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Theorem revccat 14127

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 13925	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑆 ++ 𝑇) ∈ Word 𝐴)
2		revcl 14122	. . . 4 ⊢ ((𝑆 ++ 𝑇) ∈ Word 𝐴 → (reverse‘(𝑆 ++ 𝑇)) ∈ Word 𝐴)
3		wrdfn 13875	. . . 4 ⊢ ((reverse‘(𝑆 ++ 𝑇)) ∈ Word 𝐴 → (reverse‘(𝑆 ++ 𝑇)) Fn (0..^(♯‘(reverse‘(𝑆 ++ 𝑇)))))
4	1, 2, 3	3syl 18	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (reverse‘(𝑆 ++ 𝑇)) Fn (0..^(♯‘(reverse‘(𝑆 ++ 𝑇)))))
5		revlen 14123	. . . . . . 7 ⊢ ((𝑆 ++ 𝑇) ∈ Word 𝐴 → (♯‘(reverse‘(𝑆 ++ 𝑇))) = (♯‘(𝑆 ++ 𝑇)))
6	1, 5	syl 17	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(reverse‘(𝑆 ++ 𝑇))) = (♯‘(𝑆 ++ 𝑇)))
7		ccatlen 13926	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑆) + (♯‘𝑇)))
8		lencl 13882	. . . . . . . 8 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℕ₀)
9	8	nn0cnd 11956	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℂ)
10		lencl 13882	. . . . . . . 8 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℕ₀)
11	10	nn0cnd 11956	. . . . . . 7 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℂ)
12		addcom 10825	. . . . . . 7 ⊢ (((♯‘𝑆) ∈ ℂ ∧ (♯‘𝑇) ∈ ℂ) → ((♯‘𝑆) + (♯‘𝑇)) = ((♯‘𝑇) + (♯‘𝑆)))
13	9, 11, 12	syl2an 597	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑆) + (♯‘𝑇)) = ((♯‘𝑇) + (♯‘𝑆)))
14	6, 7, 13	3eqtrd 2860	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(reverse‘(𝑆 ++ 𝑇))) = ((♯‘𝑇) + (♯‘𝑆)))
15	14	oveq2d 7171	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘(reverse‘(𝑆 ++ 𝑇)))) = (0..^((♯‘𝑇) + (♯‘𝑆))))
16	15	fneq2d 6446	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((reverse‘(𝑆 ++ 𝑇)) Fn (0..^(♯‘(reverse‘(𝑆 ++ 𝑇)))) ↔ (reverse‘(𝑆 ++ 𝑇)) Fn (0..^((♯‘𝑇) + (♯‘𝑆)))))
17	4, 16	mpbid 234	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (reverse‘(𝑆 ++ 𝑇)) Fn (0..^((♯‘𝑇) + (♯‘𝑆))))
18		revcl 14122	. . . . 5 ⊢ (𝑇 ∈ Word 𝐴 → (reverse‘𝑇) ∈ Word 𝐴)
19		revcl 14122	. . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → (reverse‘𝑆) ∈ Word 𝐴)
20		ccatcl 13925	. . . . 5 ⊢ (((reverse‘𝑇) ∈ Word 𝐴 ∧ (reverse‘𝑆) ∈ Word 𝐴) → ((reverse‘𝑇) ++ (reverse‘𝑆)) ∈ Word 𝐴)
21	18, 19, 20	syl2anr 598	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((reverse‘𝑇) ++ (reverse‘𝑆)) ∈ Word 𝐴)
22		wrdfn 13875	. . . 4 ⊢ (((reverse‘𝑇) ++ (reverse‘𝑆)) ∈ Word 𝐴 → ((reverse‘𝑇) ++ (reverse‘𝑆)) Fn (0..^(♯‘((reverse‘𝑇) ++ (reverse‘𝑆)))))
23	21, 22	syl 17	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((reverse‘𝑇) ++ (reverse‘𝑆)) Fn (0..^(♯‘((reverse‘𝑇) ++ (reverse‘𝑆)))))
24		ccatlen 13926	. . . . . . 7 ⊢ (((reverse‘𝑇) ∈ Word 𝐴 ∧ (reverse‘𝑆) ∈ Word 𝐴) → (♯‘((reverse‘𝑇) ++ (reverse‘𝑆))) = ((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆))))
25	18, 19, 24	syl2anr 598	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘((reverse‘𝑇) ++ (reverse‘𝑆))) = ((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆))))
26		revlen 14123	. . . . . . 7 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘(reverse‘𝑇)) = (♯‘𝑇))
27		revlen 14123	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘(reverse‘𝑆)) = (♯‘𝑆))
28	26, 27	oveqan12rd 7175	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆))) = ((♯‘𝑇) + (♯‘𝑆)))
29	25, 28	eqtrd 2856	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘((reverse‘𝑇) ++ (reverse‘𝑆))) = ((♯‘𝑇) + (♯‘𝑆)))
