ Word 𝐴) → ((reverse‘𝑇) ++ (reverse‘𝑆)) ∈ Word 𝐴)
23	20, 21, 22	syl2anr 494	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((reverse‘𝑇) ++ (reverse‘𝑆)) ∈ Word 𝐴)
24		wrdf 13165	. . . 4 ⊢ (((reverse‘𝑇) ++ (reverse‘𝑆)) ∈ Word 𝐴 → ((reverse‘𝑇) ++ (reverse‘𝑆)):(0..^(#‘((reverse‘𝑇) ++ (reverse‘𝑆))))⟶𝐴)
25		ffn 5958	. . . 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 13213	. . . . . . 7 ⊢ (((reverse‘𝑇) ∈ Word 𝐴 ∧ (reverse‘𝑆) ∈ Word 𝐴) → (#‘((reverse‘𝑇) ++ (reverse‘𝑆))) = ((#‘(reverse‘𝑇)) + (#‘(reverse‘𝑆))))
28	20, 21, 27	syl2anr 494	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘((reverse‘𝑇) ++ (reverse‘𝑆))) = ((#‘(reverse‘𝑇)) + (#‘(reverse‘𝑆))))
29		revlen 13362	. . . . . . 7 ⊢ (𝑇 ∈ Word 𝐴 → (#‘(reverse‘𝑇)) = (#‘𝑇))
30		revlen 13362	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐴 → (#‘(reverse‘𝑆)) = (#‘𝑆))
31	29, 30	oveqan12rd 6569	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘(reverse‘𝑇)) + (#‘(reverse‘𝑆))) = ((#‘𝑇) + (#‘𝑆)))
32	28, 31	eqtrd 2644	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘((reverse‘𝑇) ++ (reverse‘𝑆))) = ((#‘𝑇) + (#‘𝑆)))
33	32	oveq2d 6565	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(#‘((reverse‘𝑇) ++ (reverse‘𝑆)))) = (0..^((#‘𝑇) + (#‘𝑆))))
34	33	fneq2d 5896	. . 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 221	. 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 11356	. . . . 5 ⊢ (𝑇 ∈ Word 𝐴 → (#‘𝑇) ∈ ℤ)
38	37	adantl 481	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘𝑇) ∈ ℤ)
39		fzospliti 12369	. . . 4 ⊢ ((𝑥 ∈ (0..^((#‘𝑇) + (#‘𝑆))) ∧ (#‘𝑇) ∈ ℤ) → (𝑥 ∈ (0..^(#‘𝑇)) ∨ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))))
40	36, 38, 39	syl2anr 494	. . 3 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^((#‘𝑇) + (#‘𝑆)))) → (𝑥 ∈ (0..^(#‘𝑇)) ∨ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))))
41		simpll 786	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → 𝑆 ∈ Word 𝐴)
42		simplr 788	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → 𝑇 ∈ Word 𝐴)
43		fzoval 12340	. . . . . . . . . . . 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 2673	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ (0..^(#‘𝑇)) ↔ 𝑥 ∈ (0...((#‘𝑇) − 1))))
46	45	biimpa 500	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → 𝑥 ∈ (0...((#‘𝑇) − 1)))
47		fznn0sub2 12315	. . . . . . . . 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 480	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (0..^(#‘𝑇)) = (0...((#‘𝑇) − 1)))
50	48, 49	eleqtrrd 2691	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (((#‘𝑇) − 1) − 𝑥) ∈ (0..^(#‘𝑇)))
51		ccatval3 13216	. . . . . . 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 1318	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((𝑆 ++ 𝑇)‘((((#‘𝑇) − 1) − 𝑥) + (#‘𝑆))) = (𝑇‘(((#‘𝑇) − 1) − 𝑥)))
53	15	oveq1d 6564	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘(𝑆 ++ 𝑇)) − 1) = (((#‘𝑇) + (#‘𝑆)) − 1))
54	12	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘𝑇) ∈ ℂ)
55	10	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘𝑆) ∈ ℂ)
56		1cnd 9935	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → 1 ∈ ℂ)
57	54, 55, 56	addsubd 10292	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((#‘𝑇) + (#‘𝑆)) − 1) = (((#‘𝑇) − 1) + (#‘𝑆)))
58	53, 57	eqtrd 2644	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘(𝑆 ++ 𝑇)) − 1) = (((#‘𝑇) − 1) + (#‘𝑆)))
59	58	oveq1d 6564	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((#‘𝑇) − 1) + (#‘𝑆)) − 𝑥))
60	59	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((#‘𝑇) − 1) + (#‘𝑆)) − 𝑥))
61		peano2zm 11297	. . . . . . . . . . . 12 ⊢ ((#‘𝑇) ∈ ℤ → ((#‘𝑇) − 1) ∈ ℤ)
62	37, 61	syl 17	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → ((#‘𝑇) − 1) ∈ ℤ)
63	62	zcnd 11359	. . . . . . . . . 10 ⊢ (𝑇 ∈ Word 𝐴 → ((#‘𝑇) − 1) ∈ ℂ)
64	63	ad2antlr 759	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((#‘𝑇) − 1) ∈ ℂ)
