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Theorem ccatco 14529

Description: Mapping of words commutes with concatenation. (Contributed by Stefan O'Rear, 27-Aug-2015.)

**Assertion**
Ref	Expression
ccatco	⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑆 ++ 𝑇)) = ((𝐹 ∘ 𝑆) ++ (𝐹 ∘ 𝑇)))

Proof of Theorem ccatco Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lenco 14526	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑆)) = (♯‘𝑆))
2	1	3adant2 1129	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑆)) = (♯‘𝑆))
3		lenco 14526	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑇)) = (♯‘𝑇))
4	3	3adant1 1128	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑇)) = (♯‘𝑇))
5	2, 4	oveq12d 7286	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ((♯‘(𝐹 ∘ 𝑆)) + (♯‘(𝐹 ∘ 𝑇))) = ((♯‘𝑆) + (♯‘𝑇)))
6	5	oveq2d 7284	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (0..^((♯‘(𝐹 ∘ 𝑆)) + (♯‘(𝐹 ∘ 𝑇)))) = (0..^((♯‘𝑆) + (♯‘𝑇))))
7	6	mpteq1d 5173	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝑥 ∈ (0..^((♯‘(𝐹 ∘ 𝑆)) + (♯‘(𝐹 ∘ 𝑇)))) ↦ if(𝑥 ∈ (0..^(♯‘(𝐹 ∘ 𝑆))), ((𝐹 ∘ 𝑆)‘𝑥), ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘(𝐹 ∘ 𝑆)))))) = (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘(𝐹 ∘ 𝑆))), ((𝐹 ∘ 𝑆)‘𝑥), ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘(𝐹 ∘ 𝑆)))))))
8	2	oveq2d 7284	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (0..^(♯‘(𝐹 ∘ 𝑆))) = (0..^(♯‘𝑆)))
9	8	adantr 480	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → (0..^(♯‘(𝐹 ∘ 𝑆))) = (0..^(♯‘𝑆)))
10	9	eleq2d 2825	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑥 ∈ (0..^(♯‘(𝐹 ∘ 𝑆))) ↔ 𝑥 ∈ (0..^(♯‘𝑆))))
11	10	ifbid 4487	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → if(𝑥 ∈ (0..^(♯‘(𝐹 ∘ 𝑆))), ((𝐹 ∘ 𝑆)‘𝑥), ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘(𝐹 ∘ 𝑆))))) = if(𝑥 ∈ (0..^(♯‘𝑆)), ((𝐹 ∘ 𝑆)‘𝑥), ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘(𝐹 ∘ 𝑆))))))
12		wrdf 14203	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
13	12	3ad2ant1 1131	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
14	13	adantr 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
15	14	ffnd 6597	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑆 Fn (0..^(♯‘𝑆)))
16		fvco2 6859	. . . . . . . 8 ⊢ ((𝑆 Fn (0..^(♯‘𝑆)) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝐹 ∘ 𝑆)‘𝑥) = (𝐹‘(𝑆‘𝑥)))
17	15, 16	sylan 579	. . . . . . 7 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝐹 ∘ 𝑆)‘𝑥) = (𝐹‘(𝑆‘𝑥)))
18		iftrue 4470	. . . . . . . 8 ⊢ (𝑥 ∈ (0..^(♯‘𝑆)) → if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆))))) = (𝐹‘(𝑆‘𝑥)))
19	18	adantl 481	. . . . . . 7 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆))))) = (𝐹‘(𝑆‘𝑥)))
20	17, 19	eqtr4d 2782	. . . . . 6 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝐹 ∘ 𝑆)‘𝑥) = if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆))))))
21		wrdf 14203	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
22	21	3ad2ant2 1132	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
23	22	ad2antrr 722	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
24	23	ffnd 6597	. . . . . . . 8 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑇 Fn (0..^(♯‘𝑇)))
25		lencl 14217	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℕ₀)
26	25	nn0zd 12406	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℤ)
27	26	3ad2ant1 1131	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘𝑆) ∈ ℤ)
28		fzospliti 13400	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ∧ (♯‘𝑆) ∈ ℤ) → (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))))
29	28	ancoms 458	. . . . . . . . . . 11 ⊢ (((♯‘𝑆) ∈ ℤ ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))))
30	27, 29	sylan 579	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))))
31	30	orcanai 999	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))))
32		lencl 14217	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℕ₀)
33	32	nn0zd 12406	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℤ)
34	33	3ad2ant2 1132	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘𝑇) ∈ ℤ)
35	34	ad2antrr 722	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → (♯‘𝑇) ∈ ℤ)
36		fzosubel3 13429	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ∧ (♯‘𝑇) ∈ ℤ) → (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇)))
37	31, 35, 36	syl2anc 583	. . . . . . . 8 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇)))
38		fvco2 6859	. . . . . . . 8 ⊢ ((𝑇 Fn (0..^(♯‘𝑇)) ∧ (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇))) → ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘𝑆))) = (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆)))))
39	24, 37, 38	syl2anc 583	. . . . . . 7 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘𝑆))) = (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆)))))
40	2	oveq2d 7284	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝑥 − (♯‘(𝐹 ∘ 𝑆))) = (𝑥 − (♯‘𝑆)))
41	40	fveq2d 6772	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘(𝐹 ∘ 𝑆)))) = ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘𝑆))))
