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Theorem revccat 14827

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 14631	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑆 ++ 𝑇) ∈ Word 𝐴)
2		revcl 14822	. . . 4 ⊢ ((𝑆 ++ 𝑇) ∈ Word 𝐴 → (reverse‘(𝑆 ++ 𝑇)) ∈ Word 𝐴)
3		wrdfn 14585	. . . 4 ⊢ ((reverse‘(𝑆 ++ 𝑇)) ∈ Word 𝐴 → (reverse‘(𝑆 ++ 𝑇)) Fn (0..^(♯‘(reverse‘(𝑆 ++ 𝑇)))))
4	1, 2, 3	3syl 19	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (reverse‘(𝑆 ++ 𝑇)) Fn (0..^(♯‘(reverse‘(𝑆 ++ 𝑇)))))
5		revlen 14823	. . . . . . 7 ⊢ ((𝑆 ++ 𝑇) ∈ Word 𝐴 → (♯‘(reverse‘(𝑆 ++ 𝑇))) = (♯‘(𝑆 ++ 𝑇)))
6	1, 5	syl 18	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(reverse‘(𝑆 ++ 𝑇))) = (♯‘(𝑆 ++ 𝑇)))
7		ccatlen 14632	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑆) + (♯‘𝑇)))
8		lencl 14590	. . . . . . . 8 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℕ₀)
9	8	nn0cnd 12585	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℂ)
10		lencl 14590	. . . . . . . 8 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℕ₀)
11	10	nn0cnd 12585	. . . . . . 7 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℂ)
12		addcom 11414	. . . . . . 7 ⊢ (((♯‘𝑆) ∈ ℂ ∧ (♯‘𝑇) ∈ ℂ) → ((♯‘𝑆) + (♯‘𝑇)) = ((♯‘𝑇) + (♯‘𝑆)))
13	9, 11, 12	syl2an 608	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑆) + (♯‘𝑇)) = ((♯‘𝑇) + (♯‘𝑆)))
14	6, 7, 13	3eqtrd 2805	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(reverse‘(𝑆 ++ 𝑇))) = ((♯‘𝑇) + (♯‘𝑆)))
15	14	oveq2d 7439	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘(reverse‘(𝑆 ++ 𝑇)))) = (0..^((♯‘𝑇) + (♯‘𝑆))))
16	15	fneq2d 6636	. . 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 235	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (reverse‘(𝑆 ++ 𝑇)) Fn (0..^((♯‘𝑇) + (♯‘𝑆))))
18		revcl 14822	. . . . 5 ⊢ (𝑇 ∈ Word 𝐴 → (reverse‘𝑇) ∈ Word 𝐴)
19		revcl 14822	. . . . 5 ⊢ (𝑆 ∈ Word 𝐴 → (reverse‘𝑆) ∈ Word 𝐴)
20		ccatcl 14631	. . . . 5 ⊢ (((reverse‘𝑇) ∈ Word 𝐴 ∧ (reverse‘𝑆) ∈ Word 𝐴) → ((reverse‘𝑇) ++ (reverse‘𝑆)) ∈ Word 𝐴)
21	18, 19, 20	syl2anr 609	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((reverse‘𝑇) ++ (reverse‘𝑆)) ∈ Word 𝐴)
22		wrdfn 14585	. . . 4 ⊢ (((reverse‘𝑇) ++ (reverse‘𝑆)) ∈ Word 𝐴 → ((reverse‘𝑇) ++ (reverse‘𝑆)) Fn (0..^(♯‘((reverse‘𝑇) ++ (reverse‘𝑆)))))
23	21, 22	syl 18	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((reverse‘𝑇) ++ (reverse‘𝑆)) Fn (0..^(♯‘((reverse‘𝑇) ++ (reverse‘𝑆)))))
24		ccatlen 14632	. . . . . . 7 ⊢ (((reverse‘𝑇) ∈ Word 𝐴 ∧ (reverse‘𝑆) ∈ Word 𝐴) → (♯‘((reverse‘𝑇) ++ (reverse‘𝑆))) = ((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆))))
25	18, 19, 24	syl2anr 609	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘((reverse‘𝑇) ++ (reverse‘𝑆))) = ((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆))))
26		revlen 14823	. . . . . . 7 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘(reverse‘𝑇)) = (♯‘𝑇))
27		revlen 14823	. . . . . . 7 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘(reverse‘𝑆)) = (♯‘𝑆))
28	26, 27	oveqan12rd 7443	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆))) = ((♯‘𝑇) + (♯‘𝑆)))