65	10	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (#‘𝑆) ∈ ℂ)
66		elfzoelz 12339	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0..^(#‘𝑇)) → 𝑥 ∈ ℤ)
67	66	zcnd 11359	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0..^(#‘𝑇)) → 𝑥 ∈ ℂ)
68	67	adantl 481	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → 𝑥 ∈ ℂ)
69	64, 65, 68	addsubd 10292	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((((#‘𝑇) − 1) + (#‘𝑆)) − 𝑥) = ((((#‘𝑇) − 1) − 𝑥) + (#‘𝑆)))
70	60, 69	eqtrd 2644	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((#‘𝑇) − 1) − 𝑥) + (#‘𝑆)))
71	70	fveq2d 6107	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((𝑆 ++ 𝑇)‘(((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = ((𝑆 ++ 𝑇)‘((((#‘𝑇) − 1) − 𝑥) + (#‘𝑆))))
72		revfv 13363	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((reverse‘𝑇)‘𝑥) = (𝑇‘(((#‘𝑇) − 1) − 𝑥)))
73	72	adantll 746	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((reverse‘𝑇)‘𝑥) = (𝑇‘(((#‘𝑇) − 1) − 𝑥)))
74	52, 71, 73	3eqtr4d 2654	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((𝑆 ++ 𝑇)‘(((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = ((reverse‘𝑇)‘𝑥))
75	1	adantr 480	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (𝑆 ++ 𝑇) ∈ Word 𝐴)
76		uzid 11578	. . . . . . . . . . 11 ⊢ ((#‘𝑇) ∈ ℤ → (#‘𝑇) ∈ (ℤ_≥‘(#‘𝑇)))
77	38, 76	syl 17	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘𝑇) ∈ (ℤ_≥‘(#‘𝑇)))
78	9	adantr 480	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘𝑆) ∈ ℕ₀)
79		uzaddcl 11620	. . . . . . . . . 10 ⊢ (((#‘𝑇) ∈ (ℤ_≥‘(#‘𝑇)) ∧ (#‘𝑆) ∈ ℕ₀) → ((#‘𝑇) + (#‘𝑆)) ∈ (ℤ_≥‘(#‘𝑇)))
80	77, 78, 79	syl2anc 691	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘𝑇) + (#‘𝑆)) ∈ (ℤ_≥‘(#‘𝑇)))
81	15, 80	eqeltrd 2688	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘(𝑆 ++ 𝑇)) ∈ (ℤ_≥‘(#‘𝑇)))
82		fzoss2 12365	. . . . . . . 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 3568	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → 𝑥 ∈ (0..^(#‘(𝑆 ++ 𝑇))))
85		revfv 13363	. . . . . 6 ⊢ (((𝑆 ++ 𝑇) ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(#‘(𝑆 ++ 𝑇)))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = ((𝑆 ++ 𝑇)‘(((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
86	75, 84, 85	syl2anc 691	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = ((𝑆 ++ 𝑇)‘(((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
87	20	ad2antlr 759	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (reverse‘𝑇) ∈ Word 𝐴)
88	21	ad2antrr 758	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (reverse‘𝑆) ∈ Word 𝐴)
89	29	adantl 481	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘(reverse‘𝑇)) = (#‘𝑇))
90	89	oveq2d 6565	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(#‘(reverse‘𝑇))) = (0..^(#‘𝑇)))
91	90	eleq2d 2673	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ (0..^(#‘(reverse‘𝑇))) ↔ 𝑥 ∈ (0..^(#‘𝑇))))
92	91	biimpar 501	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → 𝑥 ∈ (0..^(#‘(reverse‘𝑇))))
93		ccatval1 13214	. . . . . 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 1318	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥) = ((reverse‘𝑇)‘𝑥))
95	74, 86, 94	3eqtr4d 2654	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥))
96	8	oveq1d 6564	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘(𝑆 ++ 𝑇)) − 1) = (((#‘𝑆) + (#‘𝑇)) − 1))
97	55, 54, 56	addsubd 10292	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((#‘𝑆) + (#‘𝑇)) − 1) = (((#‘𝑆) − 1) + (#‘𝑇)))
98	96, 97	eqtrd 2644	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘(𝑆 ++ 𝑇)) − 1) = (((#‘𝑆) − 1) + (#‘𝑇)))
99	98	oveq1d 6564	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((#‘𝑆) − 1) + (#‘𝑇)) − 𝑥))
100	99	adantr 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((#‘𝑆) − 1) + (#‘𝑇)) − 𝑥))
101	9	nn0zd 11356	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Word 𝐴 → (#‘𝑆) ∈ ℤ)
102		peano2zm 11297	. . . . . . . . . . . . 13 ⊢ ((#‘𝑆) ∈ ℤ → ((#‘𝑆) − 1) ∈ ℤ)
