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Theorem ccatalpha 13228

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 13211	. . . 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 2672	. . 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 13165	. . . 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 5840	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ (0..^((#‘𝐴) + (#‘𝐵))) ↦ if(𝑥 ∈ (0..^(#‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (#‘𝐴)))))
5		fzofi 12635	. . . . . . . . . . 11 ⊢ (0..^((#‘𝐴) + (#‘𝐵))) ∈ Fin
6		mptfi 8148	. . . . . . . . . . 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 13084	. . . . . . . . . 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 222	. . . . . . . 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 5549	. . . . . . . . . . 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 6113	. . . . . . . . . . . . 13 ⊢ (𝐴‘𝑥) ∈ V
13		fvex 6113	. . . . . . . . . . . . 13 ⊢ (𝐵‘(𝑥 − (#‘𝐴))) ∈ V
14	12, 13	ifex 4106	. . . . . . . . . . . 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 2910	. . . . . . . . . 10 ⊢ dom (𝑥 ∈ (0..^((#‘𝐴) + (#‘𝐵))) ↦ if(𝑥 ∈ (0..^(#‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (#‘𝐴))))) = (0..^((#‘𝐴) + (#‘𝐵)))
17	16	fveq2i 6106	. . . . . . . . 9 ⊢ (#‘dom (𝑥 ∈ (0..^((#‘𝐴) + (#‘𝐵))) ↦ if(𝑥 ∈ (0..^(#‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (#‘𝐴)))))) = (#‘(0..^((#‘𝐴) + (#‘𝐵))))
18		lencl 13179	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word V → (#‘𝐴) ∈ ℕ₀)
19		lencl 13179	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Word V → (#‘𝐵) ∈ ℕ₀)
20		nn0addcl 11205	. . . . . . . . . . 11 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → ((#‘𝐴) + (#‘𝐵)) ∈ ℕ₀)
21	18, 19, 20	syl2an 493	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((#‘𝐴) + (#‘𝐵)) ∈ ℕ₀)
22		hashfzo0 13077	. . . . . . . . . 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	syl5eq 2656	. . . . . . . 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 2644	. . . . . . 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 6565	. . . . . 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 5944	. . . . 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 2610	. . . . . . 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 6289	. . . . . 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 472	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → 𝐴 ∈ Word V)
31		nn0cn 11179	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝐴) ∈ ℕ₀ → (#‘𝐴) ∈ ℂ)
32		nn0cn 11179	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝐵) ∈ ℕ₀ → (#‘𝐵) ∈ ℂ)
33		addcom 10101	. . . . . . . . . . . . . . . . . . 19 ⊢ (((#‘𝐴) ∈ ℂ ∧ (#‘𝐵) ∈ ℂ) → ((#‘𝐴) + (#‘𝐵)) = ((#‘𝐵) + (#‘𝐴)))
34	31, 32, 33	syl2an 493	. . . . . . . . . . . . . . . . . 18 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → ((#‘𝐴) + (#‘𝐵)) = ((#‘𝐵) + (#‘𝐴)))
35		nn0z 11277	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝐴) ∈ ℕ₀ → (#‘𝐴) ∈ ℤ)
36	35	anim1i 590	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → ((#‘𝐴) ∈ ℤ ∧ (#‘𝐵) ∈ ℕ₀))
37	36	ancomd 466	. . . . . . . . . . . . . . . . . . 19 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → ((#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ∈ ℤ))
38		nn0pzuz 11621	. . . . . . . . . . . . . . . . . . 19 ⊢ (((#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ∈ ℤ) → ((#‘𝐵) + (#‘𝐴)) ∈ (ℤ_≥‘(#‘𝐴)))