29	25, 28	eqtrd 2801	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘((reverse‘𝑇) ++ (reverse‘𝑆))) = ((♯‘𝑇) + (♯‘𝑆)))
30	29	oveq2d 7439	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘((reverse‘𝑇) ++ (reverse‘𝑆)))) = (0..^((♯‘𝑇) + (♯‘𝑆))))
31	30	fneq2d 6636	. . 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 235	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((reverse‘𝑇) ++ (reverse‘𝑆)) Fn (0..^((♯‘𝑇) + (♯‘𝑆))))
33		id 23	. . . 4 ⊢ (𝑥 ∈ (0..^((♯‘𝑇) + (♯‘𝑆))) → 𝑥 ∈ (0..^((♯‘𝑇) + (♯‘𝑆))))
34	10	nn0zd 12634	. . . . 5 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℤ)
35	34	adantl 487	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘𝑇) ∈ ℤ)
36		fzospliti 13739	. . . 4 ⊢ ((𝑥 ∈ (0..^((♯‘𝑇) + (♯‘𝑆))) ∧ (♯‘𝑇) ∈ ℤ) → (𝑥 ∈ (0..^(♯‘𝑇)) ∨ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))))
37	33, 35, 36	syl2anr 609	. . 3 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^((♯‘𝑇) + (♯‘𝑆)))) → (𝑥 ∈ (0..^(♯‘𝑇)) ∨ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))))
38		simpll 779	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑆 ∈ Word 𝐴)
39		simplr 781	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑇 ∈ Word 𝐴)
40		fzoval 13707	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑇) ∈ ℤ → (0..^(♯‘𝑇)) = (0...((♯‘𝑇) − 1)))
41	34, 40	syl 18	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ Word 𝐴 → (0..^(♯‘𝑇)) = (0...((♯‘𝑇) − 1)))
42	41	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘𝑇)) = (0...((♯‘𝑇) − 1)))
43	42	eleq2d 2852	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ (0..^(♯‘𝑇)) ↔ 𝑥 ∈ (0...((♯‘𝑇) − 1))))
44	43	biimpa 482	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑥 ∈ (0...((♯‘𝑇) − 1)))
45		fznn0sub2 13682	. . . . . . . . 9 ⊢ (𝑥 ∈ (0...((♯‘𝑇) − 1)) → (((♯‘𝑇) − 1) − 𝑥) ∈ (0...((♯‘𝑇) − 1)))
46	44, 45	syl 18	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (((♯‘𝑇) − 1) − 𝑥) ∈ (0...((♯‘𝑇) − 1)))
47	41	ad2antlr 740	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (0..^(♯‘𝑇)) = (0...((♯‘𝑇) − 1)))
48	46, 47	eleqtrrd 2869	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (((♯‘𝑇) − 1) − 𝑥) ∈ (0..^(♯‘𝑇)))
49		ccatval3 14636	. . . . . . 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 1398	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((𝑆 ++ 𝑇)‘((((♯‘𝑇) − 1) − 𝑥) + (♯‘𝑆))) = (𝑇‘(((♯‘𝑇) − 1) − 𝑥)))
51	7, 13	eqtrd 2801	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑇) + (♯‘𝑆)))
52	51	oveq1d 7438	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(𝑆 ++ 𝑇)) − 1) = (((♯‘𝑇) + (♯‘𝑆)) − 1))
53	11	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘𝑇) ∈ ℂ)
54	9	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘𝑆) ∈ ℂ)
55		1cnd 11220	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → 1 ∈ ℂ)
56	53, 54, 55	addsubd 11608	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘𝑇) + (♯‘𝑆)) − 1) = (((♯‘𝑇) − 1) + (♯‘𝑆)))
57	52, 56	eqtrd 2801	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(𝑆 ++ 𝑇)) − 1) = (((♯‘𝑇) − 1) + (♯‘𝑆)))
58	57	oveq1d 7438	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑇) − 1) + (♯‘𝑆)) − 𝑥))
59	58	adantr 486	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑇) − 1) + (♯‘𝑆)) − 𝑥))
