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Theorem ccatass 11300

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 11285	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵) → (𝑆 ++ 𝑇) ∈ Word 𝐵)
2		ccatcl 11285	. . . . 5 ⊢ (((𝑆 ++ 𝑇) ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) ∈ Word 𝐵)
3	1, 2	stoic3 1476	. . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) ∈ Word 𝐵)
4		wrdfn 11243	. . . 4 ⊢ (((𝑆 ++ 𝑇) ++ 𝑈) ∈ Word 𝐵 → ((𝑆 ++ 𝑇) ++ 𝑈) Fn (0..^(♯‘((𝑆 ++ 𝑇) ++ 𝑈))))
5	3, 4	syl 14	. . 3 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) Fn (0..^(♯‘((𝑆 ++ 𝑇) ++ 𝑈))))
6		ccatlen 11287	. . . . . . 7 ⊢ (((𝑆 ++ 𝑇) ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘((𝑆 ++ 𝑇) ++ 𝑈)) = ((♯‘(𝑆 ++ 𝑇)) + (♯‘𝑈)))
7	1, 6	stoic3 1476	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘((𝑆 ++ 𝑇) ++ 𝑈)) = ((♯‘(𝑆 ++ 𝑇)) + (♯‘𝑈)))
8		ccatlen 11287	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑆) + (♯‘𝑇)))
9	8	3adant3 1044	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘(𝑆 ++ 𝑇)) = ((♯‘𝑆) + (♯‘𝑇)))
10	9	oveq1d 6067	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘(𝑆 ++ 𝑇)) + (♯‘𝑈)) = (((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))
11	7, 10	eqtrd 2267	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘((𝑆 ++ 𝑇) ++ 𝑈)) = (((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))
12	11	oveq2d 6068	. . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (0..^(♯‘((𝑆 ++ 𝑇) ++ 𝑈))) = (0..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
13	12	fneq2d 5449	. . 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 147	. 2 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) Fn (0..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
15		simp1 1024	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → 𝑆 ∈ Word 𝐵)
16		ccatcl 11285	. . . . . 6 ⊢ ((𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑇 ++ 𝑈) ∈ Word 𝐵)
17	16	3adant1 1042	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑇 ++ 𝑈) ∈ Word 𝐵)
18		ccatcl 11285	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ (𝑇 ++ 𝑈) ∈ Word 𝐵) → (𝑆 ++ (𝑇 ++ 𝑈)) ∈ Word 𝐵)
19	15, 17, 18	syl2anc 411	. . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑆 ++ (𝑇 ++ 𝑈)) ∈ Word 𝐵)
20		wrdfn 11243	. . . 4 ⊢ ((𝑆 ++ (𝑇 ++ 𝑈)) ∈ Word 𝐵 → (𝑆 ++ (𝑇 ++ 𝑈)) Fn (0..^(♯‘(𝑆 ++ (𝑇 ++ 𝑈)))))
21	19, 20	syl 14	. . 3 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑆 ++ (𝑇 ++ 𝑈)) Fn (0..^(♯‘(𝑆 ++ (𝑇 ++ 𝑈)))))
22		ccatlen 11287	. . . . . . . 8 ⊢ ((𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘(𝑇 ++ 𝑈)) = ((♯‘𝑇) + (♯‘𝑈)))
23	22	3adant1 1042	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘(𝑇 ++ 𝑈)) = ((♯‘𝑇) + (♯‘𝑈)))
24	23	oveq2d 6068	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈))) = ((♯‘𝑆) + ((♯‘𝑇) + (♯‘𝑈))))
25		ccatlen 11287	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ (𝑇 ++ 𝑈) ∈ Word 𝐵) → (♯‘(𝑆 ++ (𝑇 ++ 𝑈))) = ((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈))))
26	15, 17, 25	syl2anc 411	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘(𝑆 ++ (𝑇 ++ 𝑈))) = ((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈))))
27		lencl 11232	. . . . . . . . 9 ⊢ (𝑆 ∈ Word 𝐵 → (♯‘𝑆) ∈ ℕ₀)
28	27	3ad2ant1 1045	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑆) ∈ ℕ₀)
29	28	nn0cnd 9557	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑆) ∈ ℂ)
30		lencl 11232	. . . . . . . . 9 ⊢ (𝑇 ∈ Word 𝐵 → (♯‘𝑇) ∈ ℕ₀)
31	30	3ad2ant2 1046	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑇) ∈ ℕ₀)
32	31	nn0cnd 9557	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑇) ∈ ℂ)
33		lencl 11232	. . . . . . . . 9 ⊢ (𝑈 ∈ Word 𝐵 → (♯‘𝑈) ∈ ℕ₀)
34	33	3ad2ant3 1047	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑈) ∈ ℕ₀)
35	34	nn0cnd 9557	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑈) ∈ ℂ)
36	29, 32, 35	addassd 8298	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)) = ((♯‘𝑆) + ((♯‘𝑇) + (♯‘𝑈))))
37	24, 26, 36	3eqtr4d 2277	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘(𝑆 ++ (𝑇 ++ 𝑈))) = (((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))
38	37	oveq2d 6068	. . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (0..^(♯‘(𝑆 ++ (𝑇 ++ 𝑈)))) = (0..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