39	37, 38	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → ((#‘𝐵) + (#‘𝐴)) ∈ (ℤ_≥‘(#‘𝐴)))
40	34, 39	eqeltrd 2688	. . . . . . . . . . . . . . . . 17 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → ((#‘𝐴) + (#‘𝐵)) ∈ (ℤ_≥‘(#‘𝐴)))
41	18, 19, 40	syl2an 493	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((#‘𝐴) + (#‘𝐵)) ∈ (ℤ_≥‘(#‘𝐴)))
42		fzoss2 12365	. . . . . . . . . . . . . . . 16 ⊢ (((#‘𝐴) + (#‘𝐵)) ∈ (ℤ_≥‘(#‘𝐴)) → (0..^(#‘𝐴)) ⊆ (0..^((#‘𝐴) + (#‘𝐵))))
43	41, 42	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (0..^(#‘𝐴)) ⊆ (0..^((#‘𝐴) + (#‘𝐵))))
44	43	sselda 3568	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐴))) → 𝑦 ∈ (0..^((#‘𝐴) + (#‘𝐵))))
45		eleq1 2676	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (0..^(#‘𝐴)) ↔ 𝑦 ∈ (0..^(#‘𝐴))))
46		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝐴‘𝑥) = (𝐴‘𝑦))
47		oveq1 6556	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 − (#‘𝐴)) = (𝑦 − (#‘𝐴)))
48	47	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝐵‘(𝑥 − (#‘𝐴))) = (𝐵‘(𝑦 − (#‘𝐴))))
49	45, 46, 48	ifbieq12d 4063	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → if(𝑥 ∈ (0..^(#‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (#‘𝐴)))) = if(𝑦 ∈ (0..^(#‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (#‘𝐴)))))
50	49	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (if(𝑥 ∈ (0..^(#‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (#‘𝐴)))) ∈ 𝑆 ↔ if(𝑦 ∈ (0..^(#‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (#‘𝐴)))) ∈ 𝑆))
51	50	rspcv 3278	. . . . . . . . . . . . . 14 ⊢ (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))
52	44, 51	syl 17	. . . . . . . . . . . . 13 ⊢ (((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))
53		iftrue 4042	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0..^(#‘𝐴)) → if(𝑦 ∈ (0..^(#‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (#‘𝐴)))) = (𝐴‘𝑦))
54	53	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐴))) → if(𝑦 ∈ (0..^(#‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (#‘𝐴)))) = (𝐴‘𝑦))
55	54	eleq1d 2672	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐴))) → (if(𝑦 ∈ (0..^(#‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (#‘𝐴)))) ∈ 𝑆 ↔ (𝐴‘𝑦) ∈ 𝑆))
56	52, 55	sylibd 228	. . . . . . . . . . . 12 ⊢ (((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))
57	56	impancom 455	. . . . . . . . . . 11 ⊢ (((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))
58	57	imp 444	. . . . . . . . . 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)
59	58	ralrimiva 2949	. . . . . . . . 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)
60		iswrdsymb 13177	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ ∀𝑦 ∈ (0..^(#‘𝐴))(𝐴‘𝑦) ∈ 𝑆) → 𝐴 ∈ Word 𝑆)
61	30, 59, 60	syl2an2r 872	. . . . . . . 8 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ ∀𝑥 ∈ (0..^((#‘𝐴) + (#‘𝐵)))if(𝑥 ∈ (0..^(#‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (#‘𝐴)))) ∈ 𝑆) → 𝐴 ∈ Word 𝑆)
62		simpr 476	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → 𝐵 ∈ Word V)
63		simpr 476	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → 𝑦 ∈ (0..^(#‘𝐵)))
64	18	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (#‘𝐴) ∈ ℕ₀)
65	64	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → (#‘𝐴) ∈ ℕ₀)
66		elincfzoext 12393	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ (0..^(#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ₀) → (𝑦 + (#‘𝐴)) ∈ (0..^((#‘𝐵) + (#‘𝐴))))
