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Theorem ccatass 14510

Description: Associative law for concatenation of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)

**Assertion**
Ref	Expression
ccatass	⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) = (𝑆 ++ (𝑇 ++ 𝑈)))

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

Step	Hyp	Ref	Expression
1		ccatcl 14495	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵) → (𝑆 ++ 𝑇) ∈ Word 𝐵)
2		ccatcl 14495	. . . . 5 ⊢ (((𝑆 ++ 𝑇) ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) ∈ Word 𝐵)
3	1, 2	stoic3 1777	. . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) ∈ Word 𝐵)
4		wrdfn 14449	. . . 4 ⊢ (((𝑆 ++ 𝑇) ++ 𝑈) ∈ Word 𝐵 → ((𝑆 ++ 𝑇) ++ 𝑈) Fn (0..^(♯‘((𝑆 ++ 𝑇) ++ 𝑈))))
5	3, 4	syl 17	. . 3 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) Fn (0..^(♯‘((𝑆 ++ 𝑇) ++ 𝑈))))
6		ccatlen 14496	. . . . . . 7 ⊢ (((𝑆 ++ 𝑇) ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘((𝑆 ++ 𝑇) ++ 𝑈)) = ((♯‘(𝑆 ++ 𝑇)) + (♯‘𝑈)))
7	1, 6	stoic3 1777	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘((𝑆 ++ 𝑇) ++ 𝑈)) = ((♯‘(𝑆 ++ 𝑇)) + (♯‘𝑈)))
8		ccatlen 14496	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑆) + (♯‘𝑇)))
9	8	3adant3 1132	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑆) + (♯‘𝑇)))
10	9	oveq1d 7371	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘(𝑆 ++ 𝑇)) + (♯‘𝑈)) = (((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))
11	7, 10	eqtrd 2769	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘((𝑆 ++ 𝑇) ++ 𝑈)) = (((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))
12	11	oveq2d 7372	. . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (0..^(♯‘((𝑆 ++ 𝑇) ++ 𝑈))) = (0..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
13	12	fneq2d 6584	. . 3 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (((𝑆 ++ 𝑇) ++ 𝑈) Fn (0..^(♯‘((𝑆 ++ 𝑇) ++ 𝑈))) ↔ ((𝑆 ++ 𝑇) ++ 𝑈) Fn (0..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))))
14	5, 13	mpbid 232	. 2 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) Fn (0..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
15		simp1 1136	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → 𝑆 ∈ Word 𝐵)
16		ccatcl 14495	. . . . . 6 ⊢ ((𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑇 ++ 𝑈) ∈ Word 𝐵)
17	16	3adant1 1130	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑇 ++ 𝑈) ∈ Word 𝐵)
18		ccatcl 14495	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ (𝑇 ++ 𝑈) ∈ Word 𝐵) → (𝑆 ++ (𝑇 ++ 𝑈)) ∈ Word 𝐵)
19	15, 17, 18	syl2anc 584	. . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑆 ++ (𝑇 ++ 𝑈)) ∈ Word 𝐵)
20		wrdfn 14449	. . . 4 ⊢ ((𝑆 ++ (𝑇 ++ 𝑈)) ∈ Word 𝐵 → (𝑆 ++ (𝑇 ++ 𝑈)) Fn (0..^(♯‘(𝑆 ++ (𝑇 ++ 𝑈)))))
21	19, 20	syl 17	. . 3 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑆 ++ (𝑇 ++ 𝑈)) Fn (0..^(♯‘(𝑆 ++ (𝑇 ++ 𝑈)))))
22		ccatlen 14496	. . . . . . . 8 ⊢ ((𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘(𝑇 ++ 𝑈)) = ((♯‘𝑇) + (♯‘𝑈)))
23	22	3adant1 1130	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘(𝑇 ++ 𝑈)) = ((♯‘𝑇) + (♯‘𝑈)))
24	23	oveq2d 7372	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈))) = ((♯‘𝑆) + ((♯‘𝑇) + (♯‘𝑈))))
25		ccatlen 14496	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ (𝑇 ++ 𝑈) ∈ Word 𝐵) → (♯‘(𝑆 ++ (𝑇 ++ 𝑈))) = ((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈))))
26	15, 17, 25	syl2anc 584	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘(𝑆 ++ (𝑇 ++ 𝑈))) = ((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈))))
27		lencl 14454	. . . . . . . . 9 ⊢ (𝑆 ∈ Word 𝐵 → (♯‘𝑆) ∈ ℕ₀)
28	27	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑆) ∈ ℕ₀)
29	28	nn0cnd 12462	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑆) ∈ ℂ)
