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Theorem ccatalpha 14767

Description: A concatenation of two arbitrary words is a word over an alphabet iff the symbols of both words belong to the alphabet. (Contributed by AV, 28-Feb-2021.)

**Assertion**
Ref	Expression
ccatalpha	⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝐴 ++ 𝐵) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆)))

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

Step	Hyp	Ref	Expression
1		ccatfval 14745	. . . 4 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (𝐴 ++ 𝐵) = (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))))
2	1	eleq1d 2849	. . 3 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝐴 ++ 𝐵) ∈ Word 𝑆 ↔ (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))) ∈ Word 𝑆))
3		wrdf 14690	. . . 4 ⊢ ((𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))) ∈ Word 𝑆 → (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))):(0..^(♯‘(𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))))⟶𝑆)
4		funmpt 6598	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))
5		fzofi 14144	. . . . . . . . . . 11 ⊢ (0..^((♯‘𝐴) + (♯‘𝐵))) ∈ Fin
6		mptfi 9368	. . . . . . . . . . 11 ⊢ ((0..^((♯‘𝐴) + (♯‘𝐵))) ∈ Fin → (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))) ∈ Fin)
7	5, 6	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))) ∈ Fin
8		hashfun 14609	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))) ∈ Fin → (Fun (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))) ↔ (♯‘(𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))) = (♯‘dom (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))))))
9	7, 8	mp1i 14	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (Fun (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))) ↔ (♯‘(𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))) = (♯‘dom (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))))))
10	4, 9	mpbii 236	. . . . . . . 8 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (♯‘(𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))) = (♯‘dom (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))))
11		dmmptg 6256	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵)))if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ V → dom (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))) = (0..^((♯‘𝐴) + (♯‘𝐵))))
12		fvex 6921	. . . . . . . . . . . . 13 ⊢ (𝐴‘𝑥) ∈ V
13		fvex 6921	. . . . . . . . . . . . 13 ⊢ (𝐵‘(𝑥 − (♯‘𝐴))) ∈ V
14	12, 13	ifex 4536	. . . . . . . . . . . 12 ⊢ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ V
15	14	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) → if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ V)
16	11, 15	mprg 3086	. . . . . . . . . 10 ⊢ dom (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))) = (0..^((♯‘𝐴) + (♯‘𝐵)))
17	16	fveq2i 6911	. . . . . . . . 9 ⊢ (♯‘dom (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))) = (♯‘(0..^((♯‘𝐴) + (♯‘𝐵))))
18		lencl 14705	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word V → (♯‘𝐴) ∈ ℕ₀)
19		lencl 14705	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Word V → (♯‘𝐵) ∈ ℕ₀)
20		nn0addcl 12668	. . . . . . . . . . 11 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℕ₀)
21	18, 19, 20	syl2an 608	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℕ₀)
22		hashfzo0 14602	. . . . . . . . . 10 ⊢ (((♯‘𝐴) + (♯‘𝐵)) ∈ ℕ₀ → (♯‘(0..^((♯‘𝐴) + (♯‘𝐵)))) = ((♯‘𝐴) + (♯‘𝐵)))
23	21, 22	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (♯‘(0..^((♯‘𝐴) + (♯‘𝐵)))) = ((♯‘𝐴) + (♯‘𝐵)))
24	17, 23	eqtrid 2810	. . . . . . . 8 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (♯‘dom (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))) = ((♯‘𝐴) + (♯‘𝐵)))
25	10, 24	eqtrd 2798	. . . . . . 7 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (♯‘(𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))) = ((♯‘𝐴) + (♯‘𝐵)))
26	25	oveq2d 7459	. . . . . 6 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (0..^(♯‘(𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))))) = (0..^((♯‘𝐴) + (♯‘𝐵))))
27	26	feq2d 6715	. . . . 5 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))):(0..^(♯‘(𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))))⟶𝑆 ↔ (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))):(0..^((♯‘𝐴) + (♯‘𝐵)))⟶𝑆))
28		eqid 2763	. . . . . . 7 ⊢ (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))) = (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))
29	28	fmpt 7134	. . . . . 6 ⊢ (∀𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵)))if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ 𝑆 ↔ (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))):(0..^((♯‘𝐴) + (♯‘𝐵)))⟶𝑆)
