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Theorem revccat 13515

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 13359	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑆 ++ 𝑇) ∈ Word 𝐴)
2		revcl 13510	. . . 4 ⊢ ((𝑆 ++ 𝑇) ∈ Word 𝐴 → (reverse‘(𝑆 ++ 𝑇)) ∈ Word 𝐴)
3		wrdf 13310	. . . 4 ⊢ ((reverse‘(𝑆 ++ 𝑇)) ∈ Word 𝐴 → (reverse‘(𝑆 ++ 𝑇)):(0..^(#‘(reverse‘(𝑆 ++ 𝑇))))⟶𝐴)
4		ffn 6045	. . . 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 13511	. . . . . . 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 13360	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘(𝑆 ++ 𝑇)) = ((#‘𝑆) + (#‘𝑇)))
9		lencl 13324	. . . . . . . . 9 ⊢ (𝑆 ∈ Word 𝐴 → (#‘𝑆) ∈ ℕ₀)
10	9	nn0cnd 11353	. . . . . . . 8 ⊢ (𝑆 ∈ Word 𝐴 → (#‘𝑆) ∈ ℂ)
11		lencl 13324	. . . . . . . . 9 ⊢ (𝑇 ∈ Word 𝐴 → (#‘𝑇) ∈ ℕ₀)
12	11	nn0cnd 11353	. . . . . . . 8 ⊢ (𝑇 ∈ Word 𝐴 → (#‘𝑇) ∈ ℂ)
13		addcom 10222	. . . . . . . 8 ⊢ (((#‘𝑆) ∈ ℂ ∧ (#‘𝑇) ∈ ℂ) → ((#‘𝑆) + (#‘𝑇)) = ((#‘𝑇) + (#‘𝑆)))
14	10, 12, 13	syl2an 494	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘𝑆) + (#‘𝑇)) = ((#‘𝑇) + (#‘𝑆)))
15	8, 14	eqtrd 2656	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘(𝑆 ++ 𝑇)) = ((#‘𝑇) + (#‘𝑆)))
16	7, 15	eqtrd 2656	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘(reverse‘(𝑆 ++ 𝑇))) = ((#‘𝑇) + (#‘𝑆)))
17	16	oveq2d 6666	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(#‘(reverse‘(𝑆 ++ 𝑇)))) = (0..^((#‘𝑇) + (#‘𝑆))))
18	17	fneq2d 5982	. . 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 222	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (reverse‘(𝑆 ++ 𝑇)) Fn (0..^((#‘𝑇) + (#‘𝑆))))
20		revcl 13510	. . . . 5 ⊢ (𝑇 ∈ Word 𝐴 → (reverse‘𝑇) ∈ Word 𝐴)
21		revcl 13510	. . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → (reverse‘𝑆) ∈ Word 𝐴)
22		ccatcl 13359	. . . . 5 ⊢ (((reverse‘𝑇) ∈ Word 𝐴 ∧ (reverse‘𝑆) ∈ Word 𝐴) → ((reverse‘𝑇) ++ (reverse‘𝑆)) ∈ Word 𝐴)
23	20, 21, 22	syl2anr 495	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((reverse‘𝑇) ++ (reverse‘𝑆)) ∈ Word 𝐴)
24		wrdf 13310	. . . 4 ⊢ (((reverse‘𝑇) ++ (reverse‘𝑆)) ∈ Word 𝐴 → ((reverse‘𝑇) ++ (reverse‘𝑆)):(0..^(#‘((reverse‘𝑇) ++ (reverse‘𝑆))))⟶𝐴)
25		ffn 6045	. . . 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 13360	. . . . . . 7 ⊢ (((reverse‘𝑇) ∈ Word 𝐴 ∧ (reverse‘𝑆) ∈ Word 𝐴) → (#‘((reverse‘𝑇) ++ (reverse‘𝑆))) = ((#‘(reverse‘𝑇)) + (#‘(reverse‘𝑆))))
28	20, 21, 27	syl2anr 495	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘((reverse‘𝑇) ++ (reverse‘𝑆))) = ((#‘(reverse‘𝑇)) + (#‘(reverse‘𝑆))))
29		revlen 13511	. . . . . . 7 ⊢ (𝑇 ∈ Word 𝐴 → (#‘(reverse‘𝑇)) = (#‘𝑇))
30		revlen 13511	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐴 → (#‘(reverse‘𝑆)) = (#‘𝑆))
31	29, 30	oveqan12rd 6670	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘(reverse‘𝑇)) + (#‘(reverse‘𝑆))) = ((#‘𝑇) + (#‘𝑆)))
32	28, 31	eqtrd 2656	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘((reverse‘𝑇) ++ (reverse‘𝑆))) = ((#‘𝑇) + (#‘𝑆)))
33	32	oveq2d 6666	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(#‘((reverse‘𝑇) ++ (reverse‘𝑆)))) = (0..^((#‘𝑇) + (#‘𝑆))))
34	33	fneq2d 5982	. . 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 222	. 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 11480	. . . . 5 ⊢ (𝑇 ∈ Word 𝐴 → (#‘𝑇) ∈ ℤ)
38	37	adantl 482	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘𝑇) ∈ ℤ)
39		fzospliti 12500	. . . 4 ⊢ ((𝑥 ∈ (0..^((#‘𝑇) + (#‘𝑆))) ∧ (#‘𝑇) ∈ ℤ) → (𝑥 ∈ (0..^(#‘𝑇)) ∨ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))))
