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Theorem ccatalpha 14529

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 14508	. . . 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 2822	. . 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 14453	. . . 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 6538	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))
5		fzofi 13909	. . . . . . . . . . 11 ⊢ (0..^((♯‘𝐴) + (♯‘𝐵))) ∈ Fin
6		mptfi 9263	. . . . . . . . . . 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 14372	. . . . . . . . . 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 13	. . . . . . . . 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 233	. . . . . . . 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 6208	. . . . . . . . . . 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 6855	. . . . . . . . . . . . 13 ⊢ (𝐴‘𝑥) ∈ V
13		fvex 6855	. . . . . . . . . . . . 13 ⊢ (𝐵‘(𝑥 − (♯‘𝐴))) ∈ V
14	12, 13	ifex 4532	. . . . . . . . . . . 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 3058	. . . . . . . . . 10 ⊢ dom (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))) = (0..^((♯‘𝐴) + (♯‘𝐵)))
17	16	fveq2i 6845	. . . . . . . . 9 ⊢ (♯‘dom (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))) = (♯‘(0..^((♯‘𝐴) + (♯‘𝐵))))
18		lencl 14468	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word V → (♯‘𝐴) ∈ ℕ₀)
19		lencl 14468	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Word V → (♯‘𝐵) ∈ ℕ₀)
20		nn0addcl 12448	. . . . . . . . . . 11 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℕ₀)
21	18, 19, 20	syl2an 597	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℕ₀)
22		hashfzo0 14365	. . . . . . . . . 10 ⊢ (((♯‘𝐴) + (♯‘𝐵)) ∈ ℕ₀ → (♯‘(0..^((♯‘𝐴) + (♯‘𝐵)))) = ((♯‘𝐴) + (♯‘𝐵)))
23	21, 22	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (♯‘(0..^((♯‘𝐴) + (♯‘𝐵)))) = ((♯‘𝐴) + (♯‘𝐵)))
24	17, 23	eqtrid 2784	. . . . . . . 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 2772	. . . . . . 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 7384	. . . . . 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 6654	. . . . 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 2737	. . . . . . 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 7064	. . . . . 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 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → 𝐴 ∈ Word V)
31		nn0cn 12423	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (♯‘𝐴) ∈ ℂ)
32		nn0cn 12423	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐵) ∈ ℕ₀ → (♯‘𝐵) ∈ ℂ)
33		addcom 11331	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝐴) ∈ ℂ ∧ (♯‘𝐵) ∈ ℂ) → ((♯‘𝐴) + (♯‘𝐵)) = ((♯‘𝐵) + (♯‘𝐴)))
34	31, 32, 33	syl2an 597	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀) → ((♯‘𝐴) + (♯‘𝐵)) = ((♯‘𝐵) + (♯‘𝐴)))
35		nn0z 12524	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (♯‘𝐴) ∈ ℤ)
36	35	anim1ci 617	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀) → ((♯‘𝐵) ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℤ))
37		nn0pzuz 12830	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝐵) ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℤ) → ((♯‘𝐵) + (♯‘𝐴)) ∈ (ℤ_≥‘(♯‘𝐴)))
38	36, 37	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀) → ((♯‘𝐵) + (♯‘𝐴)) ∈ (ℤ_≥‘(♯‘𝐴)))
39	34, 38	eqeltrd 2837	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀) → ((♯‘𝐴) + (♯‘𝐵)) ∈ (ℤ_≥‘(♯‘𝐴)))
40	18, 19, 39	syl2an 597	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((♯‘𝐴) + (♯‘𝐵)) ∈ (ℤ_≥‘(♯‘𝐴)))
41		fzoss2 13615	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝐴) + (♯‘𝐵)) ∈ (ℤ_≥‘(♯‘𝐴)) → (0..^(♯‘𝐴)) ⊆ (0..^((♯‘𝐴) + (♯‘𝐵))))
42	40, 41	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (0..^(♯‘𝐴)) ⊆ (0..^((♯‘𝐴) + (♯‘𝐵))))
43	42	sselda 3935	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐴))) → 𝑦 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))))
44		eleq1 2825	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (0..^(♯‘𝐴)) ↔ 𝑦 ∈ (0..^(♯‘𝐴))))
45		fveq2 6842	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝐴‘𝑥) = (𝐴‘𝑦))