60		peano2zm 12655	. . . . . . . . . . . 12 ⊢ ((♯‘𝑇) ∈ ℤ → ((♯‘𝑇) − 1) ∈ ℤ)
61	34, 60	syl 18	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → ((♯‘𝑇) − 1) ∈ ℤ)
62	61	zcnd 12719	. . . . . . . . . 10 ⊢ (𝑇 ∈ Word 𝐴 → ((♯‘𝑇) − 1) ∈ ℂ)
63	62	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((♯‘𝑇) − 1) ∈ ℂ)
64	9	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (♯‘𝑆) ∈ ℂ)
65		elfzoelz 13706	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0..^(♯‘𝑇)) → 𝑥 ∈ ℤ)
66	65	zcnd 12719	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0..^(♯‘𝑇)) → 𝑥 ∈ ℂ)
67	66	adantl 487	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑥 ∈ ℂ)
68	63, 64, 67	addsubd 11608	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((((♯‘𝑇) − 1) + (♯‘𝑆)) − 𝑥) = ((((♯‘𝑇) − 1) − 𝑥) + (♯‘𝑆)))
69	59, 68	eqtrd 2801	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑇) − 1) − 𝑥) + (♯‘𝑆)))
70	69	fveq2d 6892	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = ((𝑆 ++ 𝑇)‘((((♯‘𝑇) − 1) − 𝑥) + (♯‘𝑆))))
71		revfv 14824	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((reverse‘𝑇)‘𝑥) = (𝑇‘(((♯‘𝑇) − 1) − 𝑥)))
72	71	adantll 727	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((reverse‘𝑇)‘𝑥) = (𝑇‘(((♯‘𝑇) − 1) − 𝑥)))
73	50, 70, 72	3eqtr4d 2811	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = ((reverse‘𝑇)‘𝑥))
74	34	uzidd 12896	. . . . . . . . . 10 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ (ℤ_≥‘(♯‘𝑇)))
75		uzaddcl 12946	. . . . . . . . . 10 ⊢ (((♯‘𝑇) ∈ (ℤ_≥‘(♯‘𝑇)) ∧ (♯‘𝑆) ∈ ℕ₀) → ((♯‘𝑇) + (♯‘𝑆)) ∈ (ℤ_≥‘(♯‘𝑇)))
76	74, 8, 75	syl2anr 609	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇) + (♯‘𝑆)) ∈ (ℤ_≥‘(♯‘𝑇)))
77	51, 76	eqeltrd 2866	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(𝑆 ++ 𝑇)) ∈ (ℤ_≥‘(♯‘𝑇)))
78		fzoss2 13735	. . . . . . . 8 ⊢ ((♯‘(𝑆 ++ 𝑇)) ∈ (ℤ_≥‘(♯‘𝑇)) → (0..^(♯‘𝑇)) ⊆ (0..^(♯‘(𝑆 ++ 𝑇))))
79	77, 78	syl 18	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘𝑇)) ⊆ (0..^(♯‘(𝑆 ++ 𝑇))))
80	79	sselda 3940	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑥 ∈ (0..^(♯‘(𝑆 ++ 𝑇))))
81		revfv 14824	. . . . . 6 ⊢ (((𝑆 ++ 𝑇) ∈ Word 𝐴 ∧ 𝑥 ∈ (0..^(♯‘(𝑆 ++ 𝑇)))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
82	1, 80, 81	syl2an2r 698	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
83	18	ad2antlr 740	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (reverse‘𝑇) ∈ Word 𝐴)
84	19	ad2antrr 739	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (reverse‘𝑆) ∈ Word 𝐴)
85	26	adantl 487	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘(reverse‘𝑇)) = (♯‘𝑇))
86	85	oveq2d 7439	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘(reverse‘𝑇))) = (0..^(♯‘𝑇)))
87	86	eleq2d 2852	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ (0..^(♯‘(reverse‘𝑇))) ↔ 𝑥 ∈ (0..^(♯‘𝑇))))
88	87	biimpar 483	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → 𝑥 ∈ (0..^(♯‘(reverse‘𝑇))))
89		ccatval1 14634	. . . . . 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 1398	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥) = ((reverse‘𝑇)‘𝑥))
91	73, 82, 90	3eqtr4d 2811	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ (0..^(♯‘𝑇))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥))