103	101, 102	syl 17	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Word 𝐴 → ((#‘𝑆) − 1) ∈ ℤ)
104	103	zcnd 11359	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝐴 → ((#‘𝑆) − 1) ∈ ℂ)
105	104	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → ((#‘𝑆) − 1) ∈ ℂ)
106		elfzoelz 12339	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆))) → 𝑥 ∈ ℤ)
107	106	zcnd 11359	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆))) → 𝑥 ∈ ℂ)
108	107	adantl 481	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → 𝑥 ∈ ℂ)
109	12	ad2antlr 759	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (#‘𝑇) ∈ ℂ)
110	105, 108, 109	subsub3d 10301	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (((#‘𝑆) − 1) − (𝑥 − (#‘𝑇))) = ((((#‘𝑆) − 1) + (#‘𝑇)) − 𝑥))
111	100, 110	eqtr4d 2647	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = (((#‘𝑆) − 1) − (𝑥 − (#‘𝑇))))
112	89	oveq2d 6565	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 − (#‘(reverse‘𝑇))) = (𝑥 − (#‘𝑇)))
113	112	oveq2d 6565	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((#‘𝑆) − 1) − (𝑥 − (#‘(reverse‘𝑇)))) = (((#‘𝑆) − 1) − (𝑥 − (#‘𝑇))))
114	113	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (((#‘𝑆) − 1) − (𝑥 − (#‘(reverse‘𝑇)))) = (((#‘𝑆) − 1) − (𝑥 − (#‘𝑇))))
115	111, 114	eqtr4d 2647	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = (((#‘𝑆) − 1) − (𝑥 − (#‘(reverse‘𝑇)))))
116	115	fveq2d 6107	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (𝑆‘(((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = (𝑆‘(((#‘𝑆) − 1) − (𝑥 − (#‘(reverse‘𝑇))))))
117		simpll 786	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → 𝑆 ∈ Word 𝐴)
118		simplr 788	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → 𝑇 ∈ Word 𝐴)
119		zaddcl 11294	. . . . . . . . . . . 12 ⊢ (((#‘𝑇) ∈ ℤ ∧ (#‘𝑆) ∈ ℤ) → ((#‘𝑇) + (#‘𝑆)) ∈ ℤ)
120	37, 101, 119	syl2anr 494	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘𝑇) + (#‘𝑆)) ∈ ℤ)
121		peano2zm 11297	. . . . . . . . . . 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 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (((#‘𝑇) + (#‘𝑆)) − 1) ∈ ℤ)
124		fzoval 12340	. . . . . . . . . . . 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 2673	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆))) ↔ 𝑥 ∈ ((#‘𝑇)...(((#‘𝑇) + (#‘𝑆)) − 1))))
127	126	biimpa 500	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → 𝑥 ∈ ((#‘𝑇)...(((#‘𝑇) + (#‘𝑆)) − 1)))
128		fzrev2i 12275	. . . . . . . . 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 691	. . . . . . . 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 6564	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((#‘𝑇) + (#‘𝑆)) − 1) − 𝑥))
131	130	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((#‘𝑇) + (#‘𝑆)) − 1) − 𝑥))
132	101	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘𝑆) ∈ ℤ)
133		fzoval 12340	. . . . . . . . . . 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 11359	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((#‘𝑇) + (#‘𝑆)) − 1) ∈ ℂ)
136	135	subidd 10259	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((#‘𝑇) + (#‘𝑆)) − 1) − (((#‘𝑇) + (#‘𝑆)) − 1)) = 0)
137		addcl 9897	. . . . . . . . . . . . . 14 ⊢ (((#‘𝑇) ∈ ℂ ∧ (#‘𝑆) ∈ ℂ) → ((#‘𝑇) + (#‘𝑆)) ∈ ℂ)
138	12, 10, 137	syl2anr 494	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘𝑇) + (#‘𝑆)) ∈ ℂ)
139	138, 56, 54	sub32d 10303	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((#‘𝑇) + (#‘𝑆)) − 1) − (#‘𝑇)) = ((((#‘𝑇) + (#‘𝑆)) − (#‘𝑇)) − 1))
140		pncan2 10167	. . . . . . . . . . . . . 14 ⊢ (((#‘𝑇) ∈ ℂ ∧ (#‘𝑆) ∈ ℂ) → (((#‘𝑇) + (#‘𝑆)) − (#‘𝑇)) = (#‘𝑆))
141	12, 10, 140	syl2anr 494	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((#‘𝑇) + (#‘𝑆)) − (#‘𝑇)) = (#‘𝑆))
142	141	oveq1d 6564	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((#‘𝑇) + (#‘𝑆)) − (#‘𝑇)) − 1) = ((#‘𝑆) − 1))
143	139, 142	eqtrd 2644	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((#‘𝑇) + (#‘𝑆)) − 1) − (#‘𝑇)) = ((#‘𝑆) − 1))