30	29	oveq2d 7171	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘((reverse‘𝑇) ++ (reverse‘𝑆)))) = (0..^((♯‘𝑇) + (♯‘𝑆))))
31	30	fneq2d 6446	. . 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)))))
32	23, 31	mpbid 234	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((reverse‘𝑇) ++ (reverse‘𝑆)) Fn (0..^((♯‘𝑇) + (♯‘𝑆))))
33		id 22	. . . 4 ⊢ (𝑥 ∈ (0..^((♯‘𝑇) + (♯‘𝑆))) → 𝑥 ∈ (0..^((♯‘𝑇) + (♯‘𝑆))))
34	10	nn0zd 12084	. . . . 5 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℤ)
35	34	adantl 484	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘𝑇) ∈ ℤ)
36		fzospliti 13068	. . . 4 ⊢ ((𝑥 ∈ (0..^((♯‘𝑇) + (♯‘𝑆))) ∧ (♯‘𝑇) ∈ ℤ) → (𝑥 ∈ (0..^(♯‘𝑇)) ∨ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))))
37	33, 35, 36	syl2anr 598	. . 3 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^((♯‘𝑇) + (♯‘𝑆)))) → (𝑥 ∈ (0..^(♯‘𝑇)) ∨ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))))
38		simpll 765	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑆 ∈ Word 𝐴)
39		simplr 767	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑇 ∈ Word 𝐴)
40		fzoval 13038	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑇) ∈ ℤ → (0..^(♯‘𝑇)) = (0...((♯‘𝑇) − 1)))
41	34, 40	syl 17	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ Word 𝐴 → (0..^(♯‘𝑇)) = (0...((♯‘𝑇) − 1)))
42	41	adantl 484	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘𝑇)) = (0...((♯‘𝑇) − 1)))
43	42	eleq2d 2898	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ (0..^(♯‘𝑇)) ↔ 𝑥 ∈ (0...((♯‘𝑇) − 1))))
44	43	biimpa 479	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑥 ∈ (0...((♯‘𝑇) − 1)))
45		fznn0sub2 13013	. . . . . . . . 9 ⊢ (𝑥 ∈ (0...((♯‘𝑇) − 1)) → (((♯‘𝑇) − 1) − 𝑥) ∈ (0...((♯‘𝑇) − 1)))
46	44, 45	syl 17	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (((♯‘𝑇) − 1) − 𝑥) ∈ (0...((♯‘𝑇) − 1)))
47	41	ad2antlr 725	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (0..^(♯‘𝑇)) = (0...((♯‘𝑇) − 1)))
48	46, 47	eleqtrrd 2916	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (((♯‘𝑇) − 1) − 𝑥) ∈ (0..^(♯‘𝑇)))
49		ccatval3 13932	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ (((♯‘𝑇) − 1) − 𝑥) ∈ (0..^(♯‘𝑇))) → ((𝑆 ++ 𝑇)‘((((♯‘𝑇) − 1) − 𝑥) + (♯‘𝑆))) = (𝑇‘(((♯‘𝑇) − 1) − 𝑥)))
50	38, 39, 48, 49	syl3anc 1367	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((𝑆 ++ 𝑇)‘((((♯‘𝑇) − 1) − 𝑥) + (♯‘𝑆))) = (𝑇‘(((♯‘𝑇) − 1) − 𝑥)))
51	7, 13	eqtrd 2856	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑇) + (♯‘𝑆)))
52	51	oveq1d 7170	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(𝑆 ++ 𝑇)) − 1) = (((♯‘𝑇) + (♯‘𝑆)) − 1))
53	11	adantl 484	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘𝑇) ∈ ℂ)
54	9	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘𝑆) ∈ ℂ)
55		1cnd 10635	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → 1 ∈ ℂ)
56	53, 54, 55	addsubd 11017	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘𝑇) + (♯‘𝑆)) − 1) = (((♯‘𝑇) − 1) + (♯‘𝑆)))
57	52, 56	eqtrd 2856	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(𝑆 ++ 𝑇)) − 1) = (((♯‘𝑇) − 1) + (♯‘𝑆)))
58	57	oveq1d 7170	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑇) − 1) + (♯‘𝑆)) − 𝑥))
59	58	adantr 483	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑇) − 1) + (♯‘𝑆)) − 𝑥))
60		peano2zm 12024	. . . . . . . . . . . 12 ⊢ ((♯‘𝑇) ∈ ℤ → ((♯‘𝑇) − 1) ∈ ℤ)
