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 14632

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 14612	. . . 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 2825	. . 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 14558	. . . 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 6603	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))
5		fzofi 14016	. . . . . . . . . . 11 ⊢ (0..^((♯‘𝐴) + (♯‘𝐵))) ∈ Fin
6		mptfi 9392	. . . . . . . . . . 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 14477	. . . . . . . . . 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 6261	. . . . . . . . . . 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 6918	. . . . . . . . . . . . 13 ⊢ (𝐴‘𝑥) ∈ V
13		fvex 6918	. . . . . . . . . . . . 13 ⊢ (𝐵‘(𝑥 − (♯‘𝐴))) ∈ V
14	12, 13	ifex 4575	. . . . . . . . . . . 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 3066	. . . . . . . . . 10 ⊢ dom (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴))))) = (0..^((♯‘𝐴) + (♯‘𝐵)))
17	16	fveq2i 6908	. . . . . . . . 9 ⊢ (♯‘dom (𝑥 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))) ↦ if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))))) = (♯‘(0..^((♯‘𝐴) + (♯‘𝐵))))
18		lencl 14572	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word V → (♯‘𝐴) ∈ ℕ₀)
19		lencl 14572	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Word V → (♯‘𝐵) ∈ ℕ₀)
20		nn0addcl 12563	. . . . . . . . . . 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 14470	. . . . . . . . . 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 2788	. . . . . . . 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 2776	. . . . . . 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 7448	. . . . . 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 6721	. . . . 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 2736	. . . . . . 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 7129	. . . . . 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 12538	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (♯‘𝐴) ∈ ℂ)
32		nn0cn 12538	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝐵) ∈ ℕ₀ → (♯‘𝐵) ∈ ℂ)
33		addcom 11448	. . . . . . . . . . . . . . . . . 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 12640	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (♯‘𝐴) ∈ ℤ)
36	35	anim1ci 616	. . . . . . . . . . . . . . . . . 18 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀) → ((♯‘𝐵) ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℤ))
37		nn0pzuz 12948	. . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . 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 13728	. . . . . . . . . . . . . . 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 3982	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐴))) → 𝑦 ∈ (0..^((♯‘𝐴) + (♯‘𝐵))))
44		eleq1 2828	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (0..^(♯‘𝐴)) ↔ 𝑦 ∈ (0..^(♯‘𝐴))))
45		fveq2 6905	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝐴‘𝑥) = (𝐴‘𝑦))
46		fvoveq1 7455	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝐵‘(𝑥 − (♯‘𝐴))) = (𝐵‘(𝑦 − (♯‘𝐴))))
47	44, 45, 46	ifbieq12d 4553	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → if(𝑥 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (♯‘𝐴)))) = if(𝑦 ∈ (0..^(♯‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (♯‘𝐴)))))
48	47	eleq1d 2825	. . . . . . . . . . . . . 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 3617	. . . . . . . . . . . . 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 4530	. . . . . . . . . . . . . 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 2825	. . . . . . . . . . . 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 3144	. . . . . . . . 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 14570	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ ∀𝑦 ∈ (0..^(♯‘𝐴))(𝐴‘𝑦) ∈ 𝑆) → 𝐴 ∈ Word 𝑆)
58	30, 56, 57	syl2an2r 685	. . . . . . . 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 13763	. . . . . . . . . . . . . . 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 12591	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ Word V → (♯‘𝐴) ∈ ℂ)
66	19	nn0cnd 12591	. . . . . . . . . . . . . . . . . 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 7448	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (0..^((♯‘𝐴) + (♯‘𝐵))) = (0..^((♯‘𝐵) + (♯‘𝐴))))
69	68	eleq2d 2826	. . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + (♯‘𝐴)) → (𝑥 ∈ (0..^(♯‘𝐴)) ↔ (𝑦 + (♯‘𝐴)) ∈ (0..^(♯‘𝐴))))
73		fveq2 6905	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + (♯‘𝐴)) → (𝐴‘𝑥) = (𝐴‘(𝑦 + (♯‘𝐴))))
74		fvoveq1 7455	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + (♯‘𝐴)) → (𝐵‘(𝑥 − (♯‘𝐴))) = (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))))
75	72, 73, 74	ifbieq12d 4553	. . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . 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 3617	. . . . . . . . . . . . 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 12590	. . . . . . . . . . . . . . . . . . . 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 13700	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 𝑦 ∈ ℤ)
83	82	zred 12724	. . . . . . . . . . . . . . . . . . . . 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 11291	. . . . . . . . . . . . . . . . . . 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 13708	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 0 ≤ 𝑦)
89	88	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → 0 ≤ 𝑦)
90		addge02 11775	. . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (♯‘𝐴) ≤ (𝑦 + (♯‘𝐴)))
93	81, 87, 92	lensymd 11413	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ¬ (𝑦 + (♯‘𝐴)) < (♯‘𝐴))
94	93	intn3an3d 1482	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → ¬ ((𝑦 + (♯‘𝐴)) ∈ ℕ₀ ∧ (♯‘𝐴) ∈ ℕ ∧ (𝑦 + (♯‘𝐴)) < (♯‘𝐴)))
95		elfzo0 13741	. . . . . . . . . . . . . . . 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 4535	. . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . 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 12725	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0..^(♯‘𝐵)) → 𝑦 ∈ ℂ)
100	65	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (♯‘𝐴) ∈ ℂ)
101		pncan 11515	. . . . . . . . . . . . . . . . 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 6909	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(♯‘𝐵))) → (𝐵‘((𝑦 + (♯‘𝐴)) − (♯‘𝐴))) = (𝐵‘𝑦))
104	103	eleq1d 2825	. . . . . . . . . . . . . 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 3144	. . . . . . . . 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 14570	. . . . . . . . 9 ⊢ ((𝐵 ∈ Word V ∧ ∀𝑦 ∈ (0..^(♯‘𝐵))(𝐵‘𝑦) ∈ 𝑆) → 𝐵 ∈ Word 𝑆)
111	59, 109, 110	syl2an2r 685	. . . . . . . 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 14613	. 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 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 Vcvv 3479 ⊆ wss 3950 ifcif 4524 class class class wbr 5142 ↦ cmpt 5224 dom cdm 5684 Fun wfun 6554 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 Fincfn 8986 ℂcc 11154 ℝcr 11155 0cc0 11156 + caddc 11159 < clt 11296 ≤ cle 11297 − cmin 11493 ℕcn 12267 ℕ₀cn0 12528 ℤcz 12615 ℤ_≥cuz 12879 ..^cfzo 13695 ♯chash 14370 Word cword 14553 ++ cconcat 14609

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-dju 9942 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12880 df-fz 13549 df-fzo 13696 df-hash 14371 df-word 14554 df-concat 14610

This theorem is referenced by: ccatrcl1 14633 ccats1alpha 14658 clwwlkwwlksb 30074

W3C validator