ccatco - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ccatco		Structured version Visualization version GIF version

Theorem ccatco 13381

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 13378	. . . . . . 7 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (#‘(𝐹 ∘ 𝑆)) = (#‘𝑆))
2	1	3adant2 1073	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (#‘(𝐹 ∘ 𝑆)) = (#‘𝑆))
3		lenco 13378	. . . . . . 7 ⊢ ((𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (#‘(𝐹 ∘ 𝑇)) = (#‘𝑇))
4	3	3adant1 1072	. . . . . 6 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (#‘(𝐹 ∘ 𝑇)) = (#‘𝑇))
5	2, 4	oveq12d 6545	. . . . 5 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ((#‘(𝐹 ∘ 𝑆)) + (#‘(𝐹 ∘ 𝑇))) = ((#‘𝑆) + (#‘𝑇)))
6	5	oveq2d 6543	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (0..^((#‘(𝐹 ∘ 𝑆)) + (#‘(𝐹 ∘ 𝑇)))) = (0..^((#‘𝑆) + (#‘𝑇))))
7	6	mpteq1d 4661	. . 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 6543	. . . . . . . 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 2673	. . . . . 6 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇)))) → (𝑥 ∈ (0..^(#‘(𝐹 ∘ 𝑆))) ↔ 𝑥 ∈ (0..^(#‘𝑆))))
11	10	ifbid 4058	. . . . 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 13114	. . . . . . . . . . 11 ⊢ (𝑆 ∈ Word 𝐴 → 𝑆:(0..^(#‘𝑆))⟶𝐴)
13	12	3ad2ant1 1075	. . . . . . . . . 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		ffn 5944	. . . . . . . . 9 ⊢ (𝑆:(0..^(#‘𝑆))⟶𝐴 → 𝑆 Fn (0..^(#‘𝑆)))
16	14, 15	syl 17	. . . . . . . 8 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇)))) → 𝑆 Fn (0..^(#‘𝑆)))
17		fvco2 6168	. . . . . . . 8 ⊢ ((𝑆 Fn (0..^(#‘𝑆)) ∧ 𝑥 ∈ (0..^(#‘𝑆))) → ((𝐹 ∘ 𝑆)‘𝑥) = (𝐹‘(𝑆‘𝑥)))
18	16, 17	sylan 487	. . . . . . 7 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇)))) ∧ 𝑥 ∈ (0..^(#‘𝑆))) → ((𝐹 ∘ 𝑆)‘𝑥) = (𝐹‘(𝑆‘𝑥)))
19		iftrue 4042	. . . . . . . 8 ⊢ (𝑥 ∈ (0..^(#‘𝑆)) → if(𝑥 ∈ (0..^(#‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (#‘𝑆))))) = (𝐹‘(𝑆‘𝑥)))
20	19	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)))
21	18, 20	eqtr4d 2647	. . . . . 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))))))
22		wrdf 13114	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → 𝑇:(0..^(#‘𝑇))⟶𝐴)
23	22	3ad2ant2 1076	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑇:(0..^(#‘𝑇))⟶𝐴)
24	23	ad2antrr 758	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(#‘𝑆))) → 𝑇:(0..^(#‘𝑇))⟶𝐴)
25		ffn 5944	. . . . . . . . 9 ⊢ (𝑇:(0..^(#‘𝑇))⟶𝐴 → 𝑇 Fn (0..^(#‘𝑇)))
26	24, 25	syl 17	. . . . . . . 8 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(#‘𝑆))) → 𝑇 Fn (0..^(#‘𝑇)))
27		lencl 13128	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ Word 𝐴 → (#‘𝑆) ∈ ℕ₀)
28	27	nn0zd 11315	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Word 𝐴 → (#‘𝑆) ∈ ℤ)
29	28	3ad2ant1 1075	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (#‘𝑆) ∈ ℤ)
30		fzospliti 12327	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇))) ∧ (#‘𝑆) ∈ ℤ) → (𝑥 ∈ (0..^(#‘𝑆)) ∨ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))))
31	30	ancoms 468	. . . . . . . . . . 11 ⊢ (((#‘𝑆) ∈ ℤ ∧ 𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇)))) → (𝑥 ∈ (0..^(#‘𝑆)) ∨ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))))
32	29, 31	sylan 487	. . . . . . . . . 10 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇)))) → (𝑥 ∈ (0..^(#‘𝑆)) ∨ 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇)))))
33	32	orcanai 950	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(#‘𝑆))) → 𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇))))
34		lencl 13128	. . . . . . . . . . . 12 ⊢ (𝑇 ∈ Word 𝐴 → (#‘𝑇) ∈ ℕ₀)
35	34	nn0zd 11315	. . . . . . . . . . 11 ⊢ (𝑇 ∈ Word 𝐴 → (#‘𝑇) ∈ ℤ)
36	35	3ad2ant2 1076	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (#‘𝑇) ∈ ℤ)
37	36	ad2antrr 758	. . . . . . . . 9 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(#‘𝑆))) → (#‘𝑇) ∈ ℤ)
38		fzosubel3 12354	. . . . . . . . 9 ⊢ ((𝑥 ∈ ((#‘𝑆)..^((#‘𝑆) + (#‘𝑇))) ∧ (#‘𝑇) ∈ ℤ) → (𝑥 − (#‘𝑆)) ∈ (0..^(#‘𝑇)))
39	33, 37, 38	syl2anc 691	. . . . . . . 8 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(#‘𝑆))) → (𝑥 − (#‘𝑆)) ∈ (0..^(#‘𝑇)))