40	36, 38, 39	syl2anr 495	. . 3 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^((#‘𝑇) + (#‘𝑆)))) → (𝑥 ∈ (0..^(#‘𝑇)) ∨ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))))
41		simpll 790	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → 𝑆 ∈ Word 𝐴)
42		simplr 792	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → 𝑇 ∈ Word 𝐴)
43		fzoval 12471	. . . . . . . . . . . 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 2687	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ (0..^(#‘𝑇)) ↔ 𝑥 ∈ (0...((#‘𝑇) − 1))))
46	45	biimpa 501	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → 𝑥 ∈ (0...((#‘𝑇) − 1)))
47		fznn0sub2 12446	. . . . . . . . 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 481	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (0..^(#‘𝑇)) = (0...((#‘𝑇) − 1)))
50	48, 49	eleqtrrd 2704	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (((#‘𝑇) − 1) − 𝑥) ∈ (0..^(#‘𝑇)))
51		ccatval3 13363	. . . . . . 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 1326	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((𝑆 ++ 𝑇)‘((((#‘𝑇) − 1) − 𝑥) + (#‘𝑆))) = (𝑇‘(((#‘𝑇) − 1) − 𝑥)))
53	15	oveq1d 6665	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘(𝑆 ++ 𝑇)) − 1) = (((#‘𝑇) + (#‘𝑆)) − 1))
54	12	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘𝑇) ∈ ℂ)
55	10	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘𝑆) ∈ ℂ)
56		1cnd 10056	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → 1 ∈ ℂ)
57	54, 55, 56	addsubd 10413	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((#‘𝑇) + (#‘𝑆)) − 1) = (((#‘𝑇) − 1) + (#‘𝑆)))
58	53, 57	eqtrd 2656	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘(𝑆 ++ 𝑇)) − 1) = (((#‘𝑇) − 1) + (#‘𝑆)))
59	58	oveq1d 6665	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((#‘𝑇) − 1) + (#‘𝑆)) − 𝑥))
60	59	adantr 481	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((#‘𝑇) − 1) + (#‘𝑆)) − 𝑥))
61		peano2zm 11420	. . . . . . . . . . . 12 ⊢ ((#‘𝑇) ∈ ℤ → ((#‘𝑇) − 1) ∈ ℤ)
62	37, 61	syl 17	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → ((#‘𝑇) − 1) ∈ ℤ)
63	62	zcnd 11483	. . . . . . . . . 10 ⊢ (𝑇 ∈ Word 𝐴 → ((#‘𝑇) − 1) ∈ ℂ)
64	63	ad2antlr 763	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((#‘𝑇) − 1) ∈ ℂ)
65	10	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (#‘𝑆) ∈ ℂ)
66		elfzoelz 12470	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0..^(#‘𝑇)) → 𝑥 ∈ ℤ)
67	66	zcnd 11483	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0..^(#‘𝑇)) → 𝑥 ∈ ℂ)
68	67	adantl 482	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → 𝑥 ∈ ℂ)
69	64, 65, 68	addsubd 10413	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((((#‘𝑇) − 1) + (#‘𝑆)) − 𝑥) = ((((#‘𝑇) − 1) − 𝑥) + (#‘𝑆)))
70	60, 69	eqtrd 2656	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((#‘𝑇) − 1) − 𝑥) + (#‘𝑆)))
71	70	fveq2d 6195	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((𝑆 ++ 𝑇)‘(((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = ((𝑆 ++ 𝑇)‘((((#‘𝑇) − 1) − 𝑥) + (#‘𝑆))))
72		revfv 13512	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((reverse‘𝑇)‘𝑥) = (𝑇‘(((#‘𝑇) − 1) − 𝑥)))
73	72	adantll 750	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((reverse‘𝑇)‘𝑥) = (𝑇‘(((#‘𝑇) − 1) − 𝑥)))
74	52, 71, 73	3eqtr4d 2666	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((𝑆 ++ 𝑇)‘(((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = ((reverse‘𝑇)‘𝑥))
75	1	adantr 481	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (𝑆 ++ 𝑇) ∈ Word 𝐴)
