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Theorem ofccat 13708

Description: Letterwise operations on word concatenations. (Contributed by Thierry Arnoux, 28-Sep-2018.)

**Hypotheses**
Ref	Expression
ofccat.1	⊢ (𝜑 → 𝐸 ∈ Word 𝑆)
ofccat.2	⊢ (𝜑 → 𝐹 ∈ Word 𝑆)
ofccat.3	⊢ (𝜑 → 𝐺 ∈ Word 𝑇)
ofccat.4	⊢ (𝜑 → 𝐻 ∈ Word 𝑇)
ofccat.5	⊢ (𝜑 → (#‘𝐸) = (#‘𝐺))
ofccat.6	⊢ (𝜑 → (#‘𝐹) = (#‘𝐻))

**Assertion**
Ref	Expression
ofccat	⊢ (𝜑 → ((𝐸 ++ 𝐹) ∘_𝑓 𝑅(𝐺 ++ 𝐻)) = ((𝐸 ∘_𝑓 𝑅𝐺) ++ (𝐹 ∘_𝑓 𝑅𝐻)))

Proof of Theorem ofccat Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofccat.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ Word 𝑆)
2		wrdf 13310	. . . . . . . . . . 11 ⊢ (𝐸 ∈ Word 𝑆 → 𝐸:(0..^(#‘𝐸))⟶𝑆)
3		ffn 6045	. . . . . . . . . . 11 ⊢ (𝐸:(0..^(#‘𝐸))⟶𝑆 → 𝐸 Fn (0..^(#‘𝐸)))
4	1, 2, 3	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 Fn (0..^(#‘𝐸)))
5		ofccat.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ Word 𝑇)
6		wrdf 13310	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Word 𝑇 → 𝐺:(0..^(#‘𝐺))⟶𝑇)
7		ffn 6045	. . . . . . . . . . . 12 ⊢ (𝐺:(0..^(#‘𝐺))⟶𝑇 → 𝐺 Fn (0..^(#‘𝐺)))
8	5, 6, 7	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 Fn (0..^(#‘𝐺)))
9		ofccat.5	. . . . . . . . . . . . 13 ⊢ (𝜑 → (#‘𝐸) = (#‘𝐺))
10	9	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝜑 → (0..^(#‘𝐸)) = (0..^(#‘𝐺)))
11	10	fneq2d 5982	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 Fn (0..^(#‘𝐸)) ↔ 𝐺 Fn (0..^(#‘𝐺))))
12	8, 11	mpbird 247	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn (0..^(#‘𝐸)))
13		ovexd 6680	. . . . . . . . . 10 ⊢ (𝜑 → (0..^(#‘𝐸)) ∈ V)
14		inidm 3822	. . . . . . . . . 10 ⊢ ((0..^(#‘𝐸)) ∩ (0..^(#‘𝐸))) = (0..^(#‘𝐸))
15	4, 12, 13, 13, 14	offn 6908	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ∘_𝑓 𝑅𝐺) Fn (0..^(#‘𝐸)))
16		hashfn 13164	. . . . . . . . 9 ⊢ ((𝐸 ∘_𝑓 𝑅𝐺) Fn (0..^(#‘𝐸)) → (#‘(𝐸 ∘_𝑓 𝑅𝐺)) = (#‘(0..^(#‘𝐸))))
17	15, 16	syl 17	. . . . . . . 8 ⊢ (𝜑 → (#‘(𝐸 ∘_𝑓 𝑅𝐺)) = (#‘(0..^(#‘𝐸))))
18		wrdfin 13323	. . . . . . . . . 10 ⊢ (𝐸 ∈ Word 𝑆 → 𝐸 ∈ Fin)
19		hashcl 13147	. . . . . . . . . 10 ⊢ (𝐸 ∈ Fin → (#‘𝐸) ∈ ℕ₀)
20	1, 18, 19	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝐸) ∈ ℕ₀)
21		hashfzo0 13217	. . . . . . . . 9 ⊢ ((#‘𝐸) ∈ ℕ₀ → (#‘(0..^(#‘𝐸))) = (#‘𝐸))
22	20, 21	syl 17	. . . . . . . 8 ⊢ (𝜑 → (#‘(0..^(#‘𝐸))) = (#‘𝐸))
23	17, 22	eqtrd 2656	. . . . . . 7 ⊢ (𝜑 → (#‘(𝐸 ∘_𝑓 𝑅𝐺)) = (#‘𝐸))