61	34, 60	syl 17	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → ((♯‘𝑇) − 1) ∈ ℤ)
62	61	zcnd 12087	. . . . . . . . . 10 ⊢ (𝑇 ∈ Word 𝐴 → ((♯‘𝑇) − 1) ∈ ℂ)
63	62	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((♯‘𝑇) − 1) ∈ ℂ)
64	9	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (♯‘𝑆) ∈ ℂ)
65		elfzoelz 13037	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0..^(♯‘𝑇)) → 𝑥 ∈ ℤ)
66	65	zcnd 12087	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0..^(♯‘𝑇)) → 𝑥 ∈ ℂ)
67	66	adantl 484	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑥 ∈ ℂ)
68	63, 64, 67	addsubd 11017	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((((♯‘𝑇) − 1) + (♯‘𝑆)) − 𝑥) = ((((♯‘𝑇) − 1) − 𝑥) + (♯‘𝑆)))
69	59, 68	eqtrd 2856	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑇) − 1) − 𝑥) + (♯‘𝑆)))
70	69	fveq2d 6673	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = ((𝑆 ++ 𝑇)‘((((♯‘𝑇) − 1) − 𝑥) + (♯‘𝑆))))
71		revfv 14124	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((reverse‘𝑇)‘𝑥) = (𝑇‘(((♯‘𝑇) − 1) − 𝑥)))
72	71	adantll 712	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((reverse‘𝑇)‘𝑥) = (𝑇‘(((♯‘𝑇) − 1) − 𝑥)))
73	50, 70, 72	3eqtr4d 2866	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = ((reverse‘𝑇)‘𝑥))
74	34	uzidd 12258	. . . . . . . . . 10 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ (ℤ_≥‘(♯‘𝑇)))
75		uzaddcl 12303	. . . . . . . . . 10 ⊢ (((♯‘𝑇) ∈ (ℤ_≥‘(♯‘𝑇)) ∧ (♯‘𝑆) ∈ ℕ₀) → ((♯‘𝑇) + (♯‘𝑆)) ∈ (ℤ_≥‘(♯‘𝑇)))
76	74, 8, 75	syl2anr 598	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇) + (♯‘𝑆)) ∈ (ℤ_≥‘(♯‘𝑇)))
77	51, 76	eqeltrd 2913	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(𝑆 ++ 𝑇)) ∈ (ℤ_≥‘(♯‘𝑇)))
78		fzoss2 13064	. . . . . . . 8 ⊢ ((♯‘(𝑆 ++ 𝑇)) ∈ (ℤ_≥‘(♯‘𝑇)) → (0..^(♯‘𝑇)) ⊆ (0..^(♯‘(𝑆 ++ 𝑇))))
79	77, 78	syl 17	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘𝑇)) ⊆ (0..^(♯‘(𝑆 ++ 𝑇))))
80	79	sselda 3966	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑥 ∈ (0..^(♯‘(𝑆 ++ 𝑇))))
81		revfv 14124	. . . . . 6 ⊢ (((𝑆 ++ 𝑇) ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘(𝑆 ++ 𝑇)))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
82	1, 80, 81	syl2an2r 683	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
83	18	ad2antlr 725	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (reverse‘𝑇) ∈ Word 𝐴)
84	19	ad2antrr 724	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (reverse‘𝑆) ∈ Word 𝐴)
85	26	adantl 484	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(reverse‘𝑇)) = (♯‘𝑇))
86	85	oveq2d 7171	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘(reverse‘𝑇))) = (0..^(♯‘𝑇)))
87	86	eleq2d 2898	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ (0..^(♯‘(reverse‘𝑇))) ↔ 𝑥 ∈ (0..^(♯‘𝑇))))
88	87	biimpar 480	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑥 ∈ (0..^(♯‘(reverse‘𝑇))))
89		ccatval1 13929	. . . . . 6 ⊢ (((reverse‘𝑇) ∈ Word 𝐴 ∧ (reverse‘𝑆) ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘(reverse‘𝑇)))) → (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥) = ((reverse‘𝑇)‘𝑥))
90	83, 84, 88, 89	syl3anc 1367	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥) = ((reverse‘𝑇)‘𝑥))
91	73, 82, 90	3eqtr4d 2866	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥))
92	8	nn0zd 12084	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℤ)
