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Theorem ccatass 13224

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 13212	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵) → (𝑆 ++ 𝑇) ∈ Word 𝐵)
2		ccatcl 13212	. . . . 5 ⊢ (((𝑆 ++ 𝑇) ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) ∈ Word 𝐵)
3	1, 2	stoic3 1692	. . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) ∈ Word 𝐵)
4		wrdf 13165	. . . 4 ⊢ (((𝑆 ++ 𝑇) ++ 𝑈) ∈ Word 𝐵 → ((𝑆 ++ 𝑇) ++ 𝑈):(0..^(#‘((𝑆 ++ 𝑇) ++ 𝑈)))⟶𝐵)
5		ffn 5958	. . . 4 ⊢ (((𝑆 ++ 𝑇) ++ 𝑈):(0..^(#‘((𝑆 ++ 𝑇) ++ 𝑈)))⟶𝐵 → ((𝑆 ++ 𝑇) ++ 𝑈) Fn (0..^(#‘((𝑆 ++ 𝑇) ++ 𝑈))))
6	3, 4, 5	3syl 18	. . 3 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) Fn (0..^(#‘((𝑆 ++ 𝑇) ++ 𝑈))))
7		ccatlen 13213	. . . . . . 7 ⊢ (((𝑆 ++ 𝑇) ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘((𝑆 ++ 𝑇) ++ 𝑈)) = ((#‘(𝑆 ++ 𝑇)) + (#‘𝑈)))
8	1, 7	stoic3 1692	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘((𝑆 ++ 𝑇) ++ 𝑈)) = ((#‘(𝑆 ++ 𝑇)) + (#‘𝑈)))
9		ccatlen 13213	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵) → (#‘(𝑆 ++ 𝑇)) = ((#‘𝑆) + (#‘𝑇)))
10	9	3adant3 1074	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘(𝑆 ++ 𝑇)) = ((#‘𝑆) + (#‘𝑇)))
11	10	oveq1d 6564	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((#‘(𝑆 ++ 𝑇)) + (#‘𝑈)) = (((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))
12	8, 11	eqtrd 2644	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘((𝑆 ++ 𝑇) ++ 𝑈)) = (((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))
13	12	oveq2d 6565	. . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (0..^(#‘((𝑆 ++ 𝑇) ++ 𝑈))) = (0..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))))
14	13	fneq2d 5896	. . 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)))))
15	6, 14	mpbid 221	. 2 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((𝑆 ++ 𝑇) ++ 𝑈) Fn (0..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))))
16		simp1 1054	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → 𝑆 ∈ Word 𝐵)
17		ccatcl 13212	. . . . . 6 ⊢ ((𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑇 ++ 𝑈) ∈ Word 𝐵)
18	17	3adant1 1072	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑇 ++ 𝑈) ∈ Word 𝐵)
19		ccatcl 13212	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ (𝑇 ++ 𝑈) ∈ Word 𝐵) → (𝑆 ++ (𝑇 ++ 𝑈)) ∈ Word 𝐵)
20	16, 18, 19	syl2anc 691	. . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑆 ++ (𝑇 ++ 𝑈)) ∈ Word 𝐵)
21		wrdf 13165	. . . 4 ⊢ ((𝑆 ++ (𝑇 ++ 𝑈)) ∈ Word 𝐵 → (𝑆 ++ (𝑇 ++ 𝑈)):(0..^(#‘(𝑆 ++ (𝑇 ++ 𝑈))))⟶𝐵)
22		ffn 5958	. . . 4 ⊢ ((𝑆 ++ (𝑇 ++ 𝑈)):(0..^(#‘(𝑆 ++ (𝑇 ++ 𝑈))))⟶𝐵 → (𝑆 ++ (𝑇 ++ 𝑈)) Fn (0..^(#‘(𝑆 ++ (𝑇 ++ 𝑈)))))
23	20, 21, 22	3syl 18	. . 3 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑆 ++ (𝑇 ++ 𝑈)) Fn (0..^(#‘(𝑆 ++ (𝑇 ++ 𝑈)))))
24		ccatlen 13213	. . . . . . . 8 ⊢ ((𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘(𝑇 ++ 𝑈)) = ((#‘𝑇) + (#‘𝑈)))
25	24	3adant1 1072	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘(𝑇 ++ 𝑈)) = ((#‘𝑇) + (#‘𝑈)))
26	25	oveq2d 6565	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((#‘𝑆) + (#‘(𝑇 ++ 𝑈))) = ((#‘𝑆) + ((#‘𝑇) + (#‘𝑈))))