24	23	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → (#‘(𝐸 ∘_𝑓 𝑅𝐺)) = (#‘𝐸))
25	24	oveq2d 6666	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺))) = (0..^(#‘𝐸)))
26	25	eleq2d 2687	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → (𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺))) ↔ 𝑖 ∈ (0..^(#‘𝐸))))
27	4	ad2antrr 762	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → 𝐸 Fn (0..^(#‘𝐸)))
28	12	ad2antrr 762	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → 𝐺 Fn (0..^(#‘𝐸)))
29		ovexd 6680	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (0..^(#‘𝐸)) ∈ V)
30	26	biimpa 501	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → 𝑖 ∈ (0..^(#‘𝐸)))
31		fnfvof 6911	. . . . 5 ⊢ (((𝐸 Fn (0..^(#‘𝐸)) ∧ 𝐺 Fn (0..^(#‘𝐸))) ∧ ((0..^(#‘𝐸)) ∈ V ∧ 𝑖 ∈ (0..^(#‘𝐸)))) → ((𝐸 ∘_𝑓 𝑅𝐺)‘𝑖) = ((𝐸‘𝑖)𝑅(𝐺‘𝑖)))
32	27, 28, 29, 30, 31	syl22anc 1327	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → ((𝐸 ∘_𝑓 𝑅𝐺)‘𝑖) = ((𝐸‘𝑖)𝑅(𝐺‘𝑖)))
33	23	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (#‘(𝐸 ∘_𝑓 𝑅𝐺)) = (#‘𝐸))
34	33	oveq2d 6666	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (𝑖 − (#‘(𝐸 ∘_𝑓 𝑅𝐺))) = (𝑖 − (#‘𝐸)))
35	34	fveq2d 6195	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → ((𝐹 ∘_𝑓 𝑅𝐻)‘(𝑖 − (#‘(𝐸 ∘_𝑓 𝑅𝐺)))) = ((𝐹 ∘_𝑓 𝑅𝐻)‘(𝑖 − (#‘𝐸))))
36		ofccat.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ Word 𝑆)
37		wrdf 13310	. . . . . . . 8 ⊢ (𝐹 ∈ Word 𝑆 → 𝐹:(0..^(#‘𝐹))⟶𝑆)
38		ffn 6045	. . . . . . . 8 ⊢ (𝐹:(0..^(#‘𝐹))⟶𝑆 → 𝐹 Fn (0..^(#‘𝐹)))
39	36, 37, 38	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn (0..^(#‘𝐹)))
40	39	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → 𝐹 Fn (0..^(#‘𝐹)))
41		ofccat.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ Word 𝑇)
42		wrdf 13310	. . . . . . . . 9 ⊢ (𝐻 ∈ Word 𝑇 → 𝐻:(0..^(#‘𝐻))⟶𝑇)
43		ffn 6045	. . . . . . . . 9 ⊢ (𝐻:(0..^(#‘𝐻))⟶𝑇 → 𝐻 Fn (0..^(#‘𝐻)))
44	41, 42, 43	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn (0..^(#‘𝐻)))
45		ofccat.6	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝐹) = (#‘𝐻))
46	45	oveq2d 6666	. . . . . . . . 9 ⊢ (𝜑 → (0..^(#‘𝐹)) = (0..^(#‘𝐻)))
47	46	fneq2d 5982	. . . . . . . 8 ⊢ (𝜑 → (𝐻 Fn (0..^(#‘𝐹)) ↔ 𝐻 Fn (0..^(#‘𝐻))))
48	44, 47	mpbird 247	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn (0..^(#‘𝐹)))
49	48	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → 𝐻 Fn (0..^(#‘𝐹)))
50		ovexd 6680	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (0..^(#‘𝐹)) ∈ V)