76		uzid 11702	. . . . . . . . . . 11 ⊢ ((#‘𝑇) ∈ ℤ → (#‘𝑇) ∈ (ℤ_≥‘(#‘𝑇)))
77	38, 76	syl 17	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘𝑇) ∈ (ℤ_≥‘(#‘𝑇)))
78	9	adantr 481	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘𝑆) ∈ ℕ₀)
79		uzaddcl 11744	. . . . . . . . . 10 ⊢ (((#‘𝑇) ∈ (ℤ_≥‘(#‘𝑇)) ∧ (#‘𝑆) ∈ ℕ₀) → ((#‘𝑇) + (#‘𝑆)) ∈ (ℤ_≥‘(#‘𝑇)))
80	77, 78, 79	syl2anc 693	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘𝑇) + (#‘𝑆)) ∈ (ℤ_≥‘(#‘𝑇)))
81	15, 80	eqeltrd 2701	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘(𝑆 ++ 𝑇)) ∈ (ℤ_≥‘(#‘𝑇)))
82		fzoss2 12496	. . . . . . . 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 3603	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → 𝑥 ∈ (0..^(#‘(𝑆 ++ 𝑇))))
85		revfv 13512	. . . . . 6 ⊢ (((𝑆 ++ 𝑇) ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(#‘(𝑆 ++ 𝑇)))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = ((𝑆 ++ 𝑇)‘(((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
86	75, 84, 85	syl2anc 693	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = ((𝑆 ++ 𝑇)‘(((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
87	20	ad2antlr 763	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (reverse‘𝑇) ∈ Word 𝐴)
88	21	ad2antrr 762	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (reverse‘𝑆) ∈ Word 𝐴)
89	29	adantl 482	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘(reverse‘𝑇)) = (#‘𝑇))
90	89	oveq2d 6666	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(#‘(reverse‘𝑇))) = (0..^(#‘𝑇)))
91	90	eleq2d 2687	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ (0..^(#‘(reverse‘𝑇))) ↔ 𝑥 ∈ (0..^(#‘𝑇))))
92	91	biimpar 502	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → 𝑥 ∈ (0..^(#‘(reverse‘𝑇))))
93		ccatval1 13361	. . . . . 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 1326	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥) = ((reverse‘𝑇)‘𝑥))
95	74, 86, 94	3eqtr4d 2666	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(#‘𝑇))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥))
96	8	oveq1d 6665	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘(𝑆 ++ 𝑇)) − 1) = (((#‘𝑆) + (#‘𝑇)) − 1))
97	55, 54, 56	addsubd 10413	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((#‘𝑆) + (#‘𝑇)) − 1) = (((#‘𝑆) − 1) + (#‘𝑇)))
98	96, 97	eqtrd 2656	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘(𝑆 ++ 𝑇)) − 1) = (((#‘𝑆) − 1) + (#‘𝑇)))
99	98	oveq1d 6665	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((#‘𝑆) − 1) + (#‘𝑇)) − 𝑥))
100	99	adantr 481	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((#‘𝑆) − 1) + (#‘𝑇)) − 𝑥))
101	9	nn0zd 11480	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Word 𝐴 → (#‘𝑆) ∈ ℤ)
102		peano2zm 11420	. . . . . . . . . . . . 13 ⊢ ((#‘𝑆) ∈ ℤ → ((#‘𝑆) − 1) ∈ ℤ)
103	101, 102	syl 17	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Word 𝐴 → ((#‘𝑆) − 1) ∈ ℤ)
104	103	zcnd 11483	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝐴 → ((#‘𝑆) − 1) ∈ ℂ)
105	104	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → ((#‘𝑆) − 1) ∈ ℂ)
106		elfzoelz 12470	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆))) → 𝑥 ∈ ℤ)
107	106	zcnd 11483	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆))) → 𝑥 ∈ ℂ)
108	107	adantl 482	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → 𝑥 ∈ ℂ)
109	12	ad2antlr 763	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (#‘𝑇) ∈ ℂ)
110	105, 108, 109	subsub3d 10422	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (((#‘𝑆) − 1) − (𝑥 − (#‘𝑇))) = ((((#‘𝑆) − 1) + (#‘𝑇)) − 𝑥))