30		lencl 14454	. . . . . . . . 9 ⊢ (𝑇 ∈ Word 𝐵 → (♯‘𝑇) ∈ ℕ₀)
31	30	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑇) ∈ ℕ₀)
32	31	nn0cnd 12462	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑇) ∈ ℂ)
33		lencl 14454	. . . . . . . . 9 ⊢ (𝑈 ∈ Word 𝐵 → (♯‘𝑈) ∈ ℕ₀)
34	33	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑈) ∈ ℕ₀)
35	34	nn0cnd 12462	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑈) ∈ ℂ)
36	29, 32, 35	addassd 11152	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)) = ((♯‘𝑆) + ((♯‘𝑇) + (♯‘𝑈))))
37	24, 26, 36	3eqtr4d 2779	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘(𝑆 ++ (𝑇 ++ 𝑈))) = (((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))
38	37	oveq2d 7372	. . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (0..^(♯‘(𝑆 ++ (𝑇 ++ 𝑈)))) = (0..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
39	38	fneq2d 6584	. . 3 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((𝑆 ++ (𝑇 ++ 𝑈)) Fn (0..^(♯‘(𝑆 ++ (𝑇 ++ 𝑈)))) ↔ (𝑆 ++ (𝑇 ++ 𝑈)) Fn (0..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))))
40	21, 39	mpbid 232	. 2 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑆 ++ (𝑇 ++ 𝑈)) Fn (0..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
41	28	nn0zd 12511	. . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑆) ∈ ℤ)
42		fzospliti 13605	. . . . 5 ⊢ ((𝑥 ∈ (0..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) ∧ (♯‘𝑆) ∈ ℤ) → (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))))
43	42	ex 412	. . . 4 ⊢ (𝑥 ∈ (0..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) → ((♯‘𝑆) ∈ ℤ → (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))))
44	41, 43	mpan9 506	. . 3 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))))
45		simp2 1137	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → 𝑇 ∈ Word 𝐵)
46		id 22	. . . . . 6 ⊢ (𝑥 ∈ (0..^(♯‘𝑆)) → 𝑥 ∈ (0..^(♯‘𝑆)))
47		ccatval1 14498	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ 𝑇)‘𝑥) = (𝑆‘𝑥))
48	15, 45, 46, 47	syl2an3an 1424	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ 𝑇)‘𝑥) = (𝑆‘𝑥))
49	1	3adant3 1132	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑆 ++ 𝑇) ∈ Word 𝐵)
50	49	adantr 480	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → (𝑆 ++ 𝑇) ∈ Word 𝐵)
51		simpl3 1194	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑈 ∈ Word 𝐵)
52	41	uzidd 12765	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑆) ∈ (ℤ_≥‘(♯‘𝑆)))
53		uzaddcl 12815	. . . . . . . . . 10 ⊢ (((♯‘𝑆) ∈ (ℤ_≥‘(♯‘𝑆)) ∧ (♯‘𝑇) ∈ ℕ₀) → ((♯‘𝑆) + (♯‘𝑇)) ∈ (ℤ_≥‘(♯‘𝑆)))
54	52, 31, 53	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆) + (♯‘𝑇)) ∈ (ℤ_≥‘(♯‘𝑆)))
55		fzoss2 13601	. . . . . . . . 9 ⊢ (((♯‘𝑆) + (♯‘𝑇)) ∈ (ℤ_≥‘(♯‘𝑆)) → (0..^(♯‘𝑆)) ⊆ (0..^((♯‘𝑆) + (♯‘𝑇))))
56	54, 55	syl 17	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (0..^(♯‘𝑆)) ⊆ (0..^((♯‘𝑆) + (♯‘𝑇))))
57	9	oveq2d 7372	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (0..^(♯‘(𝑆 ++ 𝑇))) = (0..^((♯‘𝑆) + (♯‘𝑇))))
58	56, 57	sseqtrrd 3969	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (0..^(♯‘𝑆)) ⊆ (0..^(♯‘(𝑆 ++ 𝑇))))
59	58	sselda 3931	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑥 ∈ (0..^(♯‘(𝑆 ++ 𝑇))))
60		ccatval1 14498	. . . . . 6 ⊢ (((𝑆 ++ 𝑇) ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘(𝑆 ++ 𝑇)))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ 𝑇)‘𝑥))
61	50, 51, 59, 60	syl3anc 1373	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ 𝑇)‘𝑥))
62		ccatval1 14498	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ (𝑇 ++ 𝑈) ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥) = (𝑆‘𝑥))
63	15, 17, 46, 62	syl2an3an 1424	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥) = (𝑆‘𝑥))
