ccatalpha - Metamath Proof Explorer

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

Theorem ccatalpha 14298

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 14276	. . . 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 2823	. . 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 14222	. . . 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 6472	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))
5		fzofi 13694	. . . . . . . . . . 11 ⊢ (0..^((♯‘𝐴) + (♯‘𝐵))) ∈ Fin
6		mptfi 9118	. . . . . . . . . . 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 14152	. . . . . . . . . 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 232	. . . . . . . 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 6145	. . . . . . . . . . 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 6787	. . . . . . . . . . . . 13 ⊢ (𝐴‘𝑥) ∈ V
13		fvex 6787	. . . . . . . . . . . . 13 ⊢ (𝐵‘(𝑥 − (♯‘𝐴))) ∈ V
14	12, 13	ifex 4509	. . . . . . . . . . . 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 3078	. . . . . . . . . 10 ⊢ dom (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))) = (0..^((♯‘𝐴) + (♯‘𝐵)))
17	16	fveq2i 6777	. . . . . . . . 9 ⊢ (♯‘dom (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))) = (♯‘(0..^((♯‘𝐴) + (♯‘𝐵))))
18		lencl 14236	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word V → (♯‘𝐴) ∈ ℕ₀)
19		lencl 14236	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Word V → (♯‘𝐵) ∈ ℕ₀)
20		nn0addcl 12268	. . . . . . . . . . 11 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℕ₀)
21	18, 19, 20	syl2an 596	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℕ₀)
22		hashfzo0 14145	. . . . . . . . . 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 2790	. . . . . . . 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 2778	. . . . . . 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 7291	. . . . . 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 6586	. . . . 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 2738	. . . . . . 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 6984	. . . . . 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 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → 𝐴 ∈ Word V)
31		nn0cn 12243	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (♯‘𝐴) ∈ ℂ)
32		nn0cn 12243	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐵) ∈ ℕ₀ → (♯‘𝐵) ∈ ℂ)
33		addcom 11161	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝐴) ∈ ℂ ∧ (♯‘𝐵) ∈ ℂ) → ((♯‘𝐴) + (♯‘𝐵)) = ((♯‘𝐵) + (♯‘𝐴)))
34	31, 32, 33	syl2an 596	. . . . . . . . . . . . . . . . 17 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀) → ((♯‘𝐴) + (♯‘𝐵)) = ((♯‘𝐵) + (♯‘𝐴)))
35		nn0z 12343	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (♯‘𝐴) ∈ ℤ)
36	35	anim1ci 616	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀) → ((♯‘𝐵) ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℤ))
37		nn0pzuz 12645	. . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀) → ((♯‘𝐴) + (♯‘𝐵)) ∈ (ℤ_≥‘(♯‘𝐴)))
40	18, 19, 39	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((♯‘𝐴) + (♯‘𝐵)) ∈ (ℤ_≥‘(♯‘𝐴)))
41		fzoss2 13415	. . . . . . . . . . . . . . 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 3921	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐴))) → 𝑦 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))))
44		eleq1 2826	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (0..^(♯‘𝐴)) ↔ 𝑦 ∈ (0..^(♯‘𝐴))))
45		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝐴‘𝑥) = (𝐴‘𝑦))
46		fvoveq1 7298	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝐵‘(𝑥 − (♯‘𝐴))) = (𝐵‘(𝑦 − (♯‘𝐴))))
47	44, 45, 46	ifbieq12d 4487	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) = if(𝑦 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (♯‘𝐴)))))
48	47	eleq1d 2823	. . . . . . . . . . . . . 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 3557	. . . . . . . . . . . . 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 4465	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (0..^(♯‘𝐴)) → if(𝑦 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (♯‘𝐴)))) = (𝐴‘𝑦))
52	51	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐴))) → if(𝑦 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (♯‘𝐴)))) = (𝐴‘𝑦))
53	52	eleq1d 2823	. . . . . . . . . . . 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 238	. . . . . . . . . . 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 452	. . . . . . . . . 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 3102	. . . . . . . . 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 14234	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ ∀𝑦 ∈ (0..^(♯‘𝐴))(𝐴‘𝑦) ∈ 𝑆) → 𝐴 ∈ Word 𝑆)
58	30, 56, 57	syl2an2r 682	. . . . . . . 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 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → 𝐵 ∈ Word V)
60		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → 𝑦 ∈ (0..^(♯‘𝐵)))
61	18	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (♯‘𝐴) ∈ ℕ₀)
62	61	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (♯‘𝐴) ∈ ℕ₀)
63		elincfzoext 13445	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ (0..^(♯‘𝐵)) ∧ (♯‘𝐴) ∈ ℕ₀) → (𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐵) + (♯‘𝐴))))
64	60, 62, 63	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐵) + (♯‘𝐴))))
65	18	nn0cnd 12295	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ Word V → (♯‘𝐴) ∈ ℂ)
66	19	nn0cnd 12295	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ Word V → (♯‘𝐵) ∈ ℂ)
