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Theorem ccatalpha 14720

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 14698	. . . 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 2846	. . 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 14643	. . . 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 6570	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))
5		fzofi 14097	. . . . . . . . . . 11 ⊢ (0..^((♯‘𝐴) + (♯‘𝐵))) ∈ Fin
6		mptfi 9324	. . . . . . . . . . 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 14562	. . . . . . . . . 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 6236	. . . . . . . . . . 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 6890	. . . . . . . . . . . . 13 ⊢ (𝐴‘𝑥) ∈ V
13		fvex 6890	. . . . . . . . . . . . 13 ⊢ (𝐵‘(𝑥 − (♯‘𝐴))) ∈ V
14	12, 13	ifex 4533	. . . . . . . . . . . 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 3083	. . . . . . . . . 10 ⊢ dom (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))) = (0..^((♯‘𝐴) + (♯‘𝐵)))
17	16	fveq2i 6880	. . . . . . . . 9 ⊢ (♯‘dom (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))) = (♯‘(0..^((♯‘𝐴) + (♯‘𝐵))))
18		lencl 14658	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word V → (♯‘𝐴) ∈ ℕ₀)
19		lencl 14658	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Word V → (♯‘𝐵) ∈ ℕ₀)
20		nn0addcl 12622	. . . . . . . . . . 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 14555	. . . . . . . . . 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 2808	. . . . . . . 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 2796	. . . . . . 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 7428	. . . . . 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 6685	. . . . 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 2761	. . . . . . 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 7102	. . . . . 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 12597	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (♯‘𝐴) ∈ ℂ)
32		nn0cn 12597	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐵) ∈ ℕ₀ → (♯‘𝐵) ∈ ℂ)
33		addcom 11477	. . . . . . . . . . . . . . . . . 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 12698	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (♯‘𝐴) ∈ ℤ)
36	35	anim1ci 628	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀) → ((♯‘𝐵) ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℤ))
37		nn0pzuz 13013	. . . . . . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . . . . 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 13802	. . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐴))) → 𝑦 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))))
44		eleq1 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (0..^(♯‘𝐴)) ↔ 𝑦 ∈ (0..^(♯‘𝐴))))
45		fveq2 6877	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝐴‘𝑥) = (𝐴‘𝑦))
46		fvoveq1 7435	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝐵‘(𝑥 − (♯‘𝐴))) = (𝐵‘(𝑦 − (♯‘𝐴))))
47	44, 45, 46	ifbieq12d 4511	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) = if(𝑦 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (♯‘𝐴)))))
48	47	eleq1d 2846	. . . . . . . . . . . . . 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 3573	. . . . . . . . . . . . 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 4488	. . . . . . . . . . . . . 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 2846	. . . . . . . . . . . 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 3154	. . . . . . . . 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 14656	. . . . . . . . 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 13838	. . . . . . . . . . . . . . 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 12650	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ Word V → (♯‘𝐴) ∈ ℂ)
66	19	nn0cnd 12650	. . . . . . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (0..^((♯‘𝐴) + (♯‘𝐵))) = (0..^((♯‘𝐵) + (♯‘𝐴))))
69	68	eleq2d 2847	. . . . . . . . . . . . . . 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 2849	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + (♯‘𝐴)) → (𝑥 ∈ (0..^(♯‘𝐴)) ↔ (𝑦 + (♯‘𝐴)) ∈ (0..^(♯‘𝐴))))
73		fveq2 6877	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + (♯‘𝐴)) → (𝐴‘𝑥) = (𝐴‘(𝑦 + (♯‘𝐴))))
74		fvoveq1 7435	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + (♯‘𝐴)) → (𝐵‘(𝑥 − (♯‘𝐴))) = (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))))
75	72, 73, 74	ifbieq12d 4511	. . . . . . . . . . . . . . 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 2846	. . . . . . . . . . . . . 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 3573	. . . . . . . . . . . . 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 12649	. . . . . . . . . . . . . . . . . . . 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 13773	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 𝑦 ∈ ℤ)
83	82	zred 12784	. . . . . . . . . . . . . . . . . . . . 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 11319	. . . . . . . . . . . . . . . . . . 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 13782	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 0 ≤ 𝑦)
89	88	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → 0 ≤ 𝑦)
90		addge02 11808	. . . . . . . . . . . . . . . . . . . 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 11442	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ¬ (𝑦 + (♯‘𝐴)) < (♯‘𝐴))
94	93	intn3an3d 1512	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ¬ ((𝑦 + (♯‘𝐴)) ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ ∧ (𝑦 + (♯‘𝐴)) < (♯‘𝐴)))
95		elfzo0 13815	. . . . . . . . . . . . . . . 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 4493	. . . . . . . . . . . . . 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 2846	. . . . . . . . . . . . 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 12785	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 𝑦 ∈ ℂ)
100	65	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (♯‘𝐴) ∈ ℂ)
101		pncan 11544	. . . . . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))) = (𝐵‘𝑦))
104	103	eleq1d 2846	. . . . . . . . . . . . . 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 3154	. . . . . . . . 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 14656	. . . . . . . . 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 14699	. 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 2145 ∀wral 3077 Vcvv 3451 ⊆ wss 3899 ifcif 4482 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5651 Fun wfun 6525 ⟶wf 6527 ‘cfv 6531 (class class class)co 7412 Fincfn 8957 ℂcc 11179 ℝcr 11180 0cc0 11181 + caddc 11184 < clt 11324 ≤ cle 11325 − cmin 11522 ℕcn 12316 ℕ₀cn0 12587 ℤcz 12674 ℤ_≥cuz 12946 ..^cfzo 13768 ♯chash 14454 Word cword 14638 ++ cconcat 14695

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-oadd 8464 df-er 8701 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-dju 9963 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-n0 12588 df-xnn0 12661 df-z 12675 df-uz 12947 df-fz 13621 df-fzo 13769 df-hash 14455 df-word 14639 df-concat 14696

This theorem is used by: ccatrcl1 14721 ccats1alpha 14747 clwwlkwwlksb 30627

W3C validator