67	63, 65, 66	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → (𝑦 + (#‘𝐴)) ∈ (0..^((#‘𝐵) + (#‘𝐴))))
68	18	nn0cnd 11230	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ Word V → (#‘𝐴) ∈ ℂ)
69	19	nn0cnd 11230	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ Word V → (#‘𝐵) ∈ ℂ)
70	68, 69, 33	syl2an 493	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((#‘𝐴) + (#‘𝐵)) = ((#‘𝐵) + (#‘𝐴)))
71	70	oveq2d 6565	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (0..^((#‘𝐴) + (#‘𝐵))) = (0..^((#‘𝐵) + (#‘𝐴))))
72	71	eleq2d 2673	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝑦 + (#‘𝐴)) ∈ (0..^((#‘𝐴) + (#‘𝐵))) ↔ (𝑦 + (#‘𝐴)) ∈ (0..^((#‘𝐵) + (#‘𝐴)))))
73	72	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → ((𝑦 + (#‘𝐴)) ∈ (0..^((#‘𝐴) + (#‘𝐵))) ↔ (𝑦 + (#‘𝐴)) ∈ (0..^((#‘𝐵) + (#‘𝐴)))))
74	67, 73	mpbird 246	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → (𝑦 + (#‘𝐴)) ∈ (0..^((#‘𝐴) + (#‘𝐵))))
75		eleq1 2676	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 + (#‘𝐴)) → (𝑥 ∈ (0..^(#‘𝐴)) ↔ (𝑦 + (#‘𝐴)) ∈ (0..^(#‘𝐴))))
76		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 + (#‘𝐴)) → (𝐴‘𝑥) = (𝐴‘(𝑦 + (#‘𝐴))))
77		oveq1 6556	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑦 + (#‘𝐴)) → (𝑥 − (#‘𝐴)) = ((𝑦 + (#‘𝐴)) − (#‘𝐴)))
78	77	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 + (#‘𝐴)) → (𝐵‘(𝑥 − (#‘𝐴))) = (𝐵‘((𝑦 + (#‘𝐴)) − (#‘𝐴))))
79	75, 76, 78	ifbieq12d 4063	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + (#‘𝐴)) → if(𝑥 ∈ (0..^(#‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (#‘𝐴)))) = if((𝑦 + (#‘𝐴)) ∈ (0..^(#‘𝐴)), (𝐴‘(𝑦 + (#‘𝐴))), (𝐵‘((𝑦 + (#‘𝐴)) − (#‘𝐴)))))
80	79	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (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))
81	80	rspcv 3278	. . . . . . . . . . . . . 14 ⊢ ((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))
82	74, 81	syl 17	. . . . . . . . . . . . 13 ⊢ (((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))
83	18	nn0red 11229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ Word V → (#‘𝐴) ∈ ℝ)
84	83	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (#‘𝐴) ∈ ℝ)
85	84	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → (#‘𝐴) ∈ ℝ)
86		elfzoelz 12339	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^(#‘𝐵)) → 𝑦 ∈ ℤ)
87	86	zred 11358	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0..^(#‘𝐵)) → 𝑦 ∈ ℝ)
88	87	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ (0..^(#‘𝐵)) ∧ (𝐴 ∈ Word V ∧ 𝐵 ∈ Word V)) → 𝑦 ∈ ℝ)
89	84	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ (0..^(#‘𝐵)) ∧ (𝐴 ∈ Word V ∧ 𝐵 ∈ Word V)) → (#‘𝐴) ∈ ℝ)
90	88, 89	readdcld 9948	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ (0..^(#‘𝐵)) ∧ (𝐴 ∈ Word V ∧ 𝐵 ∈ Word V)) → (𝑦 + (#‘𝐴)) ∈ ℝ)
91	90	ancoms 468	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → (𝑦 + (#‘𝐴)) ∈ ℝ)
92		elfzole1 12347	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0..^(#‘𝐵)) → 0 ≤ 𝑦)
93	92	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → 0 ≤ 𝑦)
94		addge02 10418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((#‘𝐴) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 ≤ 𝑦 ↔ (#‘𝐴) ≤ (𝑦 + (#‘𝐴))))
95	84, 87, 94	syl2an 493	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → (0 ≤ 𝑦 ↔ (#‘𝐴) ≤ (𝑦 + (#‘𝐴))))
96	93, 95	mpbid 221	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → (#‘𝐴) ≤ (𝑦 + (#‘𝐴)))
