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Theorem ofccat 13556

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 13165	. . . . . . . . . . 11 ⊢ (𝐸 ∈ Word 𝑆 → 𝐸:(0..^(#‘𝐸))⟶𝑆)
3		ffn 5958	. . . . . . . . . . 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 13165	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Word 𝑇 → 𝐺:(0..^(#‘𝐺))⟶𝑇)
7		ffn 5958	. . . . . . . . . . . 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 6565	. . . . . . . . . . . 12 ⊢ (𝜑 → (0..^(#‘𝐸)) = (0..^(#‘𝐺)))
11	10	fneq2d 5896	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐺 Fn (0..^(#‘𝐸)) ↔ 𝐺 Fn (0..^(#‘𝐺))))
12	8, 11	mpbird 246	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn (0..^(#‘𝐸)))
13		ovex 6577	. . . . . . . . . . 11 ⊢ (0..^(#‘𝐸)) ∈ V
14	13	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (0..^(#‘𝐸)) ∈ V)
15		inidm 3784	. . . . . . . . . 10 ⊢ ((0..^(#‘𝐸)) ∩ (0..^(#‘𝐸))) = (0..^(#‘𝐸))
16	4, 12, 14, 14, 15	offn 6806	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ∘_𝑓 𝑅𝐺) Fn (0..^(#‘𝐸)))
17		hashfn 13025	. . . . . . . . 9 ⊢ ((𝐸 ∘_𝑓 𝑅𝐺) Fn (0..^(#‘𝐸)) → (#‘(𝐸 ∘_𝑓 𝑅𝐺)) = (#‘(0..^(#‘𝐸))))
18	16, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → (#‘(𝐸 ∘_𝑓 𝑅𝐺)) = (#‘(0..^(#‘𝐸))))
19		wrdfin 13178	. . . . . . . . . 10 ⊢ (𝐸 ∈ Word 𝑆 → 𝐸 ∈ Fin)
20		hashcl 13009	. . . . . . . . . 10 ⊢ (𝐸 ∈ Fin → (#‘𝐸) ∈ ℕ₀)
21	1, 19, 20	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝐸) ∈ ℕ₀)
22		hashfzo0 13077	. . . . . . . . 9 ⊢ ((#‘𝐸) ∈ ℕ₀ → (#‘(0..^(#‘𝐸))) = (#‘𝐸))
23	21, 22	syl 17	. . . . . . . 8 ⊢ (𝜑 → (#‘(0..^(#‘𝐸))) = (#‘𝐸))
24	18, 23	eqtrd 2644	. . . . . . 7 ⊢ (𝜑 → (#‘(𝐸 ∘_𝑓 𝑅𝐺)) = (#‘𝐸))
25	24	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → (#‘(𝐸 ∘_𝑓 𝑅𝐺)) = (#‘𝐸))
26	25	oveq2d 6565	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺))) = (0..^(#‘𝐸)))
27	26	eleq2d 2673	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → (𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺))) ↔ 𝑖 ∈ (0..^(#‘𝐸))))
28	4	ad2antrr 758	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → 𝐸 Fn (0..^(#‘𝐸)))
29	12	ad2antrr 758	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → 𝐺 Fn (0..^(#‘𝐸)))
30	13	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (0..^(#‘𝐸)) ∈ V)
31	27	biimpa 500	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → 𝑖 ∈ (0..^(#‘𝐸)))
32		fnfvof 6809	. . . . 5 ⊢ (((𝐸 Fn (0..^(#‘𝐸)) ∧ 𝐺 Fn (0..^(#‘𝐸))) ∧ ((0..^(#‘𝐸)) ∈ V ∧ 𝑖 ∈ (0..^(#‘𝐸)))) → ((𝐸 ∘_𝑓 𝑅𝐺)‘𝑖) = ((𝐸‘𝑖)𝑅(𝐺‘𝑖)))
33	28, 29, 30, 31, 32	syl22anc 1319	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → ((𝐸 ∘_𝑓 𝑅𝐺)‘𝑖) = ((𝐸‘𝑖)𝑅(𝐺‘𝑖)))
34	24	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (#‘(𝐸 ∘_𝑓 𝑅𝐺)) = (#‘𝐸))
