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Theorem cxplim 25231

Description: A power to a negative exponent goes to zero as the base becomes large. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Mario Carneiro, 18-May-2016.)

**Assertion**
Ref	Expression
cxplim	⊢ (𝐴 ∈ ℝ⁺ → (𝑛 ∈ ℝ⁺ ↦ (1 / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0)

Distinct variable group: 𝐴,𝑛

Proof of Theorem cxplim Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpre 12247	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
2	1	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
3		rpge0 12252	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ 𝑥)
4	3	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ 𝑥)
5		rpre 12247	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
6	5	renegcld 10917	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → -𝐴 ∈ ℝ)
7	6	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → -𝐴 ∈ ℝ)
8		rpcn 12249	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
9		rpne0 12255	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
10	8, 9	negne0d 10845	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → -𝐴 ≠ 0)
11	10	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → -𝐴 ≠ 0)
12	7, 11	rereccld 11317	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 / -𝐴) ∈ ℝ)
13	2, 4, 12	recxpcld 24987	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(1 / -𝐴)) ∈ ℝ)
14		simprl 767	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑛 ∈ ℝ⁺)
15	5	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝐴 ∈ ℝ)
16	14, 15	rpcxpcld 24996	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
17	16	rpreccld 12291	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / (𝑛↑_𝑐𝐴)) ∈ ℝ⁺)
18	17	rprege0d 12288	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / (𝑛↑_𝑐𝐴)) ∈ ℝ ∧ 0 ≤ (1 / (𝑛↑_𝑐𝐴))))
19		absid 14490	. . . . . . . 8 ⊢ (((1 / (𝑛↑_𝑐𝐴)) ∈ ℝ ∧ 0 ≤ (1 / (𝑛↑_𝑐𝐴))) → (abs‘(1 / (𝑛↑_𝑐𝐴))) = (1 / (𝑛↑_𝑐𝐴)))
20	18, 19	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (abs‘(1 / (𝑛↑_𝑐𝐴))) = (1 / (𝑛↑_𝑐𝐴)))
21		simplr 765	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑥 ∈ ℝ⁺)
22		simprr 769	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)
23		rpreccl 12265	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℝ⁺)
24	23	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝐴) ∈ ℝ⁺)
25	24	rpcnd 12283	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝐴) ∈ ℂ)
26	21, 25	cxprecd 24995	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / 𝑥)↑_𝑐(1 / 𝐴)) = (1 / (𝑥↑_𝑐(1 / 𝐴))))
27		rpcn 12249	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
28	27	ad2antlr 723	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑥 ∈ ℂ)
29		rpne0 12255	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
30	29	ad2antlr 723	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑥 ≠ 0)
31	28, 30, 25	cxpnegd 24979	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑥↑_𝑐-(1 / 𝐴)) = (1 / (𝑥↑_𝑐(1 / 𝐴))))
32		1cnd 10485	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 1 ∈ ℂ)
33	8	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝐴 ∈ ℂ)
34	9	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝐴 ≠ 0)
35	32, 33, 34	divneg2d 11280	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → -(1 / 𝐴) = (1 / -𝐴))
36	35	oveq2d 7035	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑥↑_𝑐-(1 / 𝐴)) = (𝑥↑_𝑐(1 / -𝐴)))
37	26, 31, 36	3eqtr2d 2836	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / 𝑥)↑_𝑐(1 / 𝐴)) = (𝑥↑_𝑐(1 / -𝐴)))
38	33, 34	recidd 11261	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝐴 · (1 / 𝐴)) = 1)
39	38	oveq2d 7035	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐(𝐴 · (1 / 𝐴))) = (𝑛↑_𝑐1))
40	14, 15, 25	cxpmuld 25000	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐(𝐴 · (1 / 𝐴))) = ((𝑛↑_𝑐𝐴)↑_𝑐(1 / 𝐴)))
41	14	rpcnd 12283	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑛 ∈ ℂ)
42	41	cxp1d 24970	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐1) = 𝑛)
43	39, 40, 42	3eqtr3d 2838	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((𝑛↑_𝑐𝐴)↑_𝑐(1 / 𝐴)) = 𝑛)
44	22, 37, 43	3brtr4d 4996	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / 𝑥)↑_𝑐(1 / 𝐴)) < ((𝑛↑_𝑐𝐴)↑_𝑐(1 / 𝐴)))
45		rpