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Theorem ccatco 14786

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 14783	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑆)) = (♯‘𝑆))
2	1	3adant2 1132	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑆)) = (♯‘𝑆))
3		lenco 14783	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑇)) = (♯‘𝑇))
4	3	3adant1 1131	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑇)) = (♯‘𝑇))
5	2, 4	oveq12d 7427	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ((♯‘(𝐹 ∘ 𝑆)) + (♯‘(𝐹 ∘ 𝑇))) = ((♯‘𝑆) + (♯‘𝑇)))
6	5	oveq2d 7425	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (0..^((♯‘(𝐹 ∘ 𝑆)) + (♯‘(𝐹 ∘ 𝑇)))) = (0..^((♯‘𝑆) + (♯‘𝑇))))
7	6	mpteq1d 5244	. . 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 7425	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (0..^(♯‘(𝐹 ∘ 𝑆))) = (0..^(♯‘𝑆)))
9	8	adantr 482	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → (0..^(♯‘(𝐹 ∘ 𝑆))) = (0..^(♯‘𝑆)))
10	9	eleq2d 2820	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑥 ∈ (0..^(♯‘(𝐹 ∘ 𝑆))) ↔ 𝑥 ∈ (0..^(♯‘𝑆))))
11	10	ifbid 4552	. . . . 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 14469	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
13	12	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
14	13	adantr 482	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
15	14	ffnd 6719	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑆 Fn (0..^(♯‘𝑆)))
16		fvco2 6989	. . . . . . . 8 ⊢ ((𝑆 Fn (0..^(♯‘𝑆)) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝐹 ∘ 𝑆)‘𝑥) = (𝐹‘(𝑆‘𝑥)))
17	15, 16	sylan 581	. . . . . . 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 4535	. . . . . . . 8 ⊢ (𝑥 ∈ (0..^(♯‘𝑆)) → if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆))))) = (𝐹‘(𝑆‘𝑥)))
19	18	adantl 483	. . . . . . 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 2776	. . . . . 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 14469	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
22	21	3ad2ant2 1135	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
23	22	ad2antrr 725	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
24	23	ffnd 6719	. . . . . . . 8 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑇 Fn (0..^(♯‘𝑇)))
25		lencl 14483	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℕ₀)
26	25	nn0zd 12584	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℤ)
27	26	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘𝑆) ∈ ℤ)
28		fzospliti 13664	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ∧ (♯‘𝑆) ∈ ℤ) → (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))))
29	28	ancoms 460	. . . . . . . . . . 11 ⊢ (((♯‘𝑆) ∈ ℤ ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))))
30	27, 29	sylan 581	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))))
31	30	orcanai 1002	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))))
32		lencl 14483	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℕ₀)
33	32	nn0zd 12584	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℤ)
34	33	3ad2ant2 1135	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘𝑇) ∈ ℤ)
35	34	ad2antrr 725	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → (♯‘𝑇) ∈ ℤ)
36		fzosubel3 13693	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ∧ (♯‘𝑇) ∈ ℤ) → (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇)))
37	31, 35, 36	syl2anc 585	. . . . . . . 8 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇)))
38		fvco2 6989	. . . . . . . 8 ⊢ ((𝑇 Fn (0..^(♯‘𝑇)) ∧ (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇))) → ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘𝑆))) = (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆)))))
39	24, 37, 38	syl2anc 585	. . . . . . 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 7425	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝑥 − (♯‘(𝐹 ∘ 𝑆))) = (𝑥 − (♯‘𝑆)))
41	40	fveq2d 6896	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘(𝐹 ∘ 𝑆)))) = ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘𝑆))))
42	41	ad2antrr 725	. . . . . . 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 4538	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ (0..^(♯‘𝑆)) → if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆))))) = (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆)))))
44	43	adantl 483	. . . . . . 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 2783	. . . . . 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 4565	. . . . 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 2773	. . . 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 5249	. . 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 2774	. 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 7087	. . . 4 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → (𝑆‘𝑥) ∈ 𝐴)
51	23, 37	ffvelcdmd 7088	. . . 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 4564	. . 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 14523	. . . 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 1133	. . 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 1139	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴⟶𝐵)
56	55	feqmptd 6961	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
57		fveq2 6892	. . . 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 6908	. . . 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 2789	. . 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 7127	. 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 6721	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
62	61	3ad2ant3 1136	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → Fun 𝐹)
63		simp1 1137	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑆 ∈ Word 𝐴)
64		cofunexg 7935	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑆 ∈ Word 𝐴) → (𝐹 ∘ 𝑆) ∈ V)
65	62, 63, 64	syl2anc 585	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 𝑆) ∈ V)
66		simp2 1138	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑇 ∈ Word 𝐴)
67		cofunexg 7935	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑇 ∈ Word 𝐴) → (𝐹 ∘ 𝑇) ∈ V)
68	62, 66, 67	syl2anc 585	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 𝑇) ∈ V)
69		ccatfval 14523	. . 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 585	. 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 2783	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 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ifcif 4529 ↦ cmpt 5232 ∘ ccom 5681 Fun wfun 6538 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 0cc0 11110 + caddc 11113 − cmin 11444 ℤcz 12558 ..^cfzo 13627 ♯chash 14290 Word cword 14464 ++ cconcat 14520

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 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 theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-concat 14521

This theorem is referenced by: cats1co 14807 frmdgsum 18743 frmdup1 18745 efginvrel2 19595 frgpuplem 19640 frgpup1 19643 mrsubccat 34509

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