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Theorem ccatalpha 13375

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 13358	. . . 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 2686	. . 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 13310	. . . 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 5926	. . . . . . . . 9 ⊢ Fun (𝑥 ∈ (0..^((#‘𝐴) + (#‘𝐵))) ↦ if(𝑥 ∈ (0..^(#‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (#‘𝐴)))))
5		fzofi 12773	. . . . . . . . . . 11 ⊢ (0..^((#‘𝐴) + (#‘𝐵))) ∈ Fin
6		mptfi 8265	. . . . . . . . . . 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 13224	. . . . . . . . . 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 223	. . . . . . . 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 5632	. . . . . . . . . . 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 6201	. . . . . . . . . . . . 13 ⊢ (𝐴‘𝑥) ∈ V
13		fvex 6201	. . . . . . . . . . . . 13 ⊢ (𝐵‘(𝑥 − (#‘𝐴))) ∈ V
14	12, 13	ifex 4156	. . . . . . . . . . . 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 2926	. . . . . . . . . 10 ⊢ dom (𝑥 ∈ (0..^((#‘𝐴) + (#‘𝐵))) ↦ if(𝑥 ∈ (0..^(#‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (#‘𝐴))))) = (0..^((#‘𝐴) + (#‘𝐵)))
17	16	fveq2i 6194	. . . . . . . . 9 ⊢ (#‘dom (𝑥 ∈ (0..^((#‘𝐴) + (#‘𝐵))) ↦ if(𝑥 ∈ (0..^(#‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (#‘𝐴)))))) = (#‘(0..^((#‘𝐴) + (#‘𝐵))))
18		lencl 13324	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Word V → (#‘𝐴) ∈ ℕ₀)
19		lencl 13324	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Word V → (#‘𝐵) ∈ ℕ₀)
20		nn0addcl 11328	. . . . . . . . . . 11 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → ((#‘𝐴) + (#‘𝐵)) ∈ ℕ₀)
21	18, 19, 20	syl2an 494	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((#‘𝐴) + (#‘𝐵)) ∈ ℕ₀)
22		hashfzo0 13217	. . . . . . . . . 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 2668	. . . . . . . 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 2656	. . . . . . 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 6666	. . . . . 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 6031	. . . . 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 2622	. . . . . . 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 6381	. . . . . 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 473	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → 𝐴 ∈ Word V)
31		nn0cn 11302	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝐴) ∈ ℕ₀ → (#‘𝐴) ∈ ℂ)
32		nn0cn 11302	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘𝐵) ∈ ℕ₀ → (#‘𝐵) ∈ ℂ)
33		addcom 10222	. . . . . . . . . . . . . . . . . . 19 ⊢ (((#‘𝐴) ∈ ℂ ∧ (#‘𝐵) ∈ ℂ) → ((#‘𝐴) + (#‘𝐵)) = ((#‘𝐵) + (#‘𝐴)))
34	31, 32, 33	syl2an 494	. . . . . . . . . . . . . . . . . 18 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → ((#‘𝐴) + (#‘𝐵)) = ((#‘𝐵) + (#‘𝐴)))
35		nn0z 11400	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝐴) ∈ ℕ₀ → (#‘𝐴) ∈ ℤ)
36	35	anim1i 592	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → ((#‘𝐴) ∈ ℤ ∧ (#‘𝐵) ∈ ℕ₀))
37	36	ancomd 467	. . . . . . . . . . . . . . . . . . 19 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → ((#‘𝐵) ∈ ℕ₀ ∧ (#‘𝐴) ∈ ℤ))
38		nn0pzuz 11745	. . . . . . . . . . . . . . . . . . 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 2701	. . . . . . . . . . . . . . . . 17 ⊢ (((#‘𝐴) ∈ ℕ₀ ∧ (#‘𝐵) ∈ ℕ₀) → ((#‘𝐴) + (#‘𝐵)) ∈ (ℤ_≥‘(#‘𝐴)))
41	18, 19, 40	syl2an 494	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((#‘𝐴) + (#‘𝐵)) ∈ (ℤ_≥‘(#‘𝐴)))
42		fzoss2 12496	. . . . . . . . . . . . . . . 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 3603	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐴))) → 𝑦 ∈ (0..^((#‘𝐴) + (#‘𝐵))))
45		eleq1 2689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (0..^(#‘𝐴)) ↔ 𝑦 ∈ (0..^(#‘𝐴))))
46		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝐴‘𝑥) = (𝐴‘𝑦))