93		peano2zm 12024	. . . . . . . . . . . 12 ⊢ ((♯‘𝑆) ∈ ℤ → ((♯‘𝑆) − 1) ∈ ℤ)
94	92, 93	syl 17	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝐴 → ((♯‘𝑆) − 1) ∈ ℤ)
95	94	zcnd 12087	. . . . . . . . . 10 ⊢ (𝑆 ∈ Word 𝐴 → ((♯‘𝑆) − 1) ∈ ℂ)
96	95	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((♯‘𝑆) − 1) ∈ ℂ)
97		elfzoelz 13037	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) → 𝑥 ∈ ℤ)
98	97	zcnd 12087	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) → 𝑥 ∈ ℂ)
99	98	adantl 484	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑥 ∈ ℂ)
100	11	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (♯‘𝑇) ∈ ℂ)
101	96, 99, 100	subsub3d 11026	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘𝑆) − 1) − (𝑥 − (♯‘𝑇))) = ((((♯‘𝑆) − 1) + (♯‘𝑇)) − 𝑥))
102	26	oveq2d 7171	. . . . . . . . . 10 ⊢ (𝑇 ∈ Word 𝐴 → (𝑥 − (♯‘(reverse‘𝑇))) = (𝑥 − (♯‘𝑇)))
103	102	oveq2d 7171	. . . . . . . . 9 ⊢ (𝑇 ∈ Word 𝐴 → (((♯‘𝑆) − 1) − (𝑥 − (♯‘(reverse‘𝑇)))) = (((♯‘𝑆) − 1) − (𝑥 − (♯‘𝑇))))
104	103	ad2antlr 725	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘𝑆) − 1) − (𝑥 − (♯‘(reverse‘𝑇)))) = (((♯‘𝑆) − 1) − (𝑥 − (♯‘𝑇))))
105	7	oveq1d 7170	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(𝑆 ++ 𝑇)) − 1) = (((♯‘𝑆) + (♯‘𝑇)) − 1))
106	54, 53, 55	addsubd 11017	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘𝑆) + (♯‘𝑇)) − 1) = (((♯‘𝑆) − 1) + (♯‘𝑇)))
107	105, 106	eqtrd 2856	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(𝑆 ++ 𝑇)) − 1) = (((♯‘𝑆) − 1) + (♯‘𝑇)))
108	107	oveq1d 7170	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑆) − 1) + (♯‘𝑇)) − 𝑥))
109	108	adantr 483	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑆) − 1) + (♯‘𝑇)) − 𝑥))
110	101, 104, 109	3eqtr4rd 2867	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = (((♯‘𝑆) − 1) − (𝑥 − (♯‘(reverse‘𝑇)))))
111	110	fveq2d 6673	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (𝑆‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = (𝑆‘(((♯‘𝑆) − 1) − (𝑥 − (♯‘(reverse‘𝑇))))))
112		simpll 765	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑆 ∈ Word 𝐴)
113		simplr 767	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑇 ∈ Word 𝐴)
114		zaddcl 12021	. . . . . . . . . . 11 ⊢ (((♯‘𝑇) ∈ ℤ ∧ (♯‘𝑆) ∈ ℤ) → ((♯‘𝑇) + (♯‘𝑆)) ∈ ℤ)
115	34, 92, 114	syl2anr 598	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇) + (♯‘𝑆)) ∈ ℤ)
116		peano2zm 12024	. . . . . . . . . 10 ⊢ (((♯‘𝑇) + (♯‘𝑆)) ∈ ℤ → (((♯‘𝑇) + (♯‘𝑆)) − 1) ∈ ℤ)
117	115, 116	syl 17	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘𝑇) + (♯‘𝑆)) − 1) ∈ ℤ)
118		fzoval 13038	. . . . . . . . . . . 12 ⊢ (((♯‘𝑇) + (♯‘𝑆)) ∈ ℤ → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) = ((♯‘𝑇)...(((♯‘𝑇) + (♯‘𝑆)) − 1)))
119	115, 118	syl 17	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) = ((♯‘𝑇)...(((♯‘𝑇) + (♯‘𝑆)) − 1)))
120	119	eleq2d 2898	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ↔ 𝑥 ∈ ((♯‘𝑇)...(((♯‘𝑇) + (♯‘𝑆)) − 1))))
121	120	biimpa 479	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑥 ∈ ((♯‘𝑇)...(((♯‘𝑇) + (♯‘𝑆)) − 1)))
122		fzrev2i 12971	. . . . . . . . 9 ⊢ (((((♯‘𝑇) + (♯‘𝑆)) − 1) ∈ ℤ ∧ 𝑥 ∈ ((♯‘𝑇)...(((♯‘𝑇) + (♯‘𝑆)) − 1))) → ((((♯‘𝑇) + (♯‘𝑆)) − 1) − 𝑥) ∈ (((((♯‘𝑇) + (♯‘𝑆)) − 1) − (((♯‘𝑇) + (♯‘𝑆)) − 1))...((((♯‘𝑇) + (♯‘𝑆)) − 1) − (♯‘𝑇))))
123	117, 121, 122	syl2an2r 683	. . . . . . . 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))))