51		simplr 792	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))))
52		simpr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺))))
53	25	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺))) = (0..^(#‘𝐸)))
54	52, 53	neleqtrd 2722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → ¬ 𝑖 ∈ (0..^(#‘𝐸)))
55	20	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (#‘𝐸) ∈ ℕ₀)
56	55	nn0zd 11480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (#‘𝐸) ∈ ℤ)
57		wrdfin 13323	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word 𝑆 → 𝐹 ∈ Fin)
58		hashcl 13147	. . . . . . . . . 10 ⊢ (𝐹 ∈ Fin → (#‘𝐹) ∈ ℕ₀)
59	36, 57, 58	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝐹) ∈ ℕ₀)
60	59	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (#‘𝐹) ∈ ℕ₀)
61	60	nn0zd 11480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (#‘𝐹) ∈ ℤ)
62		fzocatel 12531	. . . . . . 7 ⊢ (((𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ∧ ¬ 𝑖 ∈ (0..^(#‘𝐸))) ∧ ((#‘𝐸) ∈ ℤ ∧ (#‘𝐹) ∈ ℤ)) → (𝑖 − (#‘𝐸)) ∈ (0..^(#‘𝐹)))
63	51, 54, 56, 61, 62	syl22anc 1327	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (𝑖 − (#‘𝐸)) ∈ (0..^(#‘𝐹)))
64		fnfvof 6911	. . . . . 6 ⊢ (((𝐹 Fn (0..^(#‘𝐹)) ∧ 𝐻 Fn (0..^(#‘𝐹))) ∧ ((0..^(#‘𝐹)) ∈ V ∧ (𝑖 − (#‘𝐸)) ∈ (0..^(#‘𝐹)))) → ((𝐹 ∘_𝑓 𝑅𝐻)‘(𝑖 − (#‘𝐸))) = ((𝐹‘(𝑖 − (#‘𝐸)))𝑅(𝐻‘(𝑖 − (#‘𝐸)))))
65	40, 49, 50, 63, 64	syl22anc 1327	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → ((𝐹 ∘_𝑓 𝑅𝐻)‘(𝑖 − (#‘𝐸))) = ((𝐹‘(𝑖 − (#‘𝐸)))𝑅(𝐻‘(𝑖 − (#‘𝐸)))))
66	35, 65	eqtrd 2656	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → ((𝐹 ∘_𝑓 𝑅𝐻)‘(𝑖 − (#‘(𝐸 ∘_𝑓 𝑅𝐺)))) = ((𝐹‘(𝑖 − (#‘𝐸)))𝑅(𝐻‘(𝑖 − (#‘𝐸)))))
67	26, 32, 66	ifbieq12d2 4119	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → if(𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺))), ((𝐸 ∘_𝑓 𝑅𝐺)‘𝑖), ((𝐹 ∘_𝑓 𝑅𝐻)‘(𝑖 − (#‘(𝐸 ∘_𝑓 𝑅𝐺))))) = if(𝑖 ∈ (0..^(#‘𝐸)), ((𝐸‘𝑖)𝑅(𝐺‘𝑖)), ((𝐹‘(𝑖 − (#‘𝐸)))𝑅(𝐻‘(𝑖 − (#‘𝐸))))))
68	67	mpteq2dva 4744	. 2 ⊢ (𝜑 → (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ if(𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺))), ((𝐸 ∘_𝑓 𝑅𝐺)‘𝑖), ((𝐹 ∘_𝑓 𝑅𝐻)‘(𝑖 − (#‘(𝐸 ∘_𝑓 𝑅𝐺)))))) = (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ if(𝑖 ∈ (0..^(#‘𝐸)), ((𝐸‘𝑖)𝑅(𝐺‘𝑖)), ((𝐹‘(𝑖 − (#‘𝐸)))𝑅(𝐻‘(𝑖 − (#‘𝐸)))))))
69		ovex 6678	. . . 4 ⊢ (𝐸 ∘_𝑓 𝑅𝐺) ∈ V
70		ovex 6678	. . . 4 ⊢ (𝐹 ∘_𝑓 𝑅𝐻) ∈ V
71		ccatfval 13358	. . . 4 ⊢ (((𝐸 ∘_𝑓 𝑅𝐺) ∈ V ∧ (𝐹 ∘_𝑓 𝑅𝐻) ∈ V) → ((𝐸 ∘_𝑓 𝑅𝐺) ++ (𝐹 ∘_𝑓 𝑅𝐻)) = (𝑖 ∈ (0..^((#‘(𝐸 ∘_𝑓 𝑅𝐺)) + (#‘(𝐹 ∘_𝑓 𝑅𝐻)))) ↦ if(𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺))), ((𝐸 ∘_𝑓 𝑅𝐺)‘𝑖), ((𝐹 ∘_𝑓 𝑅𝐻)‘(𝑖 − (#‘(𝐸 ∘_𝑓 𝑅𝐺)))))))
72	69, 70, 71	mp2an 708	. . 3 ⊢ ((𝐸 ∘_𝑓 𝑅𝐺) ++ (𝐹 ∘_𝑓 𝑅𝐻)) = (𝑖 ∈ (0..^((#‘(𝐸 ∘_𝑓 𝑅𝐺)) + (#‘(𝐹 ∘_𝑓 𝑅𝐻)))) ↦ if(𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺))), ((𝐸 ∘_𝑓 𝑅𝐺)‘𝑖), ((𝐹 ∘_𝑓 𝑅𝐻)‘(𝑖 − (#‘(𝐸 ∘_𝑓 𝑅𝐺))))))
73		ovexd 6680	. . . . . . . . 9 ⊢ (𝜑 → (0..^(#‘𝐹)) ∈ V)
74		inidm 3822	. . . . . . . . 9 ⊢ ((0..^(#‘𝐹)) ∩ (0..^(#‘𝐹))) = (0..^(#‘𝐹))
75	39, 48, 73, 73, 74	offn 6908	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘_𝑓 𝑅𝐻) Fn (0..^(#‘𝐹)))
76		hashfn 13164	. . . . . . . 8 ⊢ ((𝐹 ∘_𝑓 𝑅𝐻) Fn (0..^(#‘𝐹)) → (#‘(𝐹 ∘_𝑓 𝑅𝐻)) = (#‘(0..^(#‘𝐹))))
77	75, 76	syl 17	. . . . . . 7 ⊢ (𝜑 → (#‘(𝐹 ∘_𝑓 𝑅𝐻)) = (#‘(0..^(#‘𝐹))))
78		hashfzo0 13217	. . . . . . . 8 ⊢ ((#‘𝐹) ∈ ℕ₀ → (#‘(0..^(#‘𝐹))) = (#‘𝐹))
79	59, 78	syl 17	. . . . . . 7 ⊢ (𝜑 → (#‘(0..^(#‘𝐹))) = (#‘𝐹))
80	77, 79	eqtrd 2656	. . . . . 6 ⊢ (𝜑 → (#‘(𝐹 ∘_𝑓 𝑅𝐻)) = (#‘𝐹))
81	23, 80	oveq12d 6668	. . . . 5 ⊢ (𝜑 → ((#‘(𝐸 ∘_𝑓 𝑅𝐺)) + (#‘(𝐹 ∘_𝑓 𝑅𝐻))) = ((#‘𝐸) + (#‘𝐹)))
82	81	oveq2d 6666	. . . 4 ⊢ (𝜑 → (0..^((#‘(𝐸 ∘_𝑓 𝑅𝐺)) + (#‘(𝐹 ∘_𝑓 𝑅𝐻)))) = (0..^((#‘𝐸) + (#‘𝐹))))
83	82	mpteq1d 4738	. . 3 ⊢ (𝜑 → (𝑖 ∈ (0..^((#‘(𝐸 ∘_𝑓 𝑅𝐺)) + (#‘(𝐹 ∘_𝑓 𝑅𝐻)))) ↦ if(𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺))), ((𝐸 ∘_𝑓 𝑅𝐺)‘𝑖), ((𝐹 ∘_𝑓 𝑅𝐻)‘(𝑖 − (#‘(𝐸 ∘_𝑓 𝑅𝐺)))))) = (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ if(𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺))), ((𝐸 ∘_𝑓 𝑅𝐺)‘𝑖), ((𝐹 ∘_𝑓 𝑅𝐻)‘(𝑖 − (#‘(𝐸 ∘_𝑓 𝑅𝐺)))))))
84	72, 83	syl5eq 2668	. 2 ⊢ (𝜑 → ((𝐸 ∘_𝑓 𝑅𝐺) ++ (𝐹 ∘_𝑓 𝑅𝐻)) = (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ if(𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺))), ((𝐸 ∘_𝑓 𝑅𝐺)‘𝑖), ((𝐹 ∘_𝑓 𝑅𝐻)‘(𝑖 − (#‘(𝐸 ∘_𝑓 𝑅𝐺)))))))
85		ovexd 6680	. . . . 5 ⊢ (𝜑 → (0..^((#‘𝐸) + (#‘𝐹))) ∈ V)
86		fvex 6201	. . . . . . 7 ⊢ (𝐸‘𝑖) ∈ V
87		fvex 6201	. . . . . . 7 ⊢ (𝐹‘(𝑖 − (#‘𝐸))) ∈ V
88	86, 87	ifex 4156	. . . . . 6 ⊢ if(𝑖 ∈ (0..^(#‘𝐸)), (𝐸‘𝑖), (𝐹‘(𝑖 − (#‘𝐸)))) ∈ V
89	88	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → if(𝑖 ∈ (0..^(#‘𝐸)), (𝐸‘𝑖), (𝐹‘(𝑖 − (#‘𝐸)))) ∈ V)
90		fvex 6201	. . . . . . 7 ⊢ (𝐺‘𝑖) ∈ V
91		fvex 6201	. . . . . . 7 ⊢ (𝐻‘(𝑖 − (#‘𝐺))) ∈ V
92	90, 91	ifex 4156	. . . . . 6 ⊢ if(𝑖 ∈ (0..^(#‘𝐺)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐺)))) ∈ V
93	92	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → if(𝑖 ∈ (0..^(#‘𝐺)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐺)))) ∈ V)
94		ccatfval 13358	. . . . . 6 ⊢ ((𝐸 ∈ Word 𝑆 ∧ 𝐹 ∈ Word 𝑆) → (𝐸 ++ 𝐹) = (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ if(𝑖 ∈ (0..^(#‘𝐸)), (𝐸‘𝑖), (𝐹‘(𝑖 − (#‘𝐸))))))
95	1, 36, 94	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝐸 ++ 𝐹) = (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ if(𝑖 ∈ (0..^(#‘𝐸)), (𝐸‘𝑖), (𝐹‘(𝑖 − (#‘𝐸))))))
96		ccatfval 13358	. . . . . . 7 ⊢ ((𝐺 ∈ Word 𝑇 ∧ 𝐻 ∈ Word 𝑇) → (𝐺 ++ 𝐻) = (𝑖 ∈ (0..^((#‘𝐺) + (#‘𝐻))) ↦ if(𝑖 ∈ (0..^(#‘𝐺)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐺))))))
97	5, 41, 96	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (𝐺 ++ 𝐻) = (𝑖 ∈ (0..^((#‘𝐺) + (#‘𝐻))) ↦ if(𝑖 ∈ (0..^(#‘𝐺)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐺))))))
98	9, 45	oveq12d 6668	. . . . . . . 8 ⊢ (𝜑 → ((#‘𝐸) + (#‘𝐹)) = ((#‘𝐺) + (#‘𝐻)))
99	98	oveq2d 6666	. . . . . . 7 ⊢ (𝜑 → (0..^((#‘𝐸) + (#‘𝐹))) = (0..^((#‘𝐺) + (#‘𝐻))))
100	99	mpteq1d 4738	. . . . . 6 ⊢ (𝜑 → (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ if(𝑖 ∈ (0..^(#‘𝐺)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐺))))) = (𝑖 ∈ (0..^((#‘𝐺) + (#‘𝐻))) ↦ if(𝑖 ∈ (0..^(#‘𝐺)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐺))))))
101	97, 100	eqtr4d 2659	. . . . 5 ⊢ (𝜑 → (𝐺 ++ 𝐻) = (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ if(𝑖 ∈ (0..^(#‘𝐺)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐺))))))
102	85, 89, 93, 95, 101	offval2 6914	. . . 4 ⊢ (𝜑 → ((𝐸 ++ 𝐹) ∘_𝑓 𝑅(𝐺 ++ 𝐻)) = (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ (if(𝑖 ∈ (0..^(#‘𝐸)), (𝐸‘𝑖), (𝐹‘(𝑖 − (#‘𝐸))))𝑅if(𝑖 ∈ (0..^(#‘𝐺)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐺)))))))
103	9	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → (#‘𝐸) = (#‘𝐺))
104	103	oveq2d 6666	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → (0..^(#‘𝐸)) = (0..^(#‘𝐺)))
105	104	eleq2d 2687	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → (𝑖 ∈ (0..^(#‘𝐸)) ↔ 𝑖 ∈ (0..^(#‘𝐺))))
106	103	oveq2d 6666	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → (𝑖 − (#‘𝐸)) = (𝑖 − (#‘𝐺)))
107	106	fveq2d 6195	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → (𝐻‘(𝑖 − (#‘𝐸))) = (𝐻‘(𝑖 − (#‘𝐺))))