46		fvoveq1 7391	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝐵‘(𝑥 − (♯‘𝐴))) = (𝐵‘(𝑦 − (♯‘𝐴))))
47	44, 45, 46	ifbieq12d 4510	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) = if(𝑦 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (♯‘𝐴)))))
48	47	eleq1d 2822	. . . . . . . . . . . . . 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 3574	. . . . . . . . . . . . 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 17	. . . . . . . . . . . 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 4487	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0..^(♯‘𝐴)) → if(𝑦 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (♯‘𝐴)))) = (𝐴‘𝑦))
52	51	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐴))) → if(𝑦 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (♯‘𝐴)))) = (𝐴‘𝑦))
53	52	eleq1d 2822	. . . . . . . . . . . 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 239	. . . . . . . . . . 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 451	. . . . . . . . . 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 3129	. . . . . . . . 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 14466	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ ∀𝑦 ∈ (0..^(♯‘𝐴))(𝐴‘𝑦) ∈ 𝑆) → 𝐴 ∈ Word 𝑆)
58	30, 56, 57	syl2an2r 686	. . . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → 𝐵 ∈ Word V)
60		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → 𝑦 ∈ (0..^(♯‘𝐵)))
61	18	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (♯‘𝐴) ∈ ℕ₀)
62	61	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (♯‘𝐴) ∈ ℕ₀)
63		elincfzoext 13651	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ (0..^(♯‘𝐵)) ∧ (♯‘𝐴) ∈ ℕ₀) → (𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐵) + (♯‘𝐴))))
64	60, 62, 63	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐵) + (♯‘𝐴))))
65	18	nn0cnd 12476	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ Word V → (♯‘𝐴) ∈ ℂ)
66	19	nn0cnd 12476	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ Word V → (♯‘𝐵) ∈ ℂ)
67	65, 66, 33	syl2an 597	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((♯‘𝐴) + (♯‘𝐵)) = ((♯‘𝐵) + (♯‘𝐴)))
68	67	oveq2d 7384	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (0..^((♯‘𝐴) + (♯‘𝐵))) = (0..^((♯‘𝐵) + (♯‘𝐴))))
69	68	eleq2d 2823	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↔ (𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐵) + (♯‘𝐴)))))
70	69	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ((𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↔ (𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐵) + (♯‘𝐴)))))
71	64, 70	mpbird 257	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐴) + (♯‘𝐵))))
72		eleq1 2825	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + (♯‘𝐴)) → (𝑥 ∈ (0..^(♯‘𝐴)) ↔ (𝑦 + (♯‘𝐴)) ∈ (0..^(♯‘𝐴))))
73		fveq2 6842	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + (♯‘𝐴)) → (𝐴‘𝑥) = (𝐴‘(𝑦 + (♯‘𝐴))))
74		fvoveq1 7391	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + (♯‘𝐴)) → (𝐵‘(𝑥 − (♯‘𝐴))) = (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))))
75	72, 73, 74	ifbieq12d 4510	. . . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . . 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 3574	. . . . . . . . . . . . 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 17	. . . . . . . . . . . 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 12475	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ Word V → (♯‘𝐴) ∈ ℝ)
80	79	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (♯‘𝐴) ∈ ℝ)
81	80	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (♯‘𝐴) ∈ ℝ)
82		elfzoelz 13587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 𝑦 ∈ ℤ)
83	82	zred 12608	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 𝑦 ∈ ℝ)
84	83	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ (0..^(♯‘𝐵)) ∧ (𝐴 ∈ Word V ∧ 𝐵 ∈ Word V)) → 𝑦 ∈ ℝ)
85	80	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ (0..^(♯‘𝐵)) ∧ (𝐴 ∈ Word V ∧ 𝐵 ∈ Word V)) → (♯‘𝐴) ∈ ℝ)
86	84, 85	readdcld 11173	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (0..^(♯‘𝐵)) ∧ (𝐴 ∈ Word V ∧ 𝐵 ∈ Word V)) → (𝑦 + (♯‘𝐴)) ∈ ℝ)