40		fvco2 6168	. . . . . . . 8 ⊢ ((𝑇 Fn (0..^(#‘𝑇)) ∧ (𝑥 − (#‘𝑆)) ∈ (0..^(#‘𝑇))) → ((𝐹 ∘ 𝑇)‘(𝑥 − (#‘𝑆))) = (𝐹‘(𝑇‘(𝑥 − (#‘𝑆)))))
41	26, 39, 40	syl2anc 691	. . . . . . 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)))))
42	2	oveq2d 6543	. . . . . . . . 9 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝑥 − (#‘(𝐹 ∘ 𝑆))) = (𝑥 − (#‘𝑆)))
43	42	fveq2d 6092	. . . . . . . 8 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → ((𝐹 ∘ 𝑇)‘(𝑥 − (#‘(𝐹 ∘ 𝑆)))) = ((𝐹 ∘ 𝑇)‘(𝑥 − (#‘𝑆))))
44	43	ad2antrr 758	. . . . . . 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))))
45		iffalse 4045	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ (0..^(#‘𝑆)) → if(𝑥 ∈ (0..^(#‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (#‘𝑆))))) = (𝐹‘(𝑇‘(𝑥 − (#‘𝑆)))))
46	45	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)))))
47	41, 44, 46	3eqtr4d 2654	. . . . . 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))))))
48	21, 47	ifeqda 4071	. . . . 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))))))
49	11, 48	eqtrd 2644	. . . 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))))))
50	49	mpteq2dva 4667	. . 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)))))))
51	7, 50	eqtr2d 2645	. 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)))))))
52	14	ffvelrnda 6252	. . . 4 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇)))) ∧ 𝑥 ∈ (0..^(#‘𝑆))) → (𝑆‘𝑥) ∈ 𝐴)
53	24, 39	ffvelrnd 6253	. . . 4 ⊢ ((((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇)))) ∧ ¬ 𝑥 ∈ (0..^(#‘𝑆))) → (𝑇‘(𝑥 − (#‘𝑆))) ∈ 𝐴)
54	52, 53	ifclda 4070	. . 3 ⊢ (((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) ∧ 𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇)))) → if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆)))) ∈ 𝐴)
55		ccatfval 13160	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴) → (𝑆 ++ 𝑇) = (𝑥 ∈ (0..^((#‘𝑆) + (#‘𝑇))) ↦ if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆))))))
56	55	3adant3 1074	. . 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))))))
57		simp3 1056	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴⟶𝐵)
58	57	feqmptd 6144	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
59		fveq2 6088	. . . 4 ⊢ (𝑦 = if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆)))) → (𝐹‘𝑦) = (𝐹‘if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆))))))
60		fvif 6099	. . . 4 ⊢ (𝐹‘if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆))))) = if(𝑥 ∈ (0..^(#‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (#‘𝑆)))))
61	59, 60	syl6eq 2660	. . 3 ⊢ (𝑦 = if(𝑥 ∈ (0..^(#‘𝑆)), (𝑆‘𝑥), (𝑇‘(𝑥 − (#‘𝑆)))) → (𝐹‘𝑦) = if(𝑥 ∈ (0..^(#‘𝑆)), (𝐹‘(𝑆‘𝑥)), (𝐹‘(𝑇‘(𝑥 − (#‘𝑆))))))
62	54, 56, 58, 61	fmptco 6288	. 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)))))))
63		ffun 5947	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
64	63	3ad2ant3 1077	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → Fun 𝐹)
65		simp1 1054	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑆 ∈ Word 𝐴)
66		cofunexg 7001	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑆 ∈ Word 𝐴) → (𝐹 ∘ 𝑆) ∈ V)
67	64, 65, 66	syl2anc 691	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 𝑆) ∈ V)
68		simp2 1055	. . . 4 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝑇 ∈ Word 𝐴)
69		cofunexg 7001	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑇 ∈ Word 𝐴) → (𝐹 ∘ 𝑇) ∈ V)
70	64, 68, 69	syl2anc 691	. . 3 ⊢ ((𝑆 ∈ Word 𝐴 ∧ 𝑇 ∈ Word 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 𝑇) ∈ V)
71		ccatfval 13160	. . 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)))))))
72	67, 70, 71	syl2anc 691	. 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)))))))
73	51, 62, 72	3eqtr4d 2654	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 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ifcif 4036 ↦ cmpt 4638 ∘ ccom 5032 Fun wfun 5784 Fn wfn 5785 ⟶wf 5786 ‘cfv 5790 (class class class)co 6527 0cc0 9793 + caddc 9796 − cmin 10118 ℤcz 11213 ..^cfzo 12292 #chash 12937 Word cword 13095 ++ cconcat 13097