108	105, 107	ifbieq2d 4111	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → if(𝑖 ∈ (0..^(#‘𝐸)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐸)))) = if(𝑖 ∈ (0..^(#‘𝐺)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐺)))))
109	108	oveq2d 6666	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → (if(𝑖 ∈ (0..^(#‘𝐸)), (𝐸‘𝑖), (𝐹‘(𝑖 − (#‘𝐸))))𝑅if(𝑖 ∈ (0..^(#‘𝐸)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐸))))) = (if(𝑖 ∈ (0..^(#‘𝐸)), (𝐸‘𝑖), (𝐹‘(𝑖 − (#‘𝐸))))𝑅if(𝑖 ∈ (0..^(#‘𝐺)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐺))))))
110	109	mpteq2dva 4744	. . . 4 ⊢ (𝜑 → (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ (if(𝑖 ∈ (0..^(#‘𝐸)), (𝐸‘𝑖), (𝐹‘(𝑖 − (#‘𝐸))))𝑅if(𝑖 ∈ (0..^(#‘𝐸)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐸)))))) = (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ (if(𝑖 ∈ (0..^(#‘𝐸)), (𝐸‘𝑖), (𝐹‘(𝑖 − (#‘𝐸))))𝑅if(𝑖 ∈ (0..^(#‘𝐺)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐺)))))))
111	102, 110	eqtr4d 2659	. . 3 ⊢ (𝜑 → ((𝐸 ++ 𝐹) ∘_𝑓 𝑅(𝐺 ++ 𝐻)) = (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ (if(𝑖 ∈ (0..^(#‘𝐸)), (𝐸‘𝑖), (𝐹‘(𝑖 − (#‘𝐸))))𝑅if(𝑖 ∈ (0..^(#‘𝐸)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐸)))))))
112		ovif12 6739	. . . 4 ⊢ (if(𝑖 ∈ (0..^(#‘𝐸)), (𝐸‘𝑖), (𝐹‘(𝑖 − (#‘𝐸))))𝑅if(𝑖 ∈ (0..^(#‘𝐸)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐸))))) = if(𝑖 ∈ (0..^(#‘𝐸)), ((𝐸‘𝑖)𝑅(𝐺‘𝑖)), ((𝐹‘(𝑖 − (#‘𝐸)))𝑅(𝐻‘(𝑖 − (#‘𝐸)))))
113	112	mpteq2i 4741	. . 3 ⊢ (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ (if(𝑖 ∈ (0..^(#‘𝐸)), (𝐸‘𝑖), (𝐹‘(𝑖 − (#‘𝐸))))𝑅if(𝑖 ∈ (0..^(#‘𝐸)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐸)))))) = (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ if(𝑖 ∈ (0..^(#‘𝐸)), ((𝐸‘𝑖)𝑅(𝐺‘𝑖)), ((𝐹‘(𝑖 − (#‘𝐸)))𝑅(𝐻‘(𝑖 − (#‘𝐸))))))
114	111, 113	syl6eq 2672	. 2 ⊢ (𝜑 → ((𝐸 ++ 𝐹) ∘_𝑓 𝑅(𝐺 ++ 𝐻)) = (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ if(𝑖 ∈ (0..^(#‘𝐸)), ((𝐸‘𝑖)𝑅(𝐺‘𝑖)), ((𝐹‘(𝑖 − (#‘𝐸)))𝑅(𝐻‘(𝑖 − (#‘𝐸)))))))
115	68, 84, 114	3eqtr4rd 2667	1 ⊢ (𝜑 → ((𝐸 ++ 𝐹) ∘_𝑓 𝑅(𝐺 ++ 𝐻)) = ((𝐸 ∘_𝑓 𝑅𝐺) ++ (𝐹 ∘_𝑓 𝑅𝐻)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ifcif 4086 ↦ cmpt 4729 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘_𝑓 cof 6895 Fincfn 7955 0cc0 9936 + caddc 9939 − cmin 10266 ℕ₀cn0 11292 ℤcz 11377 ..^cfzo 12465 #chash 13117 Word cword 13291 ++ cconcat 13293

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301

This theorem is referenced by: ofs2 13710 ofcccat 30620

W3C validator