67	65, 66, 33	syl2an 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((♯‘𝐴) + (♯‘𝐵)) = ((♯‘𝐵) + (♯‘𝐴)))
68	67	oveq2d 7291	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (0..^((♯‘𝐴) + (♯‘𝐵))) = (0..^((♯‘𝐵) + (♯‘𝐴))))
69	68	eleq2d 2824	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↔ (𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐵) + (♯‘𝐴)))))
70	69	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ((𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↔ (𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐵) + (♯‘𝐴)))))
71	64, 70	mpbird 256	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (𝑦 + (♯‘𝐴)) ∈ (0..^((♯‘𝐴) + (♯‘𝐵))))
72		eleq1 2826	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + (♯‘𝐴)) → (𝑥 ∈ (0..^(♯‘𝐴)) ↔ (𝑦 + (♯‘𝐴)) ∈ (0..^(♯‘𝐴))))
73		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + (♯‘𝐴)) → (𝐴‘𝑥) = (𝐴‘(𝑦 + (♯‘𝐴))))
74		fvoveq1 7298	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + (♯‘𝐴)) → (𝐵‘(𝑥 − (♯‘𝐴))) = (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))))
75	72, 73, 74	ifbieq12d 4487	. . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . 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 3557	. . . . . . . . . . . . 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 12294	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ Word V → (♯‘𝐴) ∈ ℝ)
80	79	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (♯‘𝐴) ∈ ℝ)
81	80	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (♯‘𝐴) ∈ ℝ)
82		elfzoelz 13387	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 𝑦 ∈ ℤ)
83	82	zred 12426	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 𝑦 ∈ ℝ)
84	83	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ (0..^(♯‘𝐵)) ∧ (𝐴 ∈ Word V ∧ 𝐵 ∈ Word V)) → 𝑦 ∈ ℝ)
85	80	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ (0..^(♯‘𝐵)) ∧ (𝐴 ∈ Word V ∧ 𝐵 ∈ Word V)) → (♯‘𝐴) ∈ ℝ)
86	84, 85	readdcld 11004	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ (0..^(♯‘𝐵)) ∧ (𝐴 ∈ Word V ∧ 𝐵 ∈ Word V)) → (𝑦 + (♯‘𝐴)) ∈ ℝ)
87	86	ancoms 459	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (𝑦 + (♯‘𝐴)) ∈ ℝ)
88		elfzole1 13395	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 0 ≤ 𝑦)
89	88	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → 0 ≤ 𝑦)
90		addge02 11486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((♯‘𝐴) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 ≤ 𝑦 ↔ (♯‘𝐴) ≤ (𝑦 + (♯‘𝐴))))
91	80, 83, 90	syl2an 596	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (0 ≤ 𝑦 ↔ (♯‘𝐴) ≤ (𝑦 + (♯‘𝐴))))
92	89, 91	mpbid 231	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (♯‘𝐴) ≤ (𝑦 + (♯‘𝐴)))
93	81, 87, 92	lensymd 11126	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ¬ (𝑦 + (♯‘𝐴)) < (♯‘𝐴))
94	93	intn3an3d 1480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ¬ ((𝑦 + (♯‘𝐴)) ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ ∧ (𝑦 + (♯‘𝐴)) < (♯‘𝐴)))
95		elfzo0 13428	. . . . . . . . . . . . . . . 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 4470	. . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . 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 12427	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 𝑦 ∈ ℂ)
100	65	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (♯‘𝐴) ∈ ℂ)
101		pncan 11227	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℂ ∧ (♯‘𝐴) ∈ ℂ) → ((𝑦 + (♯‘𝐴)) − (♯‘𝐴)) = 𝑦)
102	99, 100, 101	syl2anr 597	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ((𝑦 + (♯‘𝐴)) − (♯‘𝐴)) = 𝑦)
103	102	fveq2d 6778	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))) = (𝐵‘𝑦))
104	103	eleq1d 2823	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ((𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))) ∈ 𝑆 ↔ (𝐵‘𝑦) ∈ 𝑆))
105	104	biimpd 228	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ((𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))) ∈ 𝑆 → (𝐵‘𝑦) ∈ 𝑆))
106	98, 105	sylbid 239	. . . . . . . . . . . 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 452	. . . . . . . . . 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 3102	. . . . . . . . 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 14234	. . . . . . . . 9 ⊢ ((𝐵 ∈ Word V ∧ ∀𝑦 ∈ (0..^(♯‘𝐵))(𝐵‘𝑦) ∈ 𝑆) → 𝐵 ∈ Word 𝑆)
111	59, 109, 110	syl2an2r 682	. . . . . . . 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 512	. . . . . . 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 413	. . . . . 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	syl5bir 242	. . . . 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 239	. . . 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 239	. 2 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝐴 ++ 𝐵) ∈ Word 𝑆 → (𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆)))
118		ccatcl 14277	. 2 ⊢ ((𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆) → (𝐴 ++ 𝐵) ∈ Word 𝑆)
119	117, 118	impbid1 224	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 ⊆ wss 3887 ifcif 4459 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 Fun wfun 6427 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 ℂcc 10869 ℝcr 10870 0cc0 10871 + caddc 10874 < clt 11009 ≤ cle 11010 − cmin 11205 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ..^cfzo 13382 ♯chash 14044 Word cword 14217 ++ cconcat 14273

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-oadd 8301 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-hash 14045 df-word 14218 df-concat 14274

This theorem is referenced by: ccatrcl1 14299 ccats1alpha 14324 clwwlkwwlksb 28418

W3C validator