87	86	ancoms 458	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (𝑦 + (♯‘𝐴)) ∈ ℝ)
88		elfzole1 13595	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 0 ≤ 𝑦)
89	88	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → 0 ≤ 𝑦)
90		addge02 11660	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((♯‘𝐴) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 ≤ 𝑦 ↔ (♯‘𝐴) ≤ (𝑦 + (♯‘𝐴))))
91	80, 83, 90	syl2an 597	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (0 ≤ 𝑦 ↔ (♯‘𝐴) ≤ (𝑦 + (♯‘𝐴))))
92	89, 91	mpbid 232	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (♯‘𝐴) ≤ (𝑦 + (♯‘𝐴)))
93	81, 87, 92	lensymd 11296	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ¬ (𝑦 + (♯‘𝐴)) < (♯‘𝐴))
94	93	intn3an3d 1484	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ¬ ((𝑦 + (♯‘𝐴)) ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ ∧ (𝑦 + (♯‘𝐴)) < (♯‘𝐴)))
95		elfzo0 13628	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 + (♯‘𝐴)) ∈ (0..^(♯‘𝐴)) ↔ ((𝑦 + (♯‘𝐴)) ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ ∧ (𝑦 + (♯‘𝐴)) < (♯‘𝐴)))
96	94, 95	sylnibr 329	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ¬ (𝑦 + (♯‘𝐴)) ∈ (0..^(♯‘𝐴)))
97	96	iffalsed 4492	. . . . . . . . . . . . . 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 2822	. . . . . . . . . . . . 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 12609	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 𝑦 ∈ ℂ)
100	65	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (♯‘𝐴) ∈ ℂ)
101		pncan 11398	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℂ ∧ (♯‘𝐴) ∈ ℂ) → ((𝑦 + (♯‘𝐴)) − (♯‘𝐴)) = 𝑦)
102	99, 100, 101	syl2anr 598	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ((𝑦 + (♯‘𝐴)) − (♯‘𝐴)) = 𝑦)
103	102	fveq2d 6846	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))) = (𝐵‘𝑦))
104	103	eleq1d 2822	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ((𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))) ∈ 𝑆 ↔ (𝐵‘𝑦) ∈ 𝑆))
105	104	biimpd 229	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ((𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))) ∈ 𝑆 → (𝐵‘𝑦) ∈ 𝑆))
106	98, 105	sylbid 240	. . . . . . . . . . . 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 47	. . . . . . . . . . 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 451	. . . . . . . . . 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 3129	. . . . . . . . 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 14466	. . . . . . . . 9 ⊢ ((𝐵 ∈ Word V ∧ ∀𝑦 ∈ (0..^(♯‘𝐵))(𝐵‘𝑦) ∈ 𝑆) → 𝐵 ∈ Word 𝑆)
111	59, 109, 110	syl2an2r 686	. . . . . . . 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 511	. . . . . . 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 412	. . . . . 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 243	. . . . 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 240	. . . 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 34	. . 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 240	. 2 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝐴 ++ 𝐵) ∈ Word 𝑆 → (𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆)))
118		ccatcl 14509	. 2 ⊢ ((𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆) → (𝐴 ++ 𝐵) ∈ Word 𝑆)
119	117, 118	impbid1 225	1 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝐴 ++ 𝐵) ∈ Word 𝑆 ↔ (𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3442 ⊆ wss 3903 ifcif 4481 class class class wbr 5100 ↦ cmpt 5181 dom cdm 5632 Fun wfun 6494 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 Fincfn 8895 ℂcc 11036 ℝcr 11037 0cc0 11038 + caddc 11041 < clt 11178 ≤ cle 11179 − cmin 11376 ℕcn 12157 ℕ₀cn0 12413 ℤcz 12500 ℤ_≥cuz 12763 ..^cfzo 13582 ♯chash 14265 Word cword 14448 ++ cconcat 14505

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-dju 9825 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-n0 12414 df-xnn0 12487 df-z 12501 df-uz 12764 df-fz 13436 df-fzo 13583 df-hash 14266 df-word 14449 df-concat 14506

This theorem is referenced by: ccatrcl1 14530 ccats1alpha 14555 clwwlkwwlksb 30141

W3C validator