39	38	fneq2d 5449	. . 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 147	. 2 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑆 ++ (𝑇 ++ 𝑈)) Fn (0..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
41	28	nn0zd 9701	. . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑆) ∈ ℤ)
42		fzospliti 10516	. . . . 5 ⊢ ((𝑥 ∈ (0..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) ∧ (♯‘𝑆) ∈ ℤ) → (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))))
43	42	ex 115	. . . 4 ⊢ (𝑥 ∈ (0..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) → ((♯‘𝑆) ∈ ℤ → (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))))
44	41, 43	mpan9 281	. . 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 1025	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → 𝑇 ∈ Word 𝐵)
46		id 19	. . . . . 6 ⊢ (𝑥 ∈ (0..^(♯‘𝑆)) → 𝑥 ∈ (0..^(♯‘𝑆)))
47		ccatval1 11289	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ 𝑇)‘𝑥) = (𝑆‘𝑥))
48	15, 45, 46, 47	syl2an3an 1335	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ 𝑇)‘𝑥) = (𝑆‘𝑥))
49	1	3adant3 1044	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑆 ++ 𝑇) ∈ Word 𝐵)
50	49	adantr 276	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → (𝑆 ++ 𝑇) ∈ Word 𝐵)
51		simpl3 1029	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑈 ∈ Word 𝐵)
52	41	uzidd 9872	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑆) ∈ (ℤ_≥‘(♯‘𝑆)))
53		uzaddcl 9921	. . . . . . . . . 10 ⊢ (((♯‘𝑆) ∈ (ℤ_≥‘(♯‘𝑆)) ∧ (♯‘𝑇) ∈ ℕ₀) → ((♯‘𝑆) + (♯‘𝑇)) ∈ (ℤ_≥‘(♯‘𝑆)))
54	52, 31, 53	syl2anc 411	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆) + (♯‘𝑇)) ∈ (ℤ_≥‘(♯‘𝑆)))
55		fzoss2 10512	. . . . . . . . 9 ⊢ (((♯‘𝑆) + (♯‘𝑇)) ∈ (ℤ_≥‘(♯‘𝑆)) → (0..^(♯‘𝑆)) ⊆ (0..^((♯‘𝑆) + (♯‘𝑇))))
56	54, 55	syl 14	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (0..^(♯‘𝑆)) ⊆ (0..^((♯‘𝑆) + (♯‘𝑇))))
57	9	oveq2d 6068	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (0..^(♯‘(𝑆 ++ 𝑇))) = (0..^((♯‘𝑆) + (♯‘𝑇))))
58	56, 57	sseqtrrd 3279	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (0..^(♯‘𝑆)) ⊆ (0..^(♯‘(𝑆 ++ 𝑇))))
59	58	sselda 3240	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑥 ∈ (0..^(♯‘(𝑆 ++ 𝑇))))
60		ccatval1 11289	. . . . . 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 1274	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ 𝑇)‘𝑥))
62		ccatval1 11289	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ (𝑇 ++ 𝑈) ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥) = (𝑆‘𝑥))
63	15, 17, 46, 62	syl2an3an 1335	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥) = (𝑆‘𝑥))
64	48, 61, 63	3eqtr4d 2277	. . . 4 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥))
65	31	nn0zd 9701	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑇) ∈ ℤ)
66	41, 65	zaddcld 9707	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆) + (♯‘𝑇)) ∈ ℤ)
67		fzospliti 10516	. . . . . . 7 ⊢ ((𝑥 ∈ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) ∧ ((♯‘𝑆) + (♯‘𝑇)) ∈ ℤ) → (𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ∨ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))))
68	67	ex 115	. . . . . 6 ⊢ (𝑥 ∈ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) → (((♯‘𝑆) + (♯‘𝑇)) ∈ ℤ → (𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ∨ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))))
69	66, 68	mpan9 281	. . . . 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 19	. . . . . . . . 9 ⊢ (𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) → 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))))
71		ccatval2 11290	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → ((𝑆 ++ 𝑇)‘𝑥) = (𝑇‘(𝑥 − (♯‘𝑆))))
72	15, 45, 70, 71	syl2an3an 1335	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → ((𝑆 ++ 𝑇)‘𝑥) = (𝑇‘(𝑥 − (♯‘𝑆))))
73		simpl2 1028	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑇 ∈ Word 𝐵)
74		simpl3 1029	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑈 ∈ Word 𝐵)
75		fzosubel3 10545	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ∧ (♯‘𝑇) ∈ ℤ) → (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇)))
76	75	ex 115	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) → ((♯‘𝑇) ∈ ℤ → (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇))))
77	65, 76	mpan9 281	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇)))
78		ccatval1 11289	. . . . . . . . 9 ⊢ ((𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵 ∧ (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇))) → ((𝑇 ++ 𝑈)‘(𝑥 − (♯‘𝑆))) = (𝑇‘(𝑥 − (♯‘𝑆))))