92	8	nn0zd 12634	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℤ)
93		peano2zm 12655	. . . . . . . . . . . 12 ⊢ ((♯‘𝑆) ∈ ℤ → ((♯‘𝑆) − 1) ∈ ℤ)
94	92, 93	syl 18	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝐴 → ((♯‘𝑆) − 1) ∈ ℤ)
95	94	zcnd 12719	. . . . . . . . . 10 ⊢ (𝑆 ∈ Word 𝐴 → ((♯‘𝑆) − 1) ∈ ℂ)
96	95	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((♯‘𝑆) − 1) ∈ ℂ)
97		elfzoelz 13706	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) → 𝑥 ∈ ℤ)
98	97	zcnd 12719	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) → 𝑥 ∈ ℂ)
99	98	adantl 487	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑥 ∈ ℂ)
100	11	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (♯‘𝑇) ∈ ℂ)
101	96, 99, 100	subsub3d 11617	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘𝑆) − 1) − (𝑥 − (♯‘𝑇))) = ((((♯‘𝑆) − 1) + (♯‘𝑇)) − 𝑥))
102	26	oveq2d 7439	. . . . . . . . . 10 ⊢ (𝑇 ∈ Word 𝐴 → (𝑥 − (♯‘(reverse‘𝑇))) = (𝑥 − (♯‘𝑇)))
103	102	oveq2d 7439	. . . . . . . . 9 ⊢ (𝑇 ∈ Word 𝐴 → (((♯‘𝑆) − 1) − (𝑥 − (♯‘(reverse‘𝑇)))) = (((♯‘𝑆) − 1) − (𝑥 − (♯‘𝑇))))
104	103	ad2antlr 740	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘𝑆) − 1) − (𝑥 − (♯‘(reverse‘𝑇)))) = (((♯‘𝑆) − 1) − (𝑥 − (♯‘𝑇))))
105	7	oveq1d 7438	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(𝑆 ++ 𝑇)) − 1) = (((♯‘𝑆) + (♯‘𝑇)) − 1))
106	54, 53, 55	addsubd 11608	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘𝑆) + (♯‘𝑇)) − 1) = (((♯‘𝑆) − 1) + (♯‘𝑇)))
107	105, 106	eqtrd 2801	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(𝑆 ++ 𝑇)) − 1) = (((♯‘𝑆) − 1) + (♯‘𝑇)))
108	107	oveq1d 7438	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑆) − 1) + (♯‘𝑇)) − 𝑥))
109	108	adantr 486	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑆) − 1) + (♯‘𝑇)) − 𝑥))
110	101, 104, 109	3eqtr4rd 2812	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = (((♯‘𝑆) − 1) − (𝑥 − (♯‘(reverse‘𝑇)))))
111	110	fveq2d 6892	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (𝑆‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = (𝑆‘(((♯‘𝑆) − 1) − (𝑥 − (♯‘(reverse‘𝑇))))))
112		simpll 779	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑆 ∈ Word 𝐴)
113		simplr 781	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑇 ∈ Word 𝐴)
114		zaddcl 12652	. . . . . . . . . . 11 ⊢ (((♯‘𝑇) ∈ ℤ ∧ (♯‘𝑆) ∈ ℤ) → ((♯‘𝑇) + (♯‘𝑆)) ∈ ℤ)
115	34, 92, 114	syl2anr 609	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇) + (♯‘𝑆)) ∈ ℤ)
116		peano2zm 12655	. . . . . . . . . 10 ⊢ (((♯‘𝑇) + (♯‘𝑆)) ∈ ℤ → (((♯‘𝑇) + (♯‘𝑆)) − 1) ∈ ℤ)
117	115, 116	syl 18	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘𝑇) + (♯‘𝑆)) − 1) ∈ ℤ)
118		fzoval 13707	. . . . . . . . . . . 12 ⊢ (((♯‘𝑇) + (♯‘𝑆)) ∈ ℤ → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) = ((♯‘𝑇)...(((♯‘𝑇) + (♯‘𝑆)) − 1)))
119	115, 118	syl 18	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) = ((♯‘𝑇)...(((♯‘𝑇) + (♯‘𝑆)) − 1)))
120	119	eleq2d 2852	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ↔ 𝑥 ∈ ((♯‘𝑇)...(((♯‘𝑇) + (♯‘𝑆)) − 1))))
121	120	biimpa 482	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑥 ∈ ((♯‘𝑇)...(((♯‘𝑇) + (♯‘𝑆)) − 1)))
