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Theorem ccatco 14883

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 14880	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑆)) = (♯‘𝑆))
2	1	3adant2 1149	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑆)) = (♯‘𝑆))
3		lenco 14880	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑇)) = (♯‘𝑇))
4	3	3adant1 1148	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑇)) = (♯‘𝑇))
5	2, 4	oveq12d 7434	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ((♯‘(𝐹 ∘ 𝑆)) + (♯‘(𝐹 ∘ 𝑇))) = ((♯‘𝑆) + (♯‘𝑇)))
6	5	oveq2d 7432	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (0..^((♯‘(𝐹 ∘ 𝑆)) + (♯‘(𝐹 ∘ 𝑇)))) = (0..^((♯‘𝑆) + (♯‘𝑇))))
7	6	mpteq1d 5204	. . 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 7432	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (0..^(♯‘(𝐹 ∘ 𝑆))) = (0..^(♯‘𝑆)))
9	8	adantr 486	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → (0..^(♯‘(𝐹 ∘ 𝑆))) = (0..^(♯‘𝑆)))
10	9	eleq2d 2852	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑥 ∈ (0..^(♯‘(𝐹 ∘ 𝑆))) ↔ 𝑥 ∈ (0..^(♯‘𝑆))))
11	10	ifbid 4514	. . . . 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 14566	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
13	12	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
14	13	adantr 486	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
15	14	ffnd 6710	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑆 Fn (0..^(♯‘𝑆)))
16		fvco2 6982	. . . . . . . 8 ⊢ ((𝑆 Fn (0..^(♯‘𝑆)) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝐹 ∘ 𝑆)‘𝑥) = (𝐹‘(𝑆‘𝑥)))
17	15, 16	sylan 592	. . . . . . 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 4496	. . . . . . . 8 ⊢ (𝑥 ∈ (0..^(♯‘𝑆)) → if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆))))) = (𝐹‘(𝑆‘𝑥)))
19	18	adantl 487	. . . . . . 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 2804	. . . . . 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 14566	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
22	21	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
23	22	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
24	23	ffnd 6710	. . . . . . . 8 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑇 Fn (0..^(♯‘𝑇)))
25		lencl 14581	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℕ₀)
26	25	nn0zd 12626	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℤ)
27	26	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘𝑆) ∈ ℤ)
28		fzospliti 13731	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ∧ (♯‘𝑆) ∈ ℤ) → (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))))
29	28	ancoms 464	. . . . . . . . . . 11 ⊢ (((♯‘𝑆) ∈ ℤ ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))))
30	27, 29	sylan 592	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))))
31	30	orcanai 1018	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))))
32		lencl 14581	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℕ₀)
33	32	nn0zd 12626	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℤ)
34	33	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘𝑇) ∈ ℤ)
35	34	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → (♯‘𝑇) ∈ ℤ)
36		fzosubel3 13766	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ∧ (♯‘𝑇) ∈ ℤ) → (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇)))
37	31, 35, 36	syl2anc 596	. . . . . . . 8 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇)))
38		fvco2 6982	. . . . . . . 8 ⊢ ((𝑇 Fn (0..^(♯‘𝑇)) ∧ (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇))) → ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘𝑆))) = (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆)))))
39	24, 37, 38	syl2anc 596	. . . . . . 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 7432	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝑥 − (♯‘(𝐹 ∘ 𝑆))) = (𝑥 − (♯‘𝑆)))
41	40	fveq2d 6889	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘(𝐹 ∘ 𝑆)))) = ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘𝑆))))
42	41	ad2antrr 739	. . . . . . 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 4499	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ (0..^(♯‘𝑆)) → if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆))))) = (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆)))))
44	43	adantl 487	. . . . . . 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 2811	. . . . . 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 4527	. . . . 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 2801	. . . 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 5207	. . 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 2802	. 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	ffvelcdmda 7083	. . . 4 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → (𝑆‘𝑥) ∈ 𝐴)
51	23, 37	ffvelcdmd 7084	. . . 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 4526	. . 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 14621	. . . 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 1150	. . 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 1156	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴⟶𝐵)
56	55	feqmptd 6953	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
57		fveq2 6885	. . . 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 6901	. . . 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 2817	. . 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 7129	. 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 6712	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
62	61	3ad2ant3 1153	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → Fun 𝐹)
63		simp1 1154	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑆 ∈ Word 𝐴)
64		cofunexg 7948	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑆 ∈ Word 𝐴) → (𝐹 ∘ 𝑆) ∈ V)
65	62, 63, 64	syl2anc 596	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 𝑆) ∈ V)
66		simp2 1155	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑇 ∈ Word 𝐴)
67		cofunexg 7948	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑇 ∈ Word 𝐴) → (𝐹 ∘ 𝑇) ∈ V)
68	62, 66, 67	syl2anc 596	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 𝑇) ∈ V)
69		ccatfval 14621	. . 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 596	. 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 2811	1 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ (𝑆 ++ 𝑇)) = ((𝐹 ∘ 𝑆) ++ (𝐹 ∘ 𝑇)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 Vcvv 3458 ifcif 4490 ↦ cmpt 5195 ∘ ccom 5668 Fun wfun 6534 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7416 0cc0 11110 + caddc 11113 − cmin 11451 ℤcz 12601 ..^cfzo 13693 ♯chash 14377 Word cword 14561 ++ cconcat 14618

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5241 ax-sep 5260 ax-nul 5272 ax-pow 5339 ax-pr 5407 ax-un 7738 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4876 df-int 4916 df-iun 4961 df-br 5113 df-opab 5177 df-mpt 5196 df-tr 5222 df-id 5559 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7865 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-card 9936 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11453 df-neg 11454 df-nn 12244 df-n0 12515 df-z 12602 df-uz 12873 df-fz 13546 df-fzo 13694 df-hash 14378 df-word 14562 df-concat 14619

This theorem is used by: cats1co 14904 frmdgsum 18931 frmdup1 18933 efginvrel2 19807 frgpuplem 19852 frgpup1 19855 ccatws1f1olast 33285 mrsubccat 36022

W3C validator