27		ccatlen 13213	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ (𝑇 ++ 𝑈) ∈ Word 𝐵) → (#‘(𝑆 ++ (𝑇 ++ 𝑈))) = ((#‘𝑆) + (#‘(𝑇 ++ 𝑈))))
28	16, 18, 27	syl2anc 691	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘(𝑆 ++ (𝑇 ++ 𝑈))) = ((#‘𝑆) + (#‘(𝑇 ++ 𝑈))))
29		lencl 13179	. . . . . . . . 9 ⊢ (𝑆 ∈ Word 𝐵 → (#‘𝑆) ∈ ℕ₀)
30	29	3ad2ant1 1075	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘𝑆) ∈ ℕ₀)
31	30	nn0cnd 11230	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘𝑆) ∈ ℂ)
32		lencl 13179	. . . . . . . . 9 ⊢ (𝑇 ∈ Word 𝐵 → (#‘𝑇) ∈ ℕ₀)
33	32	3ad2ant2 1076	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘𝑇) ∈ ℕ₀)
34	33	nn0cnd 11230	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘𝑇) ∈ ℂ)
35		lencl 13179	. . . . . . . . 9 ⊢ (𝑈 ∈ Word 𝐵 → (#‘𝑈) ∈ ℕ₀)
36	35	3ad2ant3 1077	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘𝑈) ∈ ℕ₀)
37	36	nn0cnd 11230	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘𝑈) ∈ ℂ)
38	31, 34, 37	addassd 9941	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)) = ((#‘𝑆) + ((#‘𝑇) + (#‘𝑈))))
39	26, 28, 38	3eqtr4d 2654	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘(𝑆 ++ (𝑇 ++ 𝑈))) = (((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))
40	39	oveq2d 6565	. . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (0..^(#‘(𝑆 ++ (𝑇 ++ 𝑈)))) = (0..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))))
41	40	fneq2d 5896	. . 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)))))
42	23, 41	mpbid 221	. 2 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑆 ++ (𝑇 ++ 𝑈)) Fn (0..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))))
43	30	nn0zd 11356	. . . 4 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘𝑆) ∈ ℤ)
44		fzospliti 12369	. . . . 5 ⊢ ((𝑥 ∈ (0..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))) ∧ (#‘𝑆) ∈ ℤ) → (𝑥 ∈ (0..^(#‘𝑆)) ∨ 𝑥 ∈ ((#‘𝑆)..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))))
45	44	ancoms 468	. . . 4 ⊢ (((#‘𝑆) ∈ ℤ ∧ 𝑥 ∈ (0..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (𝑥 ∈ (0..^(#‘𝑆)) ∨ 𝑥 ∈ ((#‘𝑆)..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))))
46	43, 45	sylan 487	. . 3 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (𝑥 ∈ (0..^(#‘𝑆)) ∨ 𝑥 ∈ ((#‘𝑆)..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))))
47		simpl1 1057	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(#‘𝑆))) → 𝑆 ∈ Word 𝐵)
48		simpl2 1058	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(#‘𝑆))) → 𝑇 ∈ Word 𝐵)
49		simpr 476	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(#‘𝑆))) → 𝑥 ∈ (0..^(#‘𝑆)))
50		ccatval1 13214	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(#‘𝑆))) → ((𝑆 ++ 𝑇)‘𝑥) = (𝑆‘𝑥))
51	47, 48, 49, 50	syl3anc 1318	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(#‘𝑆))) → ((𝑆 ++ 𝑇)‘𝑥) = (𝑆‘𝑥))
52	1	3adant3 1074	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑆 ++ 𝑇) ∈ Word 𝐵)
53	52	adantr 480	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(#‘𝑆))) → (𝑆 ++ 𝑇) ∈ Word 𝐵)
54		simpl3 1059	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(#‘𝑆))) → 𝑈 ∈ Word 𝐵)
55		uzid 11578	. . . . . . . . . . 11 ⊢ ((#‘𝑆) ∈ ℤ → (#‘𝑆) ∈ (ℤ_≥‘(#‘𝑆)))