122		fzrev2i 13636	. . . . . . . . 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 698	. . . . . . . 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 7438	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑇) + (♯‘𝑆)) − 1) − 𝑥))
125	124	adantr 486	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) = ((((♯‘𝑇) + (♯‘𝑆)) − 1) − 𝑥))
126	92	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (♯‘𝑆) ∈ ℤ)
127		fzoval 13707	. . . . . . . . . . 11 ⊢ ((♯‘𝑆) ∈ ℤ → (0..^(♯‘𝑆)) = (0...((♯‘𝑆) − 1)))
128	126, 127	syl 18	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘𝑆)) = (0...((♯‘𝑆) − 1)))
129	117	zcnd 12719	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘𝑇) + (♯‘𝑆)) − 1) ∈ ℂ)
130	129	subidd 11575	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((♯‘𝑇) + (♯‘𝑆)) − 1) − (((♯‘𝑇) + (♯‘𝑆)) − 1)) = 0)
131		addcl 11200	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑆) ∈ ℂ) → ((♯‘𝑇) + (♯‘𝑆)) ∈ ℂ)
132	11, 9, 131	syl2anr 609	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇) + (♯‘𝑆)) ∈ ℂ)
133	132, 55, 53	sub32d 11619	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((♯‘𝑇) + (♯‘𝑆)) − 1) − (♯‘𝑇)) = ((((♯‘𝑇) + (♯‘𝑆)) − (♯‘𝑇)) − 1))
134		pncan2 11482	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑇) ∈ ℂ ∧ (♯‘𝑆) ∈ ℂ) → (((♯‘𝑇) + (♯‘𝑆)) − (♯‘𝑇)) = (♯‘𝑆))
135	11, 9, 134	syl2anr 609	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((♯‘𝑇) + (♯‘𝑆)) − (♯‘𝑇)) = (♯‘𝑆))
136	135	oveq1d 7438	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((♯‘𝑇) + (♯‘𝑆)) − (♯‘𝑇)) − 1) = ((♯‘𝑆) − 1))
137	133, 136	eqtrd 2801	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((((♯‘𝑇) + (♯‘𝑆)) − 1) − (♯‘𝑇)) = ((♯‘𝑆) − 1))
138	130, 137	oveq12d 7441	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (((((♯‘𝑇) + (♯‘𝑆)) − 1) − (((♯‘𝑇) + (♯‘𝑆)) − 1))...((((♯‘𝑇) + (♯‘𝑆)) − 1) − (♯‘𝑇))) = (0...((♯‘𝑆) − 1)))
139	128, 138	eqtr4d 2804	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘𝑆)) = (((((♯‘𝑇) + (♯‘𝑆)) − 1) − (((♯‘𝑇) + (♯‘𝑆)) − 1))...((((♯‘𝑇) + (♯‘𝑆)) − 1) − (♯‘𝑇))))
140	139	adantr 486	. . . . . . . 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 2881	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥) ∈ (0..^(♯‘𝑆)))
142		ccatval1 14634	. . . . . . 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 1398	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = (𝑆‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
144		simpl 488	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → 𝑆 ∈ Word 𝐴)
145	102	ad2antlr 740	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (𝑥 − (♯‘(reverse‘𝑇))) = (𝑥 − (♯‘𝑇)))
146		id 23	. . . . . . . . 9 ⊢ (𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) → 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))))
147		fzosubel3 13774	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ∧ (♯‘𝑆) ∈ ℤ) → (𝑥 − (♯‘𝑇)) ∈ (0..^(♯‘𝑆)))
148	146, 126, 147	syl2anr 609	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (𝑥 − (♯‘𝑇)) ∈ (0..^(♯‘𝑆)))
149	145, 148	eqeltrd 2866	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (𝑥 − (♯‘(reverse‘𝑇))) ∈ (0..^(♯‘𝑆)))