111	100, 110	eqtr4d 2659	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = (((#‘𝑆) − 1) − (𝑥 − (#‘𝑇))))
112	89	oveq2d 6666	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 − (#‘(reverse‘𝑇))) = (𝑥 − (#‘𝑇)))
113	112	oveq2d 6666	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((#‘𝑆) − 1) − (𝑥 − (#‘(reverse‘𝑇)))) = (((#‘𝑆) − 1) − (𝑥 − (#‘𝑇))))
114	113	adantr 481	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (((#‘𝑆) − 1) − (𝑥 − (#‘(reverse‘𝑇)))) = (((#‘𝑆) − 1) − (𝑥 − (#‘𝑇))))
115	111, 114	eqtr4d 2659	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = (((#‘𝑆) − 1) − (𝑥 − (#‘(reverse‘𝑇)))))
116	115	fveq2d 6195	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (𝑆‘(((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = (𝑆‘(((#‘𝑆) − 1) − (𝑥 − (#‘(reverse‘𝑇))))))
117		simpll 790	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → 𝑆 ∈ Word 𝐴)
118		simplr 792	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → 𝑇 ∈ Word 𝐴)
119		zaddcl 11417	. . . . . . . . . . . 12 ⊢ (((#‘𝑇) ∈ ℤ ∧ (#‘𝑆) ∈ ℤ) → ((#‘𝑇) + (#‘𝑆)) ∈ ℤ)
120	37, 101, 119	syl2anr 495	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘𝑇) + (#‘𝑆)) ∈ ℤ)
121		peano2zm 11420	. . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (((#‘𝑇) + (#‘𝑆)) − 1) ∈ ℤ)
124		fzoval 12471	. . . . . . . . . . . 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 2687	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆))) ↔ 𝑥 ∈ ((#‘𝑇)...(((#‘𝑇) + (#‘𝑆)) − 1))))
127	126	biimpa 501	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → 𝑥 ∈ ((#‘𝑇)...(((#‘𝑇) + (#‘𝑆)) − 1)))
128		fzrev2i 12405	. . . . . . . . 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 693	. . . . . . . 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 6665	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((#‘𝑇) + (#‘𝑆)) − 1) − 𝑥))
131	130	adantr 481	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((#‘𝑇) + (#‘𝑆)) − 1) − 𝑥))
132	101	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘𝑆) ∈ ℤ)
133		fzoval 12471	. . . . . . . . . . 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 11483	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((#‘𝑇) + (#‘𝑆)) − 1) ∈ ℂ)
136	135	subidd 10380	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((#‘𝑇) + (#‘𝑆)) − 1) − (((#‘𝑇) + (#‘𝑆)) − 1)) = 0)
137		addcl 10018	. . . . . . . . . . . . . 14 ⊢ (((#‘𝑇) ∈ ℂ ∧ (#‘𝑆) ∈ ℂ) → ((#‘𝑇) + (#‘𝑆)) ∈ ℂ)
138	12, 10, 137	syl2anr 495	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘𝑇) + (#‘𝑆)) ∈ ℂ)
139	138, 56, 54	sub32d 10424	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((#‘𝑇) + (#‘𝑆)) − 1) − (#‘𝑇)) = ((((#‘𝑇) + (#‘𝑆)) − (#‘𝑇)) − 1))
140		pncan2 10288	. . . . . . . . . . . . . 14 ⊢ (((#‘𝑇) ∈ ℂ ∧ (#‘𝑆) ∈ ℂ) → (((#‘𝑇) + (#‘𝑆)) − (#‘𝑇)) = (#‘𝑆))
141	12, 10, 140	syl2anr 495	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((#‘𝑇) + (#‘𝑆)) − (#‘𝑇)) = (#‘𝑆))
142	141	oveq1d 6665	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((#‘𝑇) + (#‘𝑆)) − (#‘𝑇)) − 1) = ((#‘𝑆) − 1))
143	139, 142	eqtrd 2656	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((#‘𝑇) + (#‘𝑆)) − 1) − (#‘𝑇)) = ((#‘𝑆) − 1))
144	136, 143	oveq12d 6668	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((((#‘𝑇) + (#‘𝑆)) − 1) − (((#‘𝑇) + (#‘𝑆)) − 1))...((((#‘𝑇) + (#‘𝑆)) − 1) − (#‘𝑇))) = (0...((#‘𝑆) − 1)))
145	134, 144	eqtr4d 2659	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(#‘𝑆)) = (((((#‘𝑇) + (#‘𝑆)) − 1) − (((#‘𝑇) + (#‘𝑆)) − 1))...((((#‘𝑇) + (#‘𝑆)) − 1) − (#‘𝑇))))