35	34	oveq2d 6565	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (𝑖 − (#‘(𝐸 ∘_𝑓 𝑅𝐺))) = (𝑖 − (#‘𝐸)))
36	35	fveq2d 6107	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → ((𝐹 ∘_𝑓 𝑅𝐻)‘(𝑖 − (#‘(𝐸 ∘_𝑓 𝑅𝐺)))) = ((𝐹 ∘_𝑓 𝑅𝐻)‘(𝑖 − (#‘𝐸))))
37		ofccat.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ Word 𝑆)
38		wrdf 13165	. . . . . . . 8 ⊢ (𝐹 ∈ Word 𝑆 → 𝐹:(0..^(#‘𝐹))⟶𝑆)
39		ffn 5958	. . . . . . . 8 ⊢ (𝐹:(0..^(#‘𝐹))⟶𝑆 → 𝐹 Fn (0..^(#‘𝐹)))
40	37, 38, 39	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn (0..^(#‘𝐹)))
41	40	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → 𝐹 Fn (0..^(#‘𝐹)))
42		ofccat.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ Word 𝑇)
43		wrdf 13165	. . . . . . . . 9 ⊢ (𝐻 ∈ Word 𝑇 → 𝐻:(0..^(#‘𝐻))⟶𝑇)
44		ffn 5958	. . . . . . . . 9 ⊢ (𝐻:(0..^(#‘𝐻))⟶𝑇 → 𝐻 Fn (0..^(#‘𝐻)))
45	42, 43, 44	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn (0..^(#‘𝐻)))
46		ofccat.6	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝐹) = (#‘𝐻))
47	46	oveq2d 6565	. . . . . . . . 9 ⊢ (𝜑 → (0..^(#‘𝐹)) = (0..^(#‘𝐻)))
48	47	fneq2d 5896	. . . . . . . 8 ⊢ (𝜑 → (𝐻 Fn (0..^(#‘𝐹)) ↔ 𝐻 Fn (0..^(#‘𝐻))))
49	45, 48	mpbird 246	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn (0..^(#‘𝐹)))
50	49	ad2antrr 758	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → 𝐻 Fn (0..^(#‘𝐹)))
51		ovex 6577	. . . . . . 7 ⊢ (0..^(#‘𝐹)) ∈ V
52	51	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (0..^(#‘𝐹)) ∈ V)
53		simplr 788	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))))
54		simpr 476	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺))))
55	26	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺))) = (0..^(#‘𝐸)))
56	54, 55	neleqtrd 2709	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → ¬ 𝑖 ∈ (0..^(#‘𝐸)))
57	21	ad2antrr 758	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (#‘𝐸) ∈ ℕ₀)
58	57	nn0zd 11356	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (#‘𝐸) ∈ ℤ)
59		wrdfin 13178	. . . . . . . . . 10 ⊢ (𝐹 ∈ Word 𝑆 → 𝐹 ∈ Fin)
60		hashcl 13009	. . . . . . . . . 10 ⊢ (𝐹 ∈ Fin → (#‘𝐹) ∈ ℕ₀)
61	37, 59, 60	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝐹) ∈ ℕ₀)
62	61	ad2antrr 758	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (#‘𝐹) ∈ ℕ₀)
63	62	nn0zd 11356	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (#‘𝐹) ∈ ℤ)
64		fzocatel 12399	. . . . . . 7 ⊢ (((𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ∧ ¬ 𝑖 ∈ (0..^(#‘𝐸))) ∧ ((#‘𝐸) ∈ ℤ ∧ (#‘𝐹) ∈ ℤ)) → (𝑖 − (#‘𝐸)) ∈ (0..^(#‘𝐹)))
65	53, 56, 58, 63, 64	syl22anc 1319	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → (𝑖 − (#‘𝐸)) ∈ (0..^(#‘𝐹)))
66		fnfvof 6809	. . . . . 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)))))