56	43, 55	syl 17	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘𝑆) ∈ (ℤ_≥‘(#‘𝑆)))
57		uzaddcl 11620	. . . . . . . . . 10 ⊢ (((#‘𝑆) ∈ (ℤ_≥‘(#‘𝑆)) ∧ (#‘𝑇) ∈ ℕ₀) → ((#‘𝑆) + (#‘𝑇)) ∈ (ℤ_≥‘(#‘𝑆)))
58	56, 33, 57	syl2anc 691	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((#‘𝑆) + (#‘𝑇)) ∈ (ℤ_≥‘(#‘𝑆)))
59		fzoss2 12365	. . . . . . . . 9 ⊢ (((#‘𝑆) + (#‘𝑇)) ∈ (ℤ_≥‘(#‘𝑆)) → (0..^(#‘𝑆)) ⊆ (0..^((#‘𝑆) + (#‘𝑇))))
60	58, 59	syl 17	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (0..^(#‘𝑆)) ⊆ (0..^((#‘𝑆) + (#‘𝑇))))
61	10	oveq2d 6565	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (0..^(#‘(𝑆 ++ 𝑇))) = (0..^((#‘𝑆) + (#‘𝑇))))
62	60, 61	sseqtr4d 3605	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (0..^(#‘𝑆)) ⊆ (0..^(#‘(𝑆 ++ 𝑇))))
63	62	sselda 3568	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(#‘𝑆))) → 𝑥 ∈ (0..^(#‘(𝑆 ++ 𝑇))))
64		ccatval1 13214	. . . . . 6 ⊢ (((𝑆 ++ 𝑇) ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(#‘(𝑆 ++ 𝑇)))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ 𝑇)‘𝑥))
65	53, 54, 63, 64	syl3anc 1318	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(#‘𝑆))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ 𝑇)‘𝑥))
66	18	adantr 480	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(#‘𝑆))) → (𝑇 ++ 𝑈) ∈ Word 𝐵)
67		ccatval1 13214	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ (𝑇 ++ 𝑈) ∈ Word 𝐵 ∧ 𝑥 ∈ (0..^(#‘𝑆))) → ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥) = (𝑆‘𝑥))
68	47, 66, 49, 67	syl3anc 1318	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(#‘𝑆))) → ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥) = (𝑆‘𝑥))
69	51, 65, 68	3eqtr4d 2654	. . . 4 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(#‘𝑆))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥))
70	33	nn0zd 11356	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘𝑇) ∈ ℤ)
71	43, 70	zaddcld 11362	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((#‘𝑆) + (#‘𝑇)) ∈ ℤ)
72		fzospliti 12369	. . . . . . 7 ⊢ ((𝑥 ∈ ((#‘𝑆)..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))) ∧ ((#‘𝑆) + (#‘𝑇)) ∈ ℤ) → (𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇))) ∨ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))))
73	72	ancoms 468	. . . . . 6 ⊢ ((((#‘𝑆) + (#‘𝑇)) ∈ ℤ ∧ 𝑥 ∈ ((#‘𝑆)..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇))) ∨ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))))
74	71, 73	sylan 487	. . . . 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)))))
75		simpl1 1057	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → 𝑆 ∈ Word 𝐵)
76		simpl2 1058	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → 𝑇 ∈ Word 𝐵)
77		simpr 476	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇))))
78		ccatval2 13215	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → ((𝑆 ++ 𝑇)‘𝑥) = (𝑇‘(𝑥 − (#‘𝑆))))
79	75, 76, 77, 78	syl3anc 1318	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → ((𝑆 ++ 𝑇)‘𝑥) = (𝑇‘(𝑥 − (#‘𝑆))))
80		simpl3 1059	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → 𝑈 ∈ Word 𝐵)
81		fzosubel3 12396	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇))) ∧ (#‘𝑇) ∈ ℤ) → (𝑥 − (#‘𝑆)) ∈ (0..^(#‘𝑇)))