47		oveq1 6657	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 − (#‘𝐴)) = (𝑦 − (#‘𝐴)))
48	47	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝐵‘(𝑥 − (#‘𝐴))) = (𝐵‘(𝑦 − (#‘𝐴))))
49	45, 46, 48	ifbieq12d 4113	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → if(𝑥 ∈ (0..^(#‘𝐴)), (𝐴‘𝑥), (𝐵‘(𝑥 − (#‘𝐴)))) = if(𝑦 ∈ (0..^(#‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (#‘𝐴)))))
50	49	eleq1d 2686	. . . . . . . . . . . . . . 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 3305	. . . . . . . . . . . . . 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 4092	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0..^(#‘𝐴)) → if(𝑦 ∈ (0..^(#‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (#‘𝐴)))) = (𝐴‘𝑦))
54	53	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐴))) → if(𝑦 ∈ (0..^(#‘𝐴)), (𝐴‘𝑦), (𝐵‘(𝑦 − (#‘𝐴)))) = (𝐴‘𝑦))
55	54	eleq1d 2686	. . . . . . . . . . . . 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 229	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 445	. . . . . . . . . 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 2966	. . . . . . . . 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 13322	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ ∀𝑦 ∈ (0..^(#‘𝐴))(𝐴‘𝑦) ∈ 𝑆) → 𝐴 ∈ Word 𝑆)
61	30, 59, 60	syl2an2r 876	. . . . . . . 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 477	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → 𝐵 ∈ Word V)
63		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → 𝑦 ∈ (0..^(#‘𝐵)))
64	18	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (#‘𝐴) ∈ ℕ₀)
65	64	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → (#‘𝐴) ∈ ℕ₀)
66		elincfzoext 12525	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ (0..^(#‘𝐵)) ∧ (#‘𝐴) ∈ ℕ₀) → (𝑦 + (#‘𝐴)) ∈ (0..^((#‘𝐵) + (#‘𝐴))))
67	63, 65, 66	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → (𝑦 + (#‘𝐴)) ∈ (0..^((#‘𝐵) + (#‘𝐴))))
68	18	nn0cnd 11353	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ Word V → (#‘𝐴) ∈ ℂ)
69	19	nn0cnd 11353	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∈ Word V → (#‘𝐵) ∈ ℂ)
70	68, 69, 33	syl2an 494	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((#‘𝐴) + (#‘𝐵)) = ((#‘𝐵) + (#‘𝐴)))
71	70	oveq2d 6666	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (0..^((#‘𝐴) + (#‘𝐵))) = (0..^((#‘𝐵) + (#‘𝐴))))
72	71	eleq2d 2687	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝑦 + (#‘𝐴)) ∈ (0..^((#‘𝐴) + (#‘𝐵))) ↔ (𝑦 + (#‘𝐴)) ∈ (0..^((#‘𝐵) + (#‘𝐴)))))
73	72	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → ((𝑦 + (#‘𝐴)) ∈ (0..^((#‘𝐴) + (#‘𝐵))) ↔ (𝑦 + (#‘𝐴)) ∈ (0..^((#‘𝐵) + (#‘𝐴)))))
74	67, 73	mpbird 247	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → (𝑦 + (#‘𝐴)) ∈ (0..^((#‘𝐴) + (#‘𝐵))))
75		eleq1 2689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 + (#‘𝐴)) → (𝑥 ∈ (0..^(#‘𝐴)) ↔ (𝑦 + (#‘𝐴)) ∈ (0..^(#‘𝐴))))
76		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 + (#‘𝐴)) → (𝐴‘𝑥) = (𝐴‘(𝑦 + (#‘𝐴))))
77		oveq1 6657	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑦 + (#‘𝐴)) → (𝑥 − (#‘𝐴)) = ((𝑦 + (#‘𝐴)) − (#‘𝐴)))
78	77	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 + (#‘𝐴)) → (𝐵‘(𝑥 − (#‘𝐴))) = (𝐵‘((𝑦 + (#‘𝐴)) − (#‘𝐴))))
79	75, 76, 78	ifbieq12d 4113	. . . . . . . . . . . . . . . 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 2686	. . . . . . . . . . . . . . 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 3305	. . . . . . . . . . . . . 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 11352	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ Word V → (#‘𝐴) ∈ ℝ)
84	83	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (#‘𝐴) ∈ ℝ)
85	84	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → (#‘𝐴) ∈ ℝ)
86		elfzoelz 12470	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (0..^(#‘𝐵)) → 𝑦 ∈ ℤ)
87	86	zred 11482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0..^(#‘𝐵)) → 𝑦 ∈ ℝ)
88	87	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ (0..^(#‘𝐵)) ∧ (𝐴 ∈ Word V ∧ 𝐵 ∈ Word V)) → 𝑦 ∈ ℝ)