146	145	adantr 481	. . . . . . . 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 2716	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥) ∈ (0..^(#‘𝑆)))
148		ccatval1 13361	. . . . . . 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 1326	. . . . . 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 763	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (#‘(reverse‘𝑇)) = (#‘𝑇))
151	150	oveq2d 6666	. . . . . . . 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 12528	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆))) ∧ (#‘𝑆) ∈ ℤ) → (𝑥 − (#‘𝑇)) ∈ (0..^(#‘𝑆)))
154	152, 132, 153	syl2anr 495	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (𝑥 − (#‘𝑇)) ∈ (0..^(#‘𝑆)))
155	151, 154	eqeltrd 2701	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (𝑥 − (#‘(reverse‘𝑇))) ∈ (0..^(#‘𝑆)))
156		revfv 13512	. . . . . . 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 693	. . . . . 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 2666	. . . . 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 481	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (𝑆 ++ 𝑇) ∈ Word 𝐴)
160	11	adantl 482	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (#‘𝑇) ∈ ℕ₀)
161		fzoss1 12495	. . . . . . . . . 10 ⊢ ((#‘𝑇) ∈ (ℤ_≥‘0) → ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆))) ⊆ (0..^((#‘𝑇) + (#‘𝑆))))
162		nn0uz 11722	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
163	161, 162	eleq2s 2719	. . . . . . . . 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 6666	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(#‘(𝑆 ++ 𝑇))) = (0..^((#‘𝑇) + (#‘𝑆))))
166	164, 165	sseqtr4d 3642	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆))) ⊆ (0..^(#‘(𝑆 ++ 𝑇))))
167	166	sselda 3603	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → 𝑥 ∈ (0..^(#‘(𝑆 ++ 𝑇))))
168	159, 167, 85	syl2anc 693	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = ((𝑆 ++ 𝑇)‘(((#‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
169	20	ad2antlr 763	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (reverse‘𝑇) ∈ Word 𝐴)
170	21	ad2antrr 762	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → (reverse‘𝑆) ∈ Word 𝐴)
171	89, 31	oveq12d 6668	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((#‘(reverse‘𝑇))..^((#‘(reverse‘𝑇)) + (#‘(reverse‘𝑆)))) = ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆))))
172	171	eleq2d 2687	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ ((#‘(reverse‘𝑇))..^((#‘(reverse‘𝑇)) + (#‘(reverse‘𝑆)))) ↔ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))))
173	172	biimpar 502	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → 𝑥 ∈ ((#‘(reverse‘𝑇))..^((#‘(reverse‘𝑇)) + (#‘(reverse‘𝑆)))))
174		ccatval2 13362	. . . . . 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 1326	. . . . 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 2666	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑆)))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥))
177	95, 176	jaodan 826	. . 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 487	. 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 6314	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 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 − cmin 10266 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 ++ cconcat 13293 reversecreverse 13297

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-reverse 13305

This theorem is referenced by: gsumwrev 17796 efginvrel2 18140

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