82	81	ancoms 468	. . . . . . . . . 10 ⊢ (((#‘𝑇) ∈ ℤ ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → (𝑥 − (#‘𝑆)) ∈ (0..^(#‘𝑇)))
83	70, 82	sylan 487	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → (𝑥 − (#‘𝑆)) ∈ (0..^(#‘𝑇)))
84		ccatval1 13214	. . . . . . . . 9 ⊢ ((𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵 ∧ (𝑥 − (#‘𝑆)) ∈ (0..^(#‘𝑇))) → ((𝑇 ++ 𝑈)‘(𝑥 − (#‘𝑆))) = (𝑇‘(𝑥 − (#‘𝑆))))
85	76, 80, 83, 84	syl3anc 1318	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → ((𝑇 ++ 𝑈)‘(𝑥 − (#‘𝑆))) = (𝑇‘(𝑥 − (#‘𝑆))))
86	79, 85	eqtr4d 2647	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → ((𝑆 ++ 𝑇)‘𝑥) = ((𝑇 ++ 𝑈)‘(𝑥 − (#‘𝑆))))
87	52	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → (𝑆 ++ 𝑇) ∈ Word 𝐵)
88		fzoss1 12364	. . . . . . . . . . . 12 ⊢ ((#‘𝑆) ∈ (ℤ_≥‘0) → ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇))) ⊆ (0..^((#‘𝑆) + (#‘𝑇))))
89		nn0uz 11598	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
90	88, 89	eleq2s 2706	. . . . . . . . . . 11 ⊢ ((#‘𝑆) ∈ ℕ₀ → ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇))) ⊆ (0..^((#‘𝑆) + (#‘𝑇))))
91	30, 90	syl 17	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇))) ⊆ (0..^((#‘𝑆) + (#‘𝑇))))
92	91, 61	sseqtr4d 3605	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇))) ⊆ (0..^(#‘(𝑆 ++ 𝑇))))
93	92	sselda 3568	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → 𝑥 ∈ (0..^(#‘(𝑆 ++ 𝑇))))
94	87, 80, 93, 64	syl3anc 1318	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ 𝑇)‘𝑥))
95	18	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → (𝑇 ++ 𝑈) ∈ Word 𝐵)
96		uzid 11578	. . . . . . . . . . . . 13 ⊢ (((#‘𝑆) + (#‘𝑇)) ∈ ℤ → ((#‘𝑆) + (#‘𝑇)) ∈ (ℤ_≥‘((#‘𝑆) + (#‘𝑇))))
97	71, 96	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((#‘𝑆) + (#‘𝑇)) ∈ (ℤ_≥‘((#‘𝑆) + (#‘𝑇))))
98		uzaddcl 11620	. . . . . . . . . . . 12 ⊢ ((((#‘𝑆) + (#‘𝑇)) ∈ (ℤ_≥‘((#‘𝑆) + (#‘𝑇))) ∧ (#‘𝑈) ∈ ℕ₀) → (((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)) ∈ (ℤ_≥‘((#‘𝑆) + (#‘𝑇))))
99	97, 36, 98	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)) ∈ (ℤ_≥‘((#‘𝑆) + (#‘𝑇))))
100		fzoss2 12365	. . . . . . . . . . 11 ⊢ ((((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)) ∈ (ℤ_≥‘((#‘𝑆) + (#‘𝑇))) → ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇))) ⊆ ((#‘𝑆)..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))))
101	99, 100	syl 17	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇))) ⊆ ((#‘𝑆)..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))))
102	26, 38	eqtr4d 2647	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((#‘𝑆) + (#‘(𝑇 ++ 𝑈))) = (((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))
103	102	oveq2d 6565	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((#‘𝑆)..^((#‘𝑆) + (#‘(𝑇 ++ 𝑈)))) = ((#‘𝑆)..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))))
104	101, 103	sseqtr4d 3605	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇))) ⊆ ((#‘𝑆)..^((#‘𝑆) + (#‘(𝑇 ++ 𝑈)))))
105	104	sselda 3568	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘(𝑇 ++ 𝑈)))))
106		ccatval2 13215	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐵 ∧ (𝑇 ++ 𝑈) ∈ Word 𝐵 ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘(𝑇 ++ 𝑈))))) → ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥) = ((𝑇 ++ 𝑈)‘(𝑥 − (#‘𝑆))))