150		revfv 14824	. . . . . . 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 698	. . . . . 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 2811	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)) = ((reverse‘𝑆)‘(𝑥 − (♯‘(reverse‘𝑇)))))
153		fzoss1 13734	. . . . . . . . . . 11 ⊢ ((♯‘𝑇) ∈ (ℤ_≥‘0) → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ⊆ (0..^((♯‘𝑇) + (♯‘𝑆))))
154		nn0uz 12918	. . . . . . . . . . 11 ⊢ ℕ₀ = (ℤ_≥‘0)
155	153, 154	eleq2s 2884	. . . . . . . . . 10 ⊢ ((♯‘𝑇) ∈ ℕ₀ → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ⊆ (0..^((♯‘𝑇) + (♯‘𝑆))))
156	10, 155	syl 18	. . . . . . . . 9 ⊢ (𝑇 ∈ Word 𝐴 → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ⊆ (0..^((♯‘𝑇) + (♯‘𝑆))))
157	156	adantl 487	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ⊆ (0..^((♯‘𝑇) + (♯‘𝑆))))
158	51	oveq2d 7439	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (0..^(♯‘(𝑆 ++ 𝑇))) = (0..^((♯‘𝑇) + (♯‘𝑆))))
159	157, 158	sseqtrrd 3977	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))) ⊆ (0..^(♯‘(𝑆 ++ 𝑇))))
160	159	sselda 3940	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑥 ∈ (0..^(♯‘(𝑆 ++ 𝑇))))
161	1, 160, 81	syl2an2r 698	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = ((𝑆 ++ 𝑇)‘(((♯‘(𝑆 ++ 𝑇)) − 1) − 𝑥)))
162	18	ad2antlr 740	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (reverse‘𝑇) ∈ Word 𝐴)
163	19	ad2antrr 739	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → (reverse‘𝑆) ∈ Word 𝐴)
164	85, 28	oveq12d 7441	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → ((♯‘(reverse‘𝑇))..^((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆)))) = ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆))))
165	164	eleq2d 2852	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑥 ∈ ((♯‘(reverse‘𝑇))..^((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆)))) ↔ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))))
166	165	biimpar 483	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → 𝑥 ∈ ((♯‘(reverse‘𝑇))..^((♯‘(reverse‘𝑇)) + (♯‘(reverse‘𝑆)))))
167		ccatval2 14635	. . . . . 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 1398	. . . . 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 2811	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) ∧ 𝑥 ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑆)))) → ((reverse‘(𝑆 ++ 𝑇))‘𝑥) = (((reverse‘𝑇) ++ (reverse‘𝑆))‘𝑥))
170	91, 169	jaodan 972	. . 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 603	. 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 7035	1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (reverse‘(𝑆 ++ 𝑇)) = ((reverse‘𝑇) ++ (reverse‘𝑆)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2146 ⊆ wss 3908 Fn wfn 6538 ‘cfv 6543 (class class class)co 7423 ℂcc 11116 0cc0 11118 1c1 11119 + caddc 11121 − cmin 11459 ℕ₀cn0 12522 ℤcz 12609 ℤ_≥cuz 12880 ...cfz 13553 ..^cfzo 13701 ♯chash 14386 Word cword 14570 ++ cconcat 14627 reversecreverse 14819

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-n0 12523 df-z 12610 df-uz 12881 df-fz 13554 df-fzo 13702 df-hash 14387 df-word 14571 df-concat 14628 df-reverse 14820

This theorem is used by: gsumwrev 19467 efginvrel2 19828

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