67	41, 50, 52, 65, 66	syl22anc 1319	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) ∧ ¬ 𝑖 ∈ (0..^(#‘(𝐸 ∘_𝑓 𝑅𝐺)))) → ((𝐹 ∘_𝑓 𝑅𝐻)‘(𝑖 − (#‘𝐸))) = ((𝐹‘(𝑖 − (#‘𝐸)))𝑅(𝐻‘(𝑖 − (#‘𝐸)))))
68	36, 67	eqtrd 2644	. . . 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)))))
69	27, 33, 68	ifbieq12d2 4069	. . 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))))))
70	69	mpteq2dva 4672	. 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)))))))
71		ovex 6577	. . . 4 ⊢ (𝐸 ∘_𝑓 𝑅𝐺) ∈ V
72		ovex 6577	. . . 4 ⊢ (𝐹 ∘_𝑓 𝑅𝐻) ∈ V
73		ccatfval 13211	. . . 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)))))))
74	71, 72, 73	mp2an 704	. . 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))))))
75	51	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (0..^(#‘𝐹)) ∈ V)
76		inidm 3784	. . . . . . . . 9 ⊢ ((0..^(#‘𝐹)) ∩ (0..^(#‘𝐹))) = (0..^(#‘𝐹))
77	40, 49, 75, 75, 76	offn 6806	. . . . . . . 8 ⊢ (𝜑 → (𝐹 ∘_𝑓 𝑅𝐻) Fn (0..^(#‘𝐹)))
78		hashfn 13025	. . . . . . . 8 ⊢ ((𝐹 ∘_𝑓 𝑅𝐻) Fn (0..^(#‘𝐹)) → (#‘(𝐹 ∘_𝑓 𝑅𝐻)) = (#‘(0..^(#‘𝐹))))
79	77, 78	syl 17	. . . . . . 7 ⊢ (𝜑 → (#‘(𝐹 ∘_𝑓 𝑅𝐻)) = (#‘(0..^(#‘𝐹))))
80		hashfzo0 13077	. . . . . . . 8 ⊢ ((#‘𝐹) ∈ ℕ₀ → (#‘(0..^(#‘𝐹))) = (#‘𝐹))
81	61, 80	syl 17	. . . . . . 7 ⊢ (𝜑 → (#‘(0..^(#‘𝐹))) = (#‘𝐹))
82	79, 81	eqtrd 2644	. . . . . 6 ⊢ (𝜑 → (#‘(𝐹 ∘_𝑓 𝑅𝐻)) = (#‘𝐹))
83	24, 82	oveq12d 6567	. . . . 5 ⊢ (𝜑 → ((#‘(𝐸 ∘_𝑓 𝑅𝐺)) + (#‘(𝐹 ∘_𝑓 𝑅𝐻))) = ((#‘𝐸) + (#‘𝐹)))
84	83	oveq2d 6565	. . . 4 ⊢ (𝜑 → (0..^((#‘(𝐸 ∘_𝑓 𝑅𝐺)) + (#‘(𝐹 ∘_𝑓 𝑅𝐻)))) = (0..^((#‘𝐸) + (#‘𝐹))))
85	84	mpteq1d 4666	. . 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)))))))
86	74, 85	syl5eq 2656	. 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)))))))
87		ovex 6577	. . . . . 6 ⊢ (0..^((#‘𝐸) + (#‘𝐹))) ∈ V
88	87	a1i 11	. . . . 5 ⊢ (𝜑 → (0..^((#‘𝐸) + (#‘𝐹))) ∈ V)
89		fvex 6113	. . . . . . 7 ⊢ (𝐸‘𝑖) ∈ V
90		fvex 6113	. . . . . . 7 ⊢ (𝐹‘(𝑖 − (#‘𝐸))) ∈ V
91	89, 90	ifex 4106	. . . . . 6 ⊢ if(𝑖 ∈ (0..^(#‘𝐸)), (𝐸‘𝑖), (𝐹‘(𝑖 − (#‘𝐸)))) ∈ V
92	91	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → if(𝑖 ∈ (0..^(#‘𝐸)), (𝐸‘𝑖), (𝐹‘(𝑖 − (#‘𝐸)))) ∈ V)
93		fvex 6113	. . . . . . 7 ⊢ (𝐺‘𝑖) ∈ V
94		fvex 6113	. . . . . . 7 ⊢ (𝐻‘(𝑖 − (#‘𝐺))) ∈ V
95	93, 94	ifex 4106	. . . . . 6 ⊢ if(𝑖 ∈ (0..^(#‘𝐺)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐺)))) ∈ V
96	95	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → if(𝑖 ∈ (0..^(#‘𝐺)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐺)))) ∈ V)
97		ccatfval 13211	. . . . . 6 ⊢ ((𝐸 ∈ Word 𝑆 ∧ 𝐹 ∈ Word 𝑆) → (𝐸 ++ 𝐹) = (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ if(𝑖 ∈ (0..^(#‘𝐸)), (𝐸‘𝑖), (𝐹‘(𝑖 − (#‘𝐸))))))
98	1, 37, 97	syl2anc 691	. . . . 5 ⊢ (𝜑 → (𝐸 ++ 𝐹) = (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ if(𝑖 ∈ (0..^(#‘𝐸)), (𝐸‘𝑖), (𝐹‘(𝑖 − (#‘𝐸))))))