107	75, 95, 105, 106	syl3anc 1318	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥) = ((𝑇 ++ 𝑈)‘(𝑥 − (#‘𝑆))))
108	86, 94, 107	3eqtr4d 2654	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥))
109	10	oveq2d 6565	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑥 − (#‘(𝑆 ++ 𝑇))) = (𝑥 − ((#‘𝑆) + (#‘𝑇))))
110	109	adantr 480	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (𝑥 − (#‘(𝑆 ++ 𝑇))) = (𝑥 − ((#‘𝑆) + (#‘𝑇))))
111		elfzoelz 12339	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))) → 𝑥 ∈ ℤ)
112	111	zcnd 11359	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))) → 𝑥 ∈ ℂ)
113	112	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → 𝑥 ∈ ℂ)
114	31	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (#‘𝑆) ∈ ℂ)
115	34	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (#‘𝑇) ∈ ℂ)
116	113, 114, 115	subsub4d 10302	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → ((𝑥 − (#‘𝑆)) − (#‘𝑇)) = (𝑥 − ((#‘𝑆) + (#‘𝑇))))
117	110, 116	eqtr4d 2647	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (𝑥 − (#‘(𝑆 ++ 𝑇))) = ((𝑥 − (#‘𝑆)) − (#‘𝑇)))
118	117	fveq2d 6107	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (𝑈‘(𝑥 − (#‘(𝑆 ++ 𝑇)))) = (𝑈‘((𝑥 − (#‘𝑆)) − (#‘𝑇))))
119		simpl2 1058	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → 𝑇 ∈ Word 𝐵)
120		simpl3 1059	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → 𝑈 ∈ Word 𝐵)
121	38	oveq2d 6565	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))) = (((#‘𝑆) + (#‘𝑇))..^((#‘𝑆) + ((#‘𝑇) + (#‘𝑈)))))
122	121	eleq2d 2673	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))) ↔ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^((#‘𝑆) + ((#‘𝑇) + (#‘𝑈))))))
123	122	biimpa 500	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^((#‘𝑆) + ((#‘𝑇) + (#‘𝑈)))))
124	43	adantr 480	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (#‘𝑆) ∈ ℤ)
125	70	adantr 480	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (#‘𝑇) ∈ ℤ)
126	36	nn0zd 11356	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (#‘𝑈) ∈ ℤ)
127	70, 126	zaddcld 11362	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((#‘𝑇) + (#‘𝑈)) ∈ ℤ)
128	127	adantr 480	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → ((#‘𝑇) + (#‘𝑈)) ∈ ℤ)
129		fzosubel2 12395	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^((#‘𝑆) + ((#‘𝑇) + (#‘𝑈)))) ∧ ((#‘𝑆) ∈ ℤ ∧ (#‘𝑇) ∈ ℤ ∧ ((#‘𝑇) + (#‘𝑈)) ∈ ℤ)) → (𝑥 − (#‘𝑆)) ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑈))))
130	123, 124, 125, 128, 129	syl13anc 1320	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (𝑥 − (#‘𝑆)) ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑈))))
131		ccatval2 13215	. . . . . . . . 9 ⊢ ((𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵 ∧ (𝑥 − (#‘𝑆)) ∈ ((#‘𝑇)..^((#‘𝑇) + (#‘𝑈)))) → ((𝑇 ++ 𝑈)‘(𝑥 − (#‘𝑆))) = (𝑈‘((𝑥 − (#‘𝑆)) − (#‘𝑇))))
132	119, 120, 130, 131	syl3anc 1318	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → ((𝑇 ++ 𝑈)‘(𝑥 − (#‘𝑆))) = (𝑈‘((𝑥 − (#‘𝑆)) − (#‘𝑇))))
133	118, 132	eqtr4d 2647	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (𝑈‘(𝑥 − (#‘(𝑆 ++ 𝑇)))) = ((𝑇 ++ 𝑈)‘(𝑥 − (#‘𝑆))))