124	52	oveq1d 7170	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑇) + (♯‘𝑆)) − 1) − 𝑥))
125	124	adantr 483	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑇) + (♯‘𝑆)) − 1) − 𝑥))
126	92	adantr 483	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘𝑆) ∈ ℤ)
127		fzoval 13038	. . . . . . . . . . 11 ⊢ ((♯‘𝑆) ∈ ℤ → (0..^(♯‘𝑆)) = (0...((♯‘𝑆) − 1)))
128	126, 127	syl 17	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘𝑆)) = (0...((♯‘𝑆) − 1)))
129	117	zcnd 12087	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘𝑇) + (♯‘𝑆)) − 1) ∈ ℂ)
130	129	subidd 10984	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((♯‘𝑇) + (♯‘𝑆)) − 1) − (((♯‘𝑇) + (♯‘𝑆)) − 1)) = 0)
131		addcl 10618	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑆) ∈ ℂ) → ((♯‘𝑇) + (♯‘𝑆)) ∈ ℂ)
132	11, 9, 131	syl2anr 598	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇) + (♯‘𝑆)) ∈ ℂ)
133	132, 55, 53	sub32d 11028	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((♯‘𝑇) + (♯‘𝑆)) − 1) − (♯‘𝑇)) = ((((♯‘𝑇) + (♯‘𝑆)) − (♯‘𝑇)) − 1))
134		pncan2 10892	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑆) ∈ ℂ) → (((♯‘𝑇) + (♯‘𝑆)) − (♯‘𝑇)) = (♯‘𝑆))
135	11, 9, 134	syl2anr 598	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘𝑇) + (♯‘𝑆)) − (♯‘𝑇)) = (♯‘𝑆))
136	135	oveq1d 7170	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((♯‘𝑇) + (♯‘𝑆)) − (♯‘𝑇)) − 1) = ((♯‘𝑆) − 1))
137	133, 136	eqtrd 2856	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((♯‘𝑇) + (♯‘𝑆)) − 1) − (♯‘𝑇)) = ((♯‘𝑆) − 1))
138	130, 137	oveq12d 7173	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((((♯‘𝑇) + (♯‘𝑆)) − 1) − (((♯‘𝑇) + (♯‘𝑆)) − 1))...((((♯‘𝑇) + (♯‘𝑆)) − 1) − (♯‘𝑇))) = (0...((♯‘𝑆) − 1)))
139	128, 138	eqtr4d 2859	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘𝑆)) = (((((♯‘𝑇) + (♯‘𝑆)) − 1) − (((♯‘𝑇) + (♯‘𝑆)) − 1))...((((♯‘𝑇) + (♯‘𝑆)) − 1) − (♯‘𝑇))))
140	139	adantr 483	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (0..^(♯‘𝑆)) = (((((♯‘𝑇) + (♯‘𝑆)) − 1) − (((♯‘𝑇) + (♯‘𝑆)) − 1))...((((♯‘𝑇) + (♯‘𝑆)) − 1) − (♯‘𝑇))))
141	123, 125, 140	3eltr4d 2928	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) ∈ (0..^(♯‘𝑆)))
142		ccatval1 13929	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = (𝑆‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
143	112, 113, 141, 142	syl3anc 1367	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = (𝑆‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
144		simpl 485	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → 𝑆 ∈ Word 𝐴)
145	102	ad2antlr 725	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (𝑥 − (♯‘(reverse‘𝑇))) = (𝑥 − (♯‘𝑇)))
146		id 22	. . . . . . . . 9 ⊢ (𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) → 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))))
147		fzosubel3 13097	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ∧ (♯‘𝑆) ∈ ℤ) → (𝑥 − (♯‘𝑇)) ∈ (0..^(♯‘𝑆)))
148	146, 126, 147	syl2anr 598	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (𝑥 − (♯‘𝑇)) ∈ (0..^(♯‘𝑆)))
149	145, 148	eqeltrd 2913	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (𝑥 − (♯‘(reverse‘𝑇))) ∈ (0..^(♯‘𝑆)))
150		revfv 14124	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ (𝑥 − (♯‘(reverse‘𝑇))) ∈ (0..^(♯‘𝑆))) → ((reverse‘𝑆)‘(𝑥 − (♯‘(reverse‘𝑇)))) = (𝑆‘(((♯‘𝑆) − 1) − (𝑥 − (♯‘(reverse‘𝑇))))))