30		simpl 488	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → 𝐴 ∈ Word V)
31		nn0cn 12643	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (♯‘𝐴) ∈ ℂ)
32		nn0cn 12643	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐵) ∈ ℕ₀ → (♯‘𝐵) ∈ ℂ)
33		addcom 11523	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝐴) ∈ ℂ ∧ (♯‘𝐵) ∈ ℂ) → ((♯‘𝐴) + (♯‘𝐵)) = ((♯‘𝐵) + (♯‘𝐴)))
34	31, 32, 33	syl2an 608	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀) → ((♯‘𝐴) + (♯‘𝐵)) = ((♯‘𝐵) + (♯‘𝐴)))
35		nn0z 12744	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (♯‘𝐴) ∈ ℤ)
36	35	anim1ci 628	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀) → ((♯‘𝐵) ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℤ))
37		nn0pzuz 13059	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝐵) ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℤ) → ((♯‘𝐵) + (♯‘𝐴)) ∈ (ℤ_≥‘(♯‘𝐴)))
38	36, 37	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀) → ((♯‘𝐵) + (♯‘𝐴)) ∈ (ℤ_≥‘(♯‘𝐴)))
39	34, 38	eqeltrd 2864	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀) → ((♯‘𝐴) + (♯‘𝐵)) ∈ (ℤ_≥‘(♯‘𝐴)))
40	18, 19, 39	syl2an 608	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((♯‘𝐴) + (♯‘𝐵)) ∈ (ℤ_≥‘(♯‘𝐴)))
41		fzoss2 13849	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝐴) + (♯‘𝐵)) ∈ (ℤ_≥‘(♯‘𝐴)) → (0..^(♯‘𝐴)) ⊆ (0..^((♯‘𝐴) + (♯‘𝐵))))
42	40, 41	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (0..^(♯‘𝐴)) ⊆ (0..^((♯‘𝐴) + (♯‘𝐵))))
43	42	sselda 3913	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐴))) → 𝑦 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))))
44		eleq1 2852	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (0..^(♯‘𝐴)) ↔ 𝑦 ∈ (0..^(♯‘𝐴))))
45		fveq2 6908	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝐴‘𝑥) = (𝐴‘𝑦))
46		fvoveq1 7466	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝐵‘(𝑥 − (♯‘𝐴))) = (𝐵‘(𝑦 − (♯‘𝐴))))
47	44, 45, 46	ifbieq12d 4514	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) = if(𝑦 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (♯‘𝐴)))))
48	47	eleq1d 2849	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ 𝑆 ↔ if(𝑦 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (♯‘𝐴)))) ∈ 𝑆))
49	48	rspcv 3576	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) → (∀𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵)))if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ 𝑆 → if(𝑦 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (♯‘𝐴)))) ∈ 𝑆))
50	43, 49	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐴))) → (∀𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵)))if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ 𝑆 → if(𝑦 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (♯‘𝐴)))) ∈ 𝑆))
51		iftrue 4491	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0..^(♯‘𝐴)) → if(𝑦 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (♯‘𝐴)))) = (𝐴‘𝑦))
52	51	adantl 487	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐴))) → if(𝑦 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (♯‘𝐴)))) = (𝐴‘𝑦))
53	52	eleq1d 2849	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐴))) → (if(𝑦 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (♯‘𝐴)))) ∈ 𝑆 ↔ (𝐴‘𝑦) ∈ 𝑆))
54	50, 53	sylibd 242	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐴))) → (∀𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵)))if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ 𝑆 → (𝐴‘𝑦) ∈ 𝑆))
55	54	impancom 457	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ ∀𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵)))if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ 𝑆) → (𝑦 ∈ (0..^(♯‘𝐴)) → (𝐴‘𝑦) ∈ 𝑆))
56	55	ralrimiv 3157	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ ∀𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵)))if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ 𝑆) → ∀𝑦 ∈ (0..^(♯‘𝐴))(𝐴‘𝑦) ∈ 𝑆)
57		iswrdsymb 14703	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ ∀𝑦 ∈ (0..^(♯‘𝐴))(𝐴‘𝑦) ∈ 𝑆) → 𝐴 ∈ Word 𝑆)
58	30, 56, 57	syl2an2r 698	. . . . . . . 8 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ ∀𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵)))if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ 𝑆) → 𝐴 ∈ Word 𝑆)
59		simpr 490	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → 𝐵 ∈ Word V)
60		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → 𝑦 ∈ (0..^(♯‘𝐵)))
61	18	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (♯‘𝐴) ∈ ℕ₀)
62	61	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (♯‘𝐴) ∈ ℕ₀)