99		ccatfval 13211	. . . . . . 7 ⊢ ((𝐺 ∈ Word 𝑇 ∧ 𝐻 ∈ Word 𝑇) → (𝐺 ++ 𝐻) = (𝑖 ∈ (0..^((#‘𝐺) + (#‘𝐻))) ↦ if(𝑖 ∈ (0..^(#‘𝐺)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐺))))))
100	5, 42, 99	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝐺 ++ 𝐻) = (𝑖 ∈ (0..^((#‘𝐺) + (#‘𝐻))) ↦ if(𝑖 ∈ (0..^(#‘𝐺)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐺))))))
101	9, 46	oveq12d 6567	. . . . . . . 8 ⊢ (𝜑 → ((#‘𝐸) + (#‘𝐹)) = ((#‘𝐺) + (#‘𝐻)))
102	101	oveq2d 6565	. . . . . . 7 ⊢ (𝜑 → (0..^((#‘𝐸) + (#‘𝐹))) = (0..^((#‘𝐺) + (#‘𝐻))))
103	102	mpteq1d 4666	. . . . . 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))))))
104	100, 103	eqtr4d 2647	. . . . 5 ⊢ (𝜑 → (𝐺 ++ 𝐻) = (𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹))) ↦ if(𝑖 ∈ (0..^(#‘𝐺)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐺))))))
105	88, 92, 96, 98, 104	offval2 6812	. . . 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)))))))
106	9	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → (#‘𝐸) = (#‘𝐺))
107	106	oveq2d 6565	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → (0..^(#‘𝐸)) = (0..^(#‘𝐺)))
108	107	eleq2d 2673	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → (𝑖 ∈ (0..^(#‘𝐸)) ↔ 𝑖 ∈ (0..^(#‘𝐺))))
109	106	oveq2d 6565	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → (𝑖 − (#‘𝐸)) = (𝑖 − (#‘𝐺)))
110	109	fveq2d 6107	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → (𝐻‘(𝑖 − (#‘𝐸))) = (𝐻‘(𝑖 − (#‘𝐺))))
111	108, 110	ifbieq2d 4061	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^((#‘𝐸) + (#‘𝐹)))) → if(𝑖 ∈ (0..^(#‘𝐸)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐸)))) = if(𝑖 ∈ (0..^(#‘𝐺)), (𝐺‘𝑖), (𝐻‘(𝑖 − (#‘𝐺)))))
112	111	oveq2d 6565	. . . . 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))))))
113	112	mpteq2dva 4672	. . . 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)))))))
114	105, 113	eqtr4d 2647	. . 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)))))))
115		ovif12 6637	. . . 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)))))
116	115	mpteq2i 4669	. . 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))))))
117	114, 116	syl6eq 2660	. 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)))))))
118	70, 86, 117	3eqtr4rd 2655	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 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ifcif 4036 ↦ cmpt 4643 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘_𝑓 cof 6793 Fincfn 7841 0cc0 9815 + caddc 9818 − cmin 10145 ℕ₀cn0 11169 ℤcz 11254 ..^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-of 6795 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: ofs2 13558 ofcccat 29946

W3C validator