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Theorem ccatco 14365

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 14362	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑆)) = (♯‘𝑆))
2	1	3adant2 1133	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑆)) = (♯‘𝑆))
3		lenco 14362	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑇)) = (♯‘𝑇))
4	3	3adant1 1132	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘(𝐹 ∘ 𝑇)) = (♯‘𝑇))
5	2, 4	oveq12d 7209	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ((♯‘(𝐹 ∘ 𝑆)) + (♯‘(𝐹 ∘ 𝑇))) = ((♯‘𝑆) + (♯‘𝑇)))
6	5	oveq2d 7207	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (0..^((♯‘(𝐹 ∘ 𝑆)) + (♯‘(𝐹 ∘ 𝑇)))) = (0..^((♯‘𝑆) + (♯‘𝑇))))
7	6	mpteq1d 5129	. . 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 7207	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (0..^(♯‘(𝐹 ∘ 𝑆))) = (0..^(♯‘𝑆)))
9	8	adantr 484	. . . . . . 7 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → (0..^(♯‘(𝐹 ∘ 𝑆))) = (0..^(♯‘𝑆)))
10	9	eleq2d 2816	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑥 ∈ (0..^(♯‘(𝐹 ∘ 𝑆))) ↔ 𝑥 ∈ (0..^(♯‘𝑆))))
11	10	ifbid 4448	. . . . 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 14039	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
13	12	3ad2ant1 1135	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
14	13	adantr 484	. . . . . . . . 9 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑆:(0..^(♯‘𝑆))⟶𝐴)
15	14	ffnd 6524	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → 𝑆 Fn (0..^(♯‘𝑆)))
16		fvco2 6786	. . . . . . . 8 ⊢ ((𝑆 Fn (0..^(♯‘𝑆)) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → ((𝐹 ∘ 𝑆)‘𝑥) = (𝐹‘(𝑆‘𝑥)))
17	15, 16	sylan 583	. . . . . . 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 4431	. . . . . . . 8 ⊢ (𝑥 ∈ (0..^(♯‘𝑆)) → if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆))))) = (𝐹‘(𝑆‘𝑥)))
19	18	adantl 485	. . . . . . 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 2774	. . . . . 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 14039	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
22	21	3ad2ant2 1136	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
23	22	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑇:(0..^(♯‘𝑇))⟶𝐴)
24	23	ffnd 6524	. . . . . . . 8 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑇 Fn (0..^(♯‘𝑇)))
25		lencl 14053	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℕ₀)
26	25	nn0zd 12245	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Word 𝐴 → (♯‘𝑆) ∈ ℤ)
27	26	3ad2ant1 1135	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘𝑆) ∈ ℤ)
28		fzospliti 13239	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇))) ∧ (♯‘𝑆) ∈ ℤ) → (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))))
29	28	ancoms 462	. . . . . . . . . . 11 ⊢ (((♯‘𝑆) ∈ ℤ ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))))
30	27, 29	sylan 583	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) → (𝑥 ∈ (0..^(♯‘𝑆)) ∨ 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇)))))
31	30	orcanai 1003	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → 𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))))
32		lencl 14053	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℕ₀)
33	32	nn0zd 12245	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → (♯‘𝑇) ∈ ℤ)
34	33	3ad2ant2 1136	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (♯‘𝑇) ∈ ℤ)
35	34	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → (♯‘𝑇) ∈ ℤ)
36		fzosubel3 13268	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((♯‘𝑆)..^((♯‘𝑆) + (♯‘𝑇))) ∧ (♯‘𝑇) ∈ ℤ) → (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇)))
37	31, 35, 36	syl2anc 587	. . . . . . . 8 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(♯‘𝑆))) → (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇)))
38		fvco2 6786	. . . . . . . 8 ⊢ ((𝑇 Fn (0..^(♯‘𝑇)) ∧ (𝑥 − (♯‘𝑆)) ∈ (0..^(♯‘𝑇))) → ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘𝑆))) = (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆)))))
39	24, 37, 38	syl2anc 587	. . . . . . 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 7207	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝑥 − (♯‘(𝐹 ∘ 𝑆))) = (𝑥 − (♯‘𝑆)))
41	40	fveq2d 6699	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘(𝐹 ∘ 𝑆)))) = ((𝐹 ∘ 𝑇)‘(𝑥 − (♯‘𝑆))))
42	41	ad2antrr 726	. . . . . . 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 4434	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ (0..^(♯‘𝑆)) → if(𝑥 ∈ (0..^(♯‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆))))) = (𝐹‘(𝑇‘(𝑥 − (♯‘𝑆)))))
44	43	adantl 485	. . . . . . 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 2781	. . . . . 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 4461	. . . . 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 2771	. . . 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 5135	. . 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 2772	. 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 6882	. . . 4 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((♯‘𝑆) + (♯‘𝑇)))) ∧ 𝑥 ∈ (0..^(♯‘𝑆))) → (𝑆‘𝑥) ∈ 𝐴)
51	23, 37	ffvelrnd 6883	. . . 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 4460	. . 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 14093	. . . 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 1134	. . 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 1140	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴⟶𝐵)
56	55	feqmptd 6758	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
57		fveq2 6695	. . . 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 6711	. . . 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 2787	. . 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 6922	. 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 6526	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
62	61	3ad2ant3 1137	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → Fun 𝐹)
63		simp1 1138	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑆 ∈ Word 𝐴)
64		cofunexg 7700	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑆 ∈ Word 𝐴) → (𝐹 ∘ 𝑆) ∈ V)
65	62, 63, 64	syl2anc 587	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 𝑆) ∈ V)
66		simp2 1139	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑇 ∈ Word 𝐴)
67		cofunexg 7700	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑇 ∈ Word 𝐴) → (𝐹 ∘ 𝑇) ∈ V)
68	62, 66, 67	syl2anc 587	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 𝑇) ∈ V)
69		ccatfval 14093	. . 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 587	. 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 2781	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 399 ∨ wo 847 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ifcif 4425 ↦ cmpt 5120 ∘ ccom 5540 Fun wfun 6352 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 0cc0 10694 + caddc 10697 − cmin 11027 ℤcz 12141 ..^cfzo 13203 ♯chash 13861 Word cword 14034 ++ cconcat 14090

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-hash 13862 df-word 14035 df-concat 14091

This theorem is referenced by: cats1co 14386 frmdgsum 18243 frmdup1 18245 efginvrel2 19071 frgpuplem 19116 frgpup1 19119 mrsubccat 33147

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