97	85, 91, 96	lensymd 10067	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → ¬ (𝑦 + (#‘𝐴)) < (#‘𝐴))
98	97	intn3an3d 1436	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → ¬ ((𝑦 + (#‘𝐴)) ∈ ℕ₀ ∧ (#‘𝐴) ∈ ℕ ∧ (𝑦 + (#‘𝐴)) < (#‘𝐴)))
99		elfzo0 12376	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 + (#‘𝐴)) ∈ (0..^(#‘𝐴)) ↔ ((𝑦 + (#‘𝐴)) ∈ ℕ₀ ∧ (#‘𝐴) ∈ ℕ ∧ (𝑦 + (#‘𝐴)) < (#‘𝐴)))
100	98, 99	sylnibr 318	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → ¬ (𝑦 + (#‘𝐴)) ∈ (0..^(#‘𝐴)))
101	100	iffalsed 4047	. . . . . . . . . . . . . . 15 ⊢ (((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))))
102	101	eleq1d 2672	. . . . . . . . . . . . . 14 ⊢ (((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))
103	86	zcnd 11359	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0..^(#‘𝐵)) → 𝑦 ∈ ℂ)
104	68	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (#‘𝐴) ∈ ℂ)
105		pncan 10166	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℂ ∧ (#‘𝐴) ∈ ℂ) → ((𝑦 + (#‘𝐴)) − (#‘𝐴)) = 𝑦)
106	103, 104, 105	syl2anr 494	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → ((𝑦 + (#‘𝐴)) − (#‘𝐴)) = 𝑦)
107	106	fveq2d 6107	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → (𝐵‘((𝑦 + (#‘𝐴)) − (#‘𝐴))) = (𝐵‘𝑦))
108	107	eleq1d 2672	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → ((𝐵‘((𝑦 + (#‘𝐴)) − (#‘𝐴))) ∈ 𝑆 ↔ (𝐵‘𝑦) ∈ 𝑆))
109	108	biimpd 218	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → ((𝐵‘((𝑦 + (#‘𝐴)) − (#‘𝐴))) ∈ 𝑆 → (𝐵‘𝑦) ∈ 𝑆))
110	102, 109	sylbid 229	. . . . . . . . . . . . 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) ∈ 𝑆))
111	82, 110	syld 46	. . . . . . . . . . . 12 ⊢ (((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))
112	111	impancom 455	. . . . . . . . . . 11 ⊢ (((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))
113	112	imp 444	. . . . . . . . . 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)
114	113	ralrimiva 2949	. . . . . . . . 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)
115		iswrdsymb 13177	. . . . . . . . 9 ⊢ ((𝐵 ∈ Word V ∧ ∀𝑦 ∈ (0..^(#‘𝐵))(𝐵‘𝑦) ∈ 𝑆) → 𝐵 ∈ Word 𝑆)
116	62, 114, 115	syl2an2r 872	. . . . . . . 8 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ ∀𝑥 ∈ (0..^((#‘𝐴) + (#‘𝐵)))if(𝑥 ∈ (0..^(#‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (#‘𝐴)))) ∈ 𝑆) → 𝐵 ∈ Word 𝑆)
117	61, 116	jca 553	. . . . . . 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))
118	117	ex 449	. . . . . 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)))
119	29, 118	syl5bir 232	. . . . 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)))
120	27, 119	sylbid 229	. . . 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)))
121	3, 120	syl5 33	. . 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)))
122	2, 121	sylbid 229	. 2 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝐴 ++ 𝐵) ∈ Word 𝑆 → (𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆)))
123		ccatcl 13212	. 2 ⊢ ((𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆) → (𝐴 ++ 𝐵) ∈ Word 𝑆)
124	122, 123	impbid1 214	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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℂcc 9813 ℝcr 9814 0cc0 9815 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ..^cfzo 12334 #chash 12979 Word cword 13146 ++ cconcat 13148

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156

This theorem is referenced by: ccatrcl1 13229

W3C validator