63		elincfzoext 13885	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ (0..^(♯‘𝐵)) ∧ (♯‘𝐴) ∈ ℕ₀) → (𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐵) + (♯‘𝐴))))
64	60, 62, 63	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐵) + (♯‘𝐴))))
65	18	nn0cnd 12696	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ Word V → (♯‘𝐴) ∈ ℂ)
66	19	nn0cnd 12696	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ Word V → (♯‘𝐵) ∈ ℂ)
67	65, 66, 33	syl2an 608	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((♯‘𝐴) + (♯‘𝐵)) = ((♯‘𝐵) + (♯‘𝐴)))
68	67	oveq2d 7459	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (0..^((♯‘𝐴) + (♯‘𝐵))) = (0..^((♯‘𝐵) + (♯‘𝐴))))
69	68	eleq2d 2850	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↔ (𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐵) + (♯‘𝐴)))))
70	69	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ((𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↔ (𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐵) + (♯‘𝐴)))))
71	64, 70	mpbird 260	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐴) + (♯‘𝐵))))
72		eleq1 2852	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + (♯‘𝐴)) → (𝑥 ∈ (0..^(♯‘𝐴)) ↔ (𝑦 + (♯‘𝐴)) ∈ (0..^(♯‘𝐴))))
73		fveq2 6908	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + (♯‘𝐴)) → (𝐴‘𝑥) = (𝐴‘(𝑦 + (♯‘𝐴))))
74		fvoveq1 7466	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + (♯‘𝐴)) → (𝐵‘(𝑥 − (♯‘𝐴))) = (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))))
75	72, 73, 74	ifbieq12d 4514	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 + (♯‘𝐴)) → if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) = if((𝑦 + (♯‘𝐴)) ∈ (0..^(♯‘𝐴)), (𝐴‘(𝑦 + (♯‘𝐴))), (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴)))))
76	75	eleq1d 2849	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 + (♯‘𝐴)) → (if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ 𝑆 ↔ if((𝑦 + (♯‘𝐴)) ∈ (0..^(♯‘𝐴)), (𝐴‘(𝑦 + (♯‘𝐴))), (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴)))) ∈ 𝑆))
77	76	rspcv 3576	. . . . . . . . . . . . 13 ⊢ ((𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) → (∀𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵)))if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ 𝑆 → if((𝑦 + (♯‘𝐴)) ∈ (0..^(♯‘𝐴)), (𝐴‘(𝑦 + (♯‘𝐴))), (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴)))) ∈ 𝑆))
78	71, 77	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (∀𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵)))if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ 𝑆 → if((𝑦 + (♯‘𝐴)) ∈ (0..^(♯‘𝐴)), (𝐴‘(𝑦 + (♯‘𝐴))), (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴)))) ∈ 𝑆))
79	18	nn0red 12695	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ Word V → (♯‘𝐴) ∈ ℝ)
80	79	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (♯‘𝐴) ∈ ℝ)
81	80	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (♯‘𝐴) ∈ ℝ)
82		elfzoelz 13820	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 𝑦 ∈ ℤ)
83	82	zred 12830	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 𝑦 ∈ ℝ)
84	83	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ (0..^(♯‘𝐵)) ∧ (𝐴 ∈ Word V ∧ 𝐵 ∈ Word V)) → 𝑦 ∈ ℝ)
85	80	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ (0..^(♯‘𝐵)) ∧ (𝐴 ∈ Word V ∧ 𝐵 ∈ Word V)) → (♯‘𝐴) ∈ ℝ)
86	84, 85	readdcld 11365	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (0..^(♯‘𝐵)) ∧ (𝐴 ∈ Word V ∧ 𝐵 ∈ Word V)) → (𝑦 + (♯‘𝐴)) ∈ ℝ)
87	86	ancoms 464	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (𝑦 + (♯‘𝐴)) ∈ ℝ)
88		elfzole1 13829	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 0 ≤ 𝑦)
89	88	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → 0 ≤ 𝑦)
90		addge02 11854	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((♯‘𝐴) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 ≤ 𝑦 ↔ (♯‘𝐴) ≤ (𝑦 + (♯‘𝐴))))
91	80, 83, 90	syl2an 608	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (0 ≤ 𝑦 ↔ (♯‘𝐴) ≤ (𝑦 + (♯‘𝐴))))
92	89, 91	mpbid 235	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (♯‘𝐴) ≤ (𝑦 + (♯‘𝐴)))
93	81, 87, 92	lensymd 11488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ¬ (𝑦 + (♯‘𝐴)) < (♯‘𝐴))
94	93	intn3an3d 1513	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ¬ ((𝑦 + (♯‘𝐴)) ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ ∧ (𝑦 + (♯‘𝐴)) < (♯‘𝐴)))
95		elfzo0 13862	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 + (♯‘𝐴)) ∈ (0..^(♯‘𝐴)) ↔ ((𝑦 + (♯‘𝐴)) ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ ∧ (𝑦 + (♯‘𝐴)) < (♯‘𝐴)))
96	94, 95	sylnibr 332	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ¬ (𝑦 + (♯‘𝐴)) ∈ (0..^(♯‘𝐴)))