64	48, 61, 63	3eqtr4d 2779	. . . 4 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥))
65	31	nn0zd 12511	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑇) ∈ ℤ)
66	41, 65	zaddcld 12598	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆) + (♯‘𝑇)) ∈ ℤ)
67		fzospliti 13605	. . . . . . 7 ⊢ ((𝑥 ∈ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) ∧ ((♯‘𝑆) + (♯‘𝑇)) ∈ ℤ) → (𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ∨ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))))
68	67	ex 412	. . . . . 6 ⊢ (𝑥 ∈ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) → (((♯‘𝑆) + (♯‘𝑇)) ∈ ℤ → (𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ∨ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))))
69	66, 68	mpan9 506	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ∨ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))))
70		id 22	. . . . . . . . 9 ⊢ (𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) → 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))))
71		ccatval2 14499	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → ((𝑆 ++ 𝑇)‘𝑥) = (𝑇‘(𝑥 − (♯‘𝑆))))
72	15, 45, 70, 71	syl2an3an 1424	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → ((𝑆 ++ 𝑇)‘𝑥) = (𝑇‘(𝑥 − (♯‘𝑆))))
73		simpl2 1193	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑇 ∈ Word 𝐵)
74		simpl3 1194	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑈 ∈ Word 𝐵)
75		fzosubel3 13640	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ∧ (♯‘𝑇) ∈ ℤ) → (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇)))
76	75	ex 412	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) → ((♯‘𝑇) ∈ ℤ → (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇))))
77	65, 76	mpan9 506	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇)))
78		ccatval1 14498	. . . . . . . . 9 ⊢ ((𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵 ∧ (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ 𝑈)‘(𝑥 − (♯‘𝑆))) = (𝑇‘(𝑥 − (♯‘𝑆))))
79	73, 74, 77, 78	syl3anc 1373	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → ((𝑇 ++ 𝑈)‘(𝑥 − (♯‘𝑆))) = (𝑇‘(𝑥 − (♯‘𝑆))))
80	72, 79	eqtr4d 2772	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → ((𝑆 ++ 𝑇)‘𝑥) = ((𝑇 ++ 𝑈)‘(𝑥 − (♯‘𝑆))))
81	49	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑆 ++ 𝑇) ∈ Word 𝐵)
82		fzoss1 13600	. . . . . . . . . . . 12 ⊢ ((♯‘𝑆) ∈ (ℤ_≥‘0) → ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ⊆ (0..^((♯‘𝑆) + (♯‘𝑇))))
83		nn0uz 12787	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
84	82, 83	eleq2s 2852	. . . . . . . . . . 11 ⊢ ((♯‘𝑆) ∈ ℕ₀ → ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ⊆ (0..^((♯‘𝑆) + (♯‘𝑇))))
85	28, 84	syl 17	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ⊆ (0..^((♯‘𝑆) + (♯‘𝑇))))
86	85, 57	sseqtrrd 3969	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ⊆ (0..^(♯‘(𝑆 ++ 𝑇))))
87	86	sselda 3931	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑥 ∈ (0..^(♯‘(𝑆 ++ 𝑇))))
88	81, 74, 87, 60	syl3anc 1373	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ 𝑇)‘𝑥))
89		simpl1 1192	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑆 ∈ Word 𝐵)
90	17	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑇 ++ 𝑈) ∈ Word 𝐵)
91	66	uzidd 12765	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆) + (♯‘𝑇)) ∈ (ℤ_≥‘((♯‘𝑆) + (♯‘𝑇))))
92		uzaddcl 12815	. . . . . . . . . . . 12 ⊢ ((((♯‘𝑆) + (♯‘𝑇)) ∈ (ℤ_≥‘((♯‘𝑆) + (♯‘𝑇))) ∧ (♯‘𝑈) ∈ ℕ₀) → (((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)) ∈ (ℤ_≥‘((♯‘𝑆) + (♯‘𝑇))))
93	91, 34, 92	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)) ∈ (ℤ_≥‘((♯‘𝑆) + (♯‘𝑇))))