144	136, 143	oveq12d 6567	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((((#‘𝑇) + (#‘𝑆)) − 1) − (((#‘𝑇) + (#‘𝑆)) − 1))...((((#‘𝑇) + (#‘𝑆)) − 1) − (#‘𝑇))) = (0...((#‘𝑆) − 1)))
145	134, 144	eqtr4d 2647	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(#‘𝑆)) = (((((#‘𝑇) + (#‘𝑆)) − 1) − (((#‘𝑇) + (#‘𝑆)) − 1))...((((#‘𝑇) + (#‘𝑆)) − 1) − (#‘𝑇))))
146	145	adantr 480	. . . . . . . 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 2703	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) ∈ (0..^(#‘𝑆)))
148		ccatval1 13214	. . . . . . 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 1318	. . . . . 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 759	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (#‘(reverse‘𝑇)) = (#‘𝑇))
151	150	oveq2d 6565	. . . . . . . 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 12396	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆))) ∧ (#‘𝑆) ∈ ℤ) → (𝑥 − (#‘𝑇)) ∈ (0..^(#‘𝑆)))
154	152, 132, 153	syl2anr 494	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (𝑥 − (#‘𝑇)) ∈ (0..^(#‘𝑆)))
155	151, 154	eqeltrd 2688	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (𝑥 − (#‘(reverse‘𝑇))) ∈ (0..^(#‘𝑆)))
156		revfv 13363	. . . . . . 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 691	. . . . . 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 2654	. . . . 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 480	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (𝑆 ++ 𝑇) ∈ Word 𝐴)
160	11	adantl 481	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘𝑇) ∈ ℕ₀)
161		fzoss1 12364	. . . . . . . . . 10 ⊢ ((#‘𝑇) ∈ (ℤ_≥‘0) → ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆))) ⊆ (0..^((#‘𝑇) + (#‘𝑆))))
162		nn0uz 11598	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
163	161, 162	eleq2s 2706	. . . . . . . . 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 6565	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(#‘(𝑆 ++ 𝑇))) = (0..^((#‘𝑇) + (#‘𝑆))))
166	164, 165	sseqtr4d 3605	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆))) ⊆ (0..^(#‘(𝑆 ++ 𝑇))))
167	166	sselda 3568	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → 𝑥 ∈ (0..^(#‘(𝑆 ++ 𝑇))))
168	159, 167, 85	syl2anc 691	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = ((𝑆 ++ 𝑇)‘(((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
169	20	ad2antlr 759	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (reverse‘𝑇) ∈ Word 𝐴)
170	21	ad2antrr 758	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (reverse‘𝑆) ∈ Word 𝐴)
171	89, 31	oveq12d 6567	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘(reverse‘𝑇))..^((#‘(reverse‘𝑇)) + (#‘(reverse‘𝑆)))) = ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆))))
172	171	eleq2d 2673	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ ((#‘(reverse‘𝑇))..^((#‘(reverse‘𝑇)) + (#‘(reverse‘𝑆)))) ↔ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))))
173	172	biimpar 501	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → 𝑥 ∈ ((#‘(reverse‘𝑇))..^((#‘(reverse‘𝑇)) + (#‘(reverse‘𝑆)))))
174		ccatval2 13215	. . . . . 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 1318	. . . . 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 2654	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥))
177	95, 176	jaodan 822	. . 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 486	. 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 6222	1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (reverse‘(𝑆 ++ 𝑇)) = ((reverse‘𝑇) ++ (reverse‘𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 − cmin 10145 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 ++ cconcat 13148 reversecreverse 13152

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-reverse 13160

This theorem is referenced by: gsumwrev 17619 efginvrel2 17963

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