151	144, 149, 150	syl2an2r 683	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((reverse‘𝑆)‘(𝑥 − (♯‘(reverse‘𝑇)))) = (𝑆‘(((♯‘𝑆) − 1) − (𝑥 − (♯‘(reverse‘𝑇))))))
152	111, 143, 151	3eqtr4d 2866	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = ((reverse‘𝑆)‘(𝑥 − (♯‘(reverse‘𝑇)))))
153		fzoss1 13063	. . . . . . . . . . 11 ⊢ ((♯‘𝑇) ∈ (ℤ_≥‘0) → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ⊆ (0..^((♯‘𝑇) + (♯‘𝑆))))
154		nn0uz 12279	. . . . . . . . . . 11 ⊢ ℕ₀ = (ℤ_≥‘0)
155	153, 154	eleq2s 2931	. . . . . . . . . 10 ⊢ ((♯‘𝑇) ∈ ℕ₀ → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ⊆ (0..^((♯‘𝑇) + (♯‘𝑆))))
156	10, 155	syl 17	. . . . . . . . 9 ⊢ (𝑇 ∈ Word 𝐴 → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ⊆ (0..^((♯‘𝑇) + (♯‘𝑆))))
157	156	adantl 484	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ⊆ (0..^((♯‘𝑇) + (♯‘𝑆))))
158	51	oveq2d 7171	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘(𝑆 ++ 𝑇))) = (0..^((♯‘𝑇) + (♯‘𝑆))))
159	157, 158	sseqtrrd 4007	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ⊆ (0..^(♯‘(𝑆 ++ 𝑇))))
160	159	sselda 3966	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑥 ∈ (0..^(♯‘(𝑆 ++ 𝑇))))
161	1, 160, 81	syl2an2r 683	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
162	18	ad2antlr 725	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (reverse‘𝑇) ∈ Word 𝐴)
163	19	ad2antrr 724	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (reverse‘𝑆) ∈ Word 𝐴)
164	85, 28	oveq12d 7173	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(reverse‘𝑇))..^((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆)))) = ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))))
165	164	eleq2d 2898	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ ((♯‘(reverse‘𝑇))..^((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆)))) ↔ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))))
166	165	biimpar 480	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑥 ∈ ((♯‘(reverse‘𝑇))..^((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆)))))
167		ccatval2 13931	. . . . . 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)))))
168	162, 163, 166, 167	syl3anc 1367	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥) = ((reverse‘𝑆)‘(𝑥 − (♯‘(reverse‘𝑇)))))
169	152, 161, 168	3eqtr4d 2866	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥))
170	91, 169	jaodan 954	. . 3 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ (𝑥 ∈ (0..^(♯‘𝑇)) ∨ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥))
171	37, 170	syldan 593	. 2 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^((♯‘𝑇) + (♯‘𝑆)))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥))
172	17, 32, 171	eqfnfvd 6804	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 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 Fn wfn 6349 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 0cc0 10536 1c1 10537 + caddc 10539 − cmin 10869 ℕ₀cn0 11896 ℤcz 11980 ℤ_≥cuz 12242 ...cfz 12891 ..^cfzo 13032 ♯chash 13689 Word cword 13860 ++ cconcat 13921 reversecreverse 14119

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-fz 12892 df-fzo 13033 df-hash 13690 df-word 13861 df-concat 13922 df-reverse 14120

This theorem is referenced by: gsumwrev 18493 efginvrel2 18852

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