94		fzoss2 13601	. . . . . . . . . . 11 ⊢ ((((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)) ∈ (ℤ_≥‘((♯‘𝑆) + (♯‘𝑇))) → ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ⊆ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
95	93, 94	syl 17	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ⊆ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
96	24, 36	eqtr4d 2772	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈))) = (((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))
97	96	oveq2d 7372	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆)..^((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈)))) = ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
98	95, 97	sseqtrrd 3969	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ⊆ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈)))))
99	98	sselda 3931	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈)))))
100		ccatval2 14499	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ (𝑇 ++ 𝑈) ∈ Word 𝐵 ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈))))) → ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥) = ((𝑇 ++ 𝑈)‘(𝑥 − (♯‘𝑆))))
101	89, 90, 99, 100	syl3anc 1373	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥) = ((𝑇 ++ 𝑈)‘(𝑥 − (♯‘𝑆))))
102	80, 88, 101	3eqtr4d 2779	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥))
103	9	oveq2d 7372	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑥 − (♯‘(𝑆 ++ 𝑇))) = (𝑥 − ((♯‘𝑆) + (♯‘𝑇))))
104	103	adantr 480	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (𝑥 − (♯‘(𝑆 ++ 𝑇))) = (𝑥 − ((♯‘𝑆) + (♯‘𝑇))))
105		elfzoelz 13573	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) → 𝑥 ∈ ℤ)
106	105	zcnd 12595	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) → 𝑥 ∈ ℂ)
107	106	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → 𝑥 ∈ ℂ)
108	29	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (♯‘𝑆) ∈ ℂ)
109	32	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (♯‘𝑇) ∈ ℂ)
110	107, 108, 109	subsub4d 11521	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → ((𝑥 − (♯‘𝑆)) − (♯‘𝑇)) = (𝑥 − ((♯‘𝑆) + (♯‘𝑇))))
111	104, 110	eqtr4d 2772	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (𝑥 − (♯‘(𝑆 ++ 𝑇))) = ((𝑥 − (♯‘𝑆)) − (♯‘𝑇)))
112	111	fveq2d 6836	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (𝑈‘(𝑥 − (♯‘(𝑆 ++ 𝑇)))) = (𝑈‘((𝑥 − (♯‘𝑆)) − (♯‘𝑇))))
113		simpl2 1193	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → 𝑇 ∈ Word 𝐵)
114		simpl3 1194	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → 𝑈 ∈ Word 𝐵)
115	36	oveq2d 7372	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) = (((♯‘𝑆) + (♯‘𝑇))..^((♯‘𝑆) + ((♯‘𝑇) + (♯‘𝑈)))))
116	115	eleq2d 2820	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) ↔ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^((♯‘𝑆) + ((♯‘𝑇) + (♯‘𝑈))))))
117	116	biimpa 476	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^((♯‘𝑆) + ((♯‘𝑇) + (♯‘𝑈)))))
118	34	nn0zd 12511	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑈) ∈ ℤ)
119	65, 118	zaddcld 12598	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑇) + (♯‘𝑈)) ∈ ℤ)
120	41, 65, 119	3jca 1128	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆) ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ ∧ ((♯‘𝑇) + (♯‘𝑈)) ∈ ℤ))
121	120	adantr 480	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → ((♯‘𝑆) ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ ∧ ((♯‘𝑇) + (♯‘𝑈)) ∈ ℤ))
122		fzosubel2 13639	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^((♯‘𝑆) + ((♯‘𝑇) + (♯‘𝑈)))) ∧ ((♯‘𝑆) ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ ∧ ((♯‘𝑇) + (♯‘𝑈)) ∈ ℤ)) → (𝑥 − (♯‘𝑆)) ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))))
123	117, 121, 122	syl2anc 584	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (𝑥 − (♯‘𝑆)) ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))))