79	73, 74, 77, 78	syl3anc 1274	. . . . . . . 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 2270	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → ((𝑆 ++ 𝑇)‘𝑥) = ((𝑇 ++ 𝑈)‘(𝑥 − (♯‘𝑆))))
81	49	adantr 276	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑆 ++ 𝑇) ∈ Word 𝐵)
82		fzoss1 10511	. . . . . . . . . . . 12 ⊢ ((♯‘𝑆) ∈ (ℤ_≥‘0) → ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ⊆ (0..^((♯‘𝑆) + (♯‘𝑇))))
83		nn0uz 9892	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
84	82, 83	eleq2s 2329	. . . . . . . . . . 11 ⊢ ((♯‘𝑆) ∈ ℕ₀ → ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ⊆ (0..^((♯‘𝑆) + (♯‘𝑇))))
85	28, 84	syl 14	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ⊆ (0..^((♯‘𝑆) + (♯‘𝑇))))
86	85, 57	sseqtrrd 3279	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ⊆ (0..^(♯‘(𝑆 ++ 𝑇))))
87	86	sselda 3240	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑥 ∈ (0..^(♯‘(𝑆 ++ 𝑇))))
88	81, 74, 87, 60	syl3anc 1274	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ 𝑇)‘𝑥))
89		simpl1 1027	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑆 ∈ Word 𝐵)
90	17	adantr 276	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑇 ++ 𝑈) ∈ Word 𝐵)
91	66	uzidd 9872	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆) + (♯‘𝑇)) ∈ (ℤ_≥‘((♯‘𝑆) + (♯‘𝑇))))
92		uzaddcl 9921	. . . . . . . . . . . 12 ⊢ ((((♯‘𝑆) + (♯‘𝑇)) ∈ (ℤ_≥‘((♯‘𝑆) + (♯‘𝑇))) ∧ (♯‘𝑈) ∈ ℕ₀) → (((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)) ∈ (ℤ_≥‘((♯‘𝑆) + (♯‘𝑇))))
93	91, 34, 92	syl2anc 411	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)) ∈ (ℤ_≥‘((♯‘𝑆) + (♯‘𝑇))))
94		fzoss2 10512	. . . . . . . . . . 11 ⊢ ((((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)) ∈ (ℤ_≥‘((♯‘𝑆) + (♯‘𝑇))) → ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ⊆ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
95	93, 94	syl 14	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ⊆ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
96	24, 36	eqtr4d 2270	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈))) = (((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))
97	96	oveq2d 6068	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆)..^((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈)))) = ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
98	95, 97	sseqtrrd 3279	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ⊆ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈)))))
99	98	sselda 3240	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈)))))
100		ccatval2 11290	. . . . . . . 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 1274	. . . . . . 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 2277	. . . . . 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 6068	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑥 − (♯‘(𝑆 ++ 𝑇))) = (𝑥 − ((♯‘𝑆) + (♯‘𝑇))))
104	103	adantr 276	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (𝑥 − (♯‘(𝑆 ++ 𝑇))) = (𝑥 − ((♯‘𝑆) + (♯‘𝑇))))
105		elfzoelz 10485	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) → 𝑥 ∈ ℤ)
106	105	zcnd 9704	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) → 𝑥 ∈ ℂ)
107	106	adantl 277	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → 𝑥 ∈ ℂ)
108	29	adantr 276	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (♯‘𝑆) ∈ ℂ)
109	32	adantr 276	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (♯‘𝑇) ∈ ℂ)
110	107, 108, 109	subsub4d 8617	. . . . . . . . . 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 2270	. . . . . . . . 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 5676	. . . . . . . 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 1028	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → 𝑇 ∈ Word 𝐵)
114		simpl3 1029	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → 𝑈 ∈ Word 𝐵)
115	36	oveq2d 6068	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) = (((♯‘𝑆) + (♯‘𝑇))..^((♯‘𝑆) + ((♯‘𝑇) + (♯‘𝑈)))))
116	115	eleq2d 2304	. . . . . . . . . . 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 296	. . . . . . . . . 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 9701	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (♯‘𝑈) ∈ ℤ)
119	65, 118	zaddcld 9707	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑇) + (♯‘𝑈)) ∈ ℤ)