42	41	ad2antrr 722	. . . . . . 7 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘(𝐹 ∘ 𝑆)))) = ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘𝑆))))
43		iffalse 4473	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ (0..^(♯‘𝑆)) → if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆))))) = (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆)))))
44	43	adantl 481	. . . . . . 7 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆))))) = (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆)))))
45	39, 42, 44	3eqtr4d 2789	. . . . . 6 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘(𝐹 ∘ 𝑆)))) = if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆))))))
46	20, 45	ifeqda 4500	. . . . 5 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → if(𝑥 ∈ (0..^(♯‘𝑆)), ((𝐹 ∘ 𝑆)‘𝑥), ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘(𝐹 ∘ 𝑆))))) = if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆))))))
47	11, 46	eqtrd 2779	. . . 4 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → if(𝑥 ∈ (0..^(♯‘(𝐹 ∘ 𝑆))), ((𝐹 ∘ 𝑆)‘𝑥), ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘(𝐹 ∘ 𝑆))))) = if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆))))))
48	47	mpteq2dva 5178	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘(𝐹 ∘ 𝑆))), ((𝐹 ∘ 𝑆)‘𝑥), ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘(𝐹 ∘ 𝑆)))))) = (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆)))))))
49	7, 48	eqtr2d 2780	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆)))))) = (𝑥 ∈ (0..^((♯‘(𝐹 ∘ 𝑆)) + (♯‘(𝐹 ∘ 𝑇)))) ↦ if(𝑥 ∈ (0..^(♯‘(𝐹 ∘ 𝑆))), ((𝐹 ∘ 𝑆)‘𝑥), ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘(𝐹 ∘ 𝑆)))))))
50	14	ffvelrnda 6955	. . . 4 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → (𝑆‘𝑥) ∈ 𝐴)
51	23, 37	ffvelrnd 6956	. . . 4 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → (𝑇‘(𝑥 − (♯‘𝑆))) ∈ 𝐴)
52	50, 51	ifclda 4499	. . 3 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆)))) ∈ 𝐴)
53		ccatfval 14257	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑆 ++ 𝑇) = (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆))))))
54	53	3adant3 1130	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝑆 ++ 𝑇) = (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆))))))
55		simp3 1136	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴⟶𝐵)
56	55	feqmptd 6831	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
57		fveq2 6768	. . . 4 ⊢ (𝑦 = if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆)))) → (𝐹‘𝑦) = (𝐹‘if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆))))))
58		fvif 6784	. . . 4 ⊢ (𝐹‘if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆))))) = if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆)))))
59	57, 58	eqtrdi 2795	. . 3 ⊢ (𝑦 = if(𝑥 ∈ (0..^(♯‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (♯‘𝑆)))) → (𝐹‘𝑦) = if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆))))))
60	52, 54, 56, 59	fmptco 6995	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑆 ++ 𝑇)) = (𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ↦ if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆)))))))
61		ffun 6599	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
62	61	3ad2ant3 1133	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → Fun 𝐹)
63		simp1 1134	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑆 ∈ Word 𝐴)
64		cofunexg 7778	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑆 ∈ Word 𝐴) → (𝐹 ∘ 𝑆) ∈ V)
65	62, 63, 64	syl2anc 583	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 𝑆) ∈ V)
66		simp2 1135	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑇 ∈ Word 𝐴)
67		cofunexg 7778	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑇 ∈ Word 𝐴) → (𝐹 ∘ 𝑇) ∈ V)
68	62, 66, 67	syl2anc 583	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 𝑇) ∈ V)
69		ccatfval 14257	. . 3 ⊢ (((𝐹 ∘ 𝑆) ∈ V ∧ (𝐹 ∘ 𝑇) ∈ V) → ((𝐹 ∘ 𝑆) ++ (𝐹 ∘ 𝑇)) = (𝑥 ∈ (0..^((♯‘(𝐹 ∘ 𝑆)) + (♯‘(𝐹 ∘ 𝑇)))) ↦ if(𝑥 ∈ (0..^(♯‘(𝐹 ∘ 𝑆))), ((𝐹 ∘ 𝑆)‘𝑥), ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘(𝐹 ∘ 𝑆)))))))
70	65, 68, 69	syl2anc 583	. 2 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ((𝐹 ∘ 𝑆) ++ (𝐹 ∘ 𝑇)) = (𝑥 ∈ (0..^((♯‘(𝐹 ∘ 𝑆)) + (♯‘(𝐹 ∘ 𝑇)))) ↦ if(𝑥 ∈ (0..^(♯‘(𝐹 ∘ 𝑆))), ((𝐹 ∘ 𝑆)‘𝑥), ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘(𝐹 ∘ 𝑆)))))))
71	49, 60, 70	3eqtr4d 2789	1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑆 ++ 𝑇)) = ((𝐹 ∘ 𝑆) ++ (𝐹 ∘ 𝑇)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 843 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 Vcvv 3430 ifcif 4464 ↦ cmpt 5161 ∘ ccom 5592 Fun wfun 6424 Fn wfn 6425 ⟶wf 6426 ‘cfv 6430 (class class class)co 7268 0cc0 10855 + caddc 10858 − cmin 11188 ℤcz 12302 ..^cfzo 13364 ♯chash 14025 Word cword 14198 ++ cconcat 14254

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-card 9681 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-n0 12217 df-z 12303 df-uz 12565 df-fz 13222 df-fzo 13365 df-hash 14026 df-word 14199 df-concat 14255

This theorem is referenced by: cats1co 14550 frmdgsum 18482 frmdup1 18484 efginvrel2 19314 frgpuplem 19359 frgpup1 19362 mrsubccat 33459

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