89	84	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ (0..^(#‘𝐵)) ∧ (𝐴 ∈ Word V ∧ 𝐵 ∈ Word V)) → (#‘𝐴) ∈ ℝ)
90	88, 89	readdcld 10069	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ (0..^(#‘𝐵)) ∧ (𝐴 ∈ Word V ∧ 𝐵 ∈ Word V)) → (𝑦 + (#‘𝐴)) ∈ ℝ)
91	90	ancoms 469	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → (𝑦 + (#‘𝐴)) ∈ ℝ)
92		elfzole1 12478	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ (0..^(#‘𝐵)) → 0 ≤ 𝑦)
93	92	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → 0 ≤ 𝑦)
94		addge02 10539	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((#‘𝐴) ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 ≤ 𝑦 ↔ (#‘𝐴) ≤ (𝑦 + (#‘𝐴))))
95	84, 87, 94	syl2an 494	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → (0 ≤ 𝑦 ↔ (#‘𝐴) ≤ (𝑦 + (#‘𝐴))))
96	93, 95	mpbid 222	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → (#‘𝐴) ≤ (𝑦 + (#‘𝐴)))
97	85, 91, 96	lensymd 10188	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → ¬ (𝑦 + (#‘𝐴)) < (#‘𝐴))
98	97	intn3an3d 1444	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → ¬ ((𝑦 + (#‘𝐴)) ∈ ℕ₀ ∧ (#‘𝐴) ∈ ℕ ∧ (𝑦 + (#‘𝐴)) < (#‘𝐴)))
99		elfzo0 12508	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 + (#‘𝐴)) ∈ (0..^(#‘𝐴)) ↔ ((𝑦 + (#‘𝐴)) ∈ ℕ₀ ∧ (#‘𝐴) ∈ ℕ ∧ (𝑦 + (#‘𝐴)) < (#‘𝐴)))
100	98, 99	sylnibr 319	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → ¬ (𝑦 + (#‘𝐴)) ∈ (0..^(#‘𝐴)))
101	100	iffalsed 4097	. . . . . . . . . . . . . . 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 2686	. . . . . . . . . . . . . 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 11483	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0..^(#‘𝐵)) → 𝑦 ∈ ℂ)
104	68	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → (#‘𝐴) ∈ ℂ)
105		pncan 10287	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℂ ∧ (#‘𝐴) ∈ ℂ) → ((𝑦 + (#‘𝐴)) − (#‘𝐴)) = 𝑦)
106	103, 104, 105	syl2anr 495	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → ((𝑦 + (#‘𝐴)) − (#‘𝐴)) = 𝑦)
107	106	fveq2d 6195	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → (𝐵‘((𝑦 + (#‘𝐴)) − (#‘𝐴))) = (𝐵‘𝑦))
108	107	eleq1d 2686	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → ((𝐵‘((𝑦 + (#‘𝐴)) − (#‘𝐴))) ∈ 𝑆 ↔ (𝐵‘𝑦) ∈ 𝑆))
109	108	biimpd 219	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) ∧ 𝑦 ∈ (0..^(#‘𝐵))) → ((𝐵‘((𝑦 + (#‘𝐴)) − (#‘𝐴))) ∈ 𝑆 → (𝐵‘𝑦) ∈ 𝑆))
110	102, 109	sylbid 230	. . . . . . . . . . . . 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 47	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 445	. . . . . . . . . 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 2966	. . . . . . . . 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 13322	. . . . . . . . 9 ⊢ ((𝐵 ∈ Word V ∧ ∀𝑦 ∈ (0..^(#‘𝐵))(𝐵‘𝑦) ∈ 𝑆) → 𝐵 ∈ Word 𝑆)
116	62, 114, 115	syl2an2r 876	. . . . . . . 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 554	. . . . . . 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 450	. . . . . 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 233	. . . . 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 230	. . . 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 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)))
122	2, 121	sylbid 230	. 2 ⊢ ((𝐴 ∈ Word V ∧ 𝐵 ∈ Word V) → ((𝐴 ++ 𝐵) ∈ Word 𝑆 → (𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆)))
123		ccatcl 13359	. 2 ⊢ ((𝐴 ∈ Word 𝑆 ∧ 𝐵 ∈ Word 𝑆) → (𝐴 ++ 𝐵) ∈ Word 𝑆)
124	122, 123	impbid1 215	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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⊆ wss 3574 ifcif 4086 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℂcc 9934 ℝcr 9935 0cc0 9936 + caddc 9939 < clt 10074 ≤ cle 10075 − cmin 10266 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ..^cfzo 12465 #chash 13117 Word cword 13291 ++ cconcat 13293

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301

This theorem is referenced by: ccatrcl1 13376

W3C validator