120	41, 65, 119	3jca 1204	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘𝑆) ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ ∧ ((♯‘𝑇) + (♯‘𝑈)) ∈ ℤ))
121	120	adantr 276	. . . . . . . . . 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 10544	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^((♯‘𝑆) + ((♯‘𝑇) + (♯‘𝑈)))) ∧ ((♯‘𝑆) ∈ ℤ ∧ (♯‘𝑇) ∈ ℤ ∧ ((♯‘𝑇) + (♯‘𝑈)) ∈ ℤ)) → (𝑥 − (♯‘𝑆)) ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))))
123	117, 121, 122	syl2anc 411	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (𝑥 − (♯‘𝑆)) ∈ ((♯‘𝑇)..^((♯‘𝑇) + (♯‘𝑈))))
124		ccatval2 11290	. . . . . . . . 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 1274	. . . . . . . 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 2270	. . . . . . 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 276	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (𝑆 ++ 𝑇) ∈ Word 𝐵)
128	9, 10	oveq12d 6070	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((♯‘(𝑆 ++ 𝑇))..^((♯‘(𝑆 ++ 𝑇)) + (♯‘𝑈))) = (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
129	128	eleq2d 2304	. . . . . . . . 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 297	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → 𝑥 ∈ ((♯‘(𝑆 ++ 𝑇))..^((♯‘(𝑆 ++ 𝑇)) + (♯‘𝑈))))
131		ccatval2 11290	. . . . . . . 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 1274	. . . . . . 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 1027	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → 𝑆 ∈ Word 𝐵)
134	17	adantr 276	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈)))) → (𝑇 ++ 𝑈) ∈ Word 𝐵)
135		fzoss1 10511	. . . . . . . . . . 11 ⊢ (((♯‘𝑆) + (♯‘𝑇)) ∈ (ℤ_≥‘(♯‘𝑆)) → (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) ⊆ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
136	54, 135	syl 14	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) ⊆ ((♯‘𝑆)..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))))
137	136, 97	sseqtrrd 3279	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (((♯‘𝑆) + (♯‘𝑇))..^(((♯‘𝑆) + (♯‘𝑇)) + (♯‘𝑈))) ⊆ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘(𝑇 ++ 𝑈)))))
138	137	sselda 3240	. . . . . . . 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 1274	. . . . . . 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 2277	. . . . . 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 805	. . . . 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 282	. . . 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 805	. . 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 282	. 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 5780	1 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) = (𝑆 ++ (𝑇 ++ 𝑈)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∨ wo 716 ∧ w3a 1005 = wceq 1398 ∈ wcel 2205 ⊆ wss 3213 Fn wfn 5349 ‘cfv 5354 (class class class)co 6052 ℂcc 8127 0cc0 8129 + caddc 8132 − cmin 8446 ℕ₀cn0 9498 ℤcz 9579 ℤ_≥cuz 9856 ..^cfzo 10480 ♯chash 11142 Word cword 11228 ++ cconcat 11282

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4227 ax-sep 4230 ax-nul 4238 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-iinf 4712 ax-cnex 8220 ax-resscn 8221 ax-1cn 8222 ax-1re 8223 ax-icn 8224 ax-addcl 8225 ax-addrcl 8226 ax-mulcl 8227 ax-addcom 8229 ax-addass 8231 ax-distr 8233 ax-i2m1 8234 ax-0lt1 8235 ax-0id 8237 ax-rnegex 8238 ax-cnre 8240 ax-pre-ltirr 8241 ax-pre-ltwlin 8242 ax-pre-lttrn 8243 ax-pre-apti 8244 ax-pre-ltadd 8245

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3045 df-csb 3141 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-nul 3511 df-if 3623 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-int 3952 df-iun 3995 df-br 4112 df-opab 4174 df-mpt 4175 df-tr 4211 df-id 4416 df-iord 4489 df-on 4491 df-ilim 4492 df-suc 4494 df-iom 4715 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-rn 4762 df-res 4763 df-ima 4764 df-iota 5314 df-fun 5356 df-fn 5357 df-f 5358 df-f1 5359 df-fo 5360 df-f1o 5361 df-fv 5362 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-1st 6336 df-2nd 6337 df-recs 6538 df-frec 6624 df-1o 6649 df-er 6769 df-en 6978 df-dom 6979 df-fin 6980 df-pnf 8312 df-mnf 8313 df-xr 8314 df-ltxr 8315 df-le 8316 df-sub 8448 df-neg 8449 df-inn 9240 df-n0 9499 df-z 9580 df-uz 9857 df-fz 10346 df-fzo 10481 df-ihash 11143 df-word 11229 df-concat 11283

This theorem is referenced by: ccat2s1fvwd 11339 cats1catd 11464 cats2catd 11465

W3C validator