134	52	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (𝑆 ++ 𝑇) ∈ Word 𝐵)
135	10, 11	oveq12d 6567	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → ((#‘(𝑆 ++ 𝑇))..^((#‘(𝑆 ++ 𝑇)) + (#‘𝑈))) = (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))))
136	135	eleq2d 2673	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (𝑥 ∈ ((#‘(𝑆 ++ 𝑇))..^((#‘(𝑆 ++ 𝑇)) + (#‘𝑈))) ↔ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))))
137	136	biimpar 501	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → 𝑥 ∈ ((#‘(𝑆 ++ 𝑇))..^((#‘(𝑆 ++ 𝑇)) + (#‘𝑈))))
138		ccatval2 13215	. . . . . . . 8 ⊢ (((𝑆 ++ 𝑇) ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵 ∧ 𝑥 ∈ ((#‘(𝑆 ++ 𝑇))..^((#‘(𝑆 ++ 𝑇)) + (#‘𝑈)))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = (𝑈‘(𝑥 − (#‘(𝑆 ++ 𝑇)))))
139	134, 120, 137, 138	syl3anc 1318	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = (𝑈‘(𝑥 − (#‘(𝑆 ++ 𝑇)))))
140		simpl1 1057	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → 𝑆 ∈ Word 𝐵)
141	18	adantr 480	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (𝑇 ++ 𝑈) ∈ Word 𝐵)
142		fzoss1 12364	. . . . . . . . . . 11 ⊢ (((#‘𝑆) + (#‘𝑇)) ∈ (ℤ_≥‘(#‘𝑆)) → (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))) ⊆ ((#‘𝑆)..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))))
143	58, 142	syl 17	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))) ⊆ ((#‘𝑆)..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))))
144	143, 103	sseqtr4d 3605	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) → (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈))) ⊆ ((#‘𝑆)..^((#‘𝑆) + (#‘(𝑇 ++ 𝑈)))))
145	144	sselda 3568	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘(𝑇 ++ 𝑈)))))
146	140, 141, 145, 106	syl3anc 1318	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥) = ((𝑇 ++ 𝑈)‘(𝑥 − (#‘𝑆))))
147	133, 139, 146	3eqtr4d 2654	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (((#‘𝑆) + (#‘𝑇))..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥))
148	108, 147	jaodan 822	. . . . 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))
149	74, 148	syldan 486	. . . 4 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ ((#‘𝑆)..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥))
150	69, 149	jaodan 822	. . 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))
151	46, 150	syldan 486	. 2 ⊢ (((𝑆 ∈ Word 𝐵 ∧ 𝑇 ∈ Word 𝐵 ∧ 𝑈 ∈ Word 𝐵) ∧ 𝑥 ∈ (0..^(((#‘𝑆) + (#‘𝑇)) + (#‘𝑈)))) → (((𝑆 ++ 𝑇) ++ 𝑈)‘𝑥) = ((𝑆 ++ (𝑇 ++ 𝑈))‘𝑥))
152	15, 42, 151	eqfnfvd 6222	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 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 + caddc 9818 − cmin 10145 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ..^cfzo 12334 #chash 12979 Word cword 13146 ++ cconcat 13148

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156

This theorem is referenced by: ccatw2s1ass 13259 cats1cat 13457 cats2cat 13458 frmdmnd 17219 efginvrel2 17963 efgredleme 17979 efgredlemc 17981 efgcpbllemb 17991 numclwlk1lem2foa 26618 numclwlk1lem2fo 26622 signstfvc 29977 av-numclwlk1lem2foa 41521 av-numclwlk1lem2fo 41525

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