124		ccatval2 14499	. . . . . . . . 9 ⊢ ((𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵 ∧ (𝑥 − (♯‘𝑆)) ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈)))) → ((𝑇 ++ 𝑈)‘(𝑥 − (♯‘𝑆))) = (𝑈‘((𝑥 − (♯‘𝑆)) − (♯‘𝑇))))
125	113, 114, 123, 124	syl3anc 1373	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → ((𝑇 ++ 𝑈)‘(𝑥 − (♯‘𝑆))) = (𝑈‘((𝑥 − (♯‘𝑆)) − (♯‘𝑇))))
126	112, 125	eqtr4d 2772	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (𝑈‘(𝑥 − (♯‘(𝑆 ++ 𝑇)))) = ((𝑇 ++ 𝑈)‘(𝑥 − (♯‘𝑆))))
127	49	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (𝑆 ++ 𝑇) ∈ Word 𝐵)
128	9, 10	oveq12d 7374	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘(𝑆 ++ 𝑇))..^((♯‘(𝑆 ++ 𝑇)) + (♯‘𝑈))) = (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
129	128	eleq2d 2820	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑥 ∈ ((♯‘(𝑆 ++ 𝑇))..^((♯‘(𝑆 ++ 𝑇)) + (♯‘𝑈))) ↔ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))))
130	129	biimpar 477	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → 𝑥 ∈ ((♯‘(𝑆 ++ 𝑇))..^((♯‘(𝑆 ++ 𝑇)) + (♯‘𝑈))))
131		ccatval2 14499	. . . . . . . 8 ⊢ (((𝑆 ++ 𝑇) ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵 ∧ 𝑥 ∈ ((♯‘(𝑆 ++ 𝑇))..^((♯‘(𝑆 ++ 𝑇)) + (♯‘𝑈)))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = (𝑈‘(𝑥 − (♯‘(𝑆 ++ 𝑇)))))
132	127, 114, 130, 131	syl3anc 1373	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = (𝑈‘(𝑥 − (♯‘(𝑆 ++ 𝑇)))))
133		simpl1 1192	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → 𝑆 ∈ Word 𝐵)
134	17	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (𝑇 ++ 𝑈) ∈ Word 𝐵)
135		fzoss1 13600	. . . . . . . . . . 11 ⊢ (((♯‘𝑆) + (♯‘𝑇)) ∈ (ℤ_≥‘(♯‘𝑆)) → (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) ⊆ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
136	54, 135	syl 17	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) ⊆ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
137	136, 97	sseqtrrd 3969	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) ⊆ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈)))))
138	137	sselda 3931	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈)))))
139	133, 134, 138, 100	syl3anc 1373	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥) = ((𝑇 ++ 𝑈)‘(𝑥 − (♯‘𝑆))))
140	126, 132, 139	3eqtr4d 2779	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥))
141	102, 140	jaodan 959	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ (𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ∨ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥))
142	69, 141	syldan 591	. . . 4 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥))
143	64, 142	jaodan 959	. . 3 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥))
144	44, 143	syldan 591	. 2 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥))
145	14, 40, 144	eqfnfvd 6977	1 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) = (𝑆 ++ (𝑇 ++ 𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ⊆ wss 3899 Fn wfn 6485 ‘cfv 6490 (class class class)co 7356 ℂcc 11022 0cc0 11024 + caddc 11027 − cmin 11362 ℕ₀cn0 12399 ℤcz 12486 ℤ_≥cuz 12749 ..^cfzo 13568 ♯chash 14251 Word cword 14434 ++ cconcat 14491

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-card 9849 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-n0 12400 df-z 12487 df-uz 12750 df-fz 13422 df-fzo 13569 df-hash 14252 df-word 14435 df-concat 14492

This theorem is referenced by: ccatw2s1ass 14553 cats1cat 14782 cats2cat 14783 frmdmnd 18782 efginvrel2 19654 efgredleme 19670 efgredlemc 19672 efgcpbllemb 19682 numclwwlk1lem2foalem 30375 numclwwlk1lem2fo 30382 signstfvc 34680

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