97	96	iffalsed 4496	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → if((𝑦 + (♯‘𝐴)) ∈ (0..^(♯‘𝐴)), (𝐴‘(𝑦 + (♯‘𝐴))), (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴)))) = (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))))
98	97	eleq1d 2849	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (if((𝑦 + (♯‘𝐴)) ∈ (0..^(♯‘𝐴)), (𝐴‘(𝑦 + (♯‘𝐴))), (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴)))) ∈ 𝑆 ↔ (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))) ∈ 𝑆))
99	82	zcnd 12831	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 𝑦 ∈ ℂ)
100	65	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (♯‘𝐴) ∈ ℂ)
101		pncan 11590	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℂ ∧ (♯‘𝐴) ∈ ℂ) → ((𝑦 + (♯‘𝐴)) − (♯‘𝐴)) = 𝑦)
102	99, 100, 101	syl2anr 609	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ((𝑦 + (♯‘𝐴)) − (♯‘𝐴)) = 𝑦)
103	102	fveq2d 6912	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))) = (𝐵‘𝑦))
104	103	eleq1d 2849	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ((𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))) ∈ 𝑆 ↔ (𝐵‘𝑦) ∈ 𝑆))
105	104	biimpd 232	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ((𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))) ∈ 𝑆 → (𝐵‘𝑦) ∈ 𝑆))
106	98, 105	sylbid 243	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (if((𝑦 + (♯‘𝐴)) ∈ (0..^(♯‘𝐴)), (𝐴‘(𝑦 + (♯‘𝐴))), (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴)))) ∈ 𝑆 → (𝐵‘𝑦) ∈ 𝑆))
107	78, 106	syld 48	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (∀𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵)))if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ 𝑆 → (𝐵‘𝑦) ∈ 𝑆))
108	107	impancom 457	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ ∀𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵)))if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ 𝑆) → (𝑦 ∈ (0..^(♯‘𝐵)) → (𝐵‘𝑦) ∈ 𝑆))
109	108	ralrimiv 3157	. . . . . . . . 9 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ ∀𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵)))if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ 𝑆) → ∀𝑦 ∈ (0..^(♯‘𝐵))(𝐵‘𝑦) ∈ 𝑆)
110		iswrdsymb 14703	. . . . . . . . 9 ⊢ ((𝐵 ∈ Word V ∧ ∀𝑦 ∈ (0..^(♯‘𝐵))(𝐵‘𝑦) ∈ 𝑆) → 𝐵 ∈ Word 𝑆)
111	59, 109, 110	syl2an2r 698	. . . . . . . 8 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ ∀𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵)))if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ 𝑆) → 𝐵 ∈ Word 𝑆)
112	58, 111	jca 521	. . . . . . 7 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ ∀𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵)))if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ 𝑆) → (𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆))
113	112	ex 418	. . . . . 6 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (∀𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵)))if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) ∈ 𝑆 → (𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆)))
114	29, 113	biimtrrid 246	. . . . 5 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))):(0..^((♯‘𝐴) + (♯‘𝐵)))⟶𝑆 → (𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆)))
115	27, 114	sylbid 243	. . . 4 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))):(0..^(♯‘(𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))))⟶𝑆 → (𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆)))
116	3, 115	syl5 35	. . 3 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))) ∈ Word 𝑆 → (𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆)))
117	2, 116	sylbid 243	. 2 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝐴 ++ 𝐵) ∈ Word 𝑆 → (𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆)))
118		ccatcl 14746	. 2 ⊢ ((𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆) → (𝐴 ++ 𝐵) ∈ Word 𝑆)
119	117, 118	impbid1 228	1 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝐴 ++ 𝐵) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 Vcvv 3454 ⊆ wss 3900 ifcif 4485 class class class wbr 5106 ↦ cmpt 5189 dom cdm 5655 Fun wfun 6552 ⟶wf 6554 ‘cfv 6558 (class class class)co 7443 Fincfn 9001 ℂcc 11225 ℝcr 11226 0cc0 11227 + caddc 11230 < clt 11370 ≤ cle 11371 − cmin 11568 ℕcn 12362 ℕ₀cn0 12633 ℤcz 12720 ℤ_≥cuz 12992 ..^cfzo 13815 ♯chash 14501 Word cword 14685 ++ cconcat 14742

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-oadd 8508 df-er 8745 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-dju 10009 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-2 12432 df-n0 12634 df-xnn0 12707 df-z 12721 df-uz 12993 df-fz 13667 df-fzo 13816 df-hash 14502 df-word 14686 df-concat 14743

This theorem is used by: ccatrcl1 14768 ccats1alpha 14795 clwwlkwwlksb 30675

W3C validator