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 4694 ax-sep 4704 ax-nul 4712 ax-pow 4764 ax-pr 4828 ax-un 6825 ax-cnex 9849 ax-resscn 9850 ax-1cn 9851 ax-icn 9852 ax-addcl 9853 ax-addrcl 9854 ax-mulcl 9855 ax-mulrcl 9856 ax-mulcom 9857 ax-addass 9858 ax-mulass 9859 ax-distr 9860 ax-i2m1 9861 ax-1ne0 9862 ax-1rid 9863 ax-rnegex 9864 ax-rrecex 9865 ax-cnre 9866 ax-pre-lttri 9867 ax-pre-lttrn 9868 ax-pre-ltadd 9869 ax-pre-mulgt0 9870

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 4368 df-int 4406 df-iun 4452 df-br 4579 df-opab 4639 df-mpt 4640 df-tr 4676 df-eprel 4939 df-id 4943 df-po 4949 df-so 4950 df-fr 4987 df-we 4989 df-xp 5034 df-rel 5035 df-cnv 5036 df-co 5037 df-dm 5038 df-rn 5039 df-res 5040 df-ima 5041 df-pred 5583 df-ord 5629 df-on 5630 df-lim 5631 df-suc 5632 df-iota 5754 df-fun 5792 df-fn 5793 df-f 5794 df-f1 5795 df-fo 5796 df-f1o 5797 df-fv 5798 df-riota 6489 df-ov 6530 df-oprab 6531 df-mpt2 6532 df-om 6936 df-1st 7037 df-2nd 7038 df-wrecs 7272 df-recs 7333 df-rdg 7371 df-1o 7425 df-oadd 7429 df-er 7607 df-en 7820 df-dom 7821 df-sdom 7822 df-fin 7823 df-card 8626 df-pnf 9933 df-mnf 9934 df-xr 9935 df-ltxr 9936 df-le 9937 df-sub 10120 df-neg 10121 df-nn 10871 df-n0 11143 df-z 11214 df-uz 11523 df-fz 12156 df-fzo 12293 df-hash 12938 df-word 13103 df-concat 13105

This theorem is referenced by: cats1co 13401 frmdgsum 17171 frmdup1 17173 efginvrel2 17912 frgpuplem 17957 frgpup1 17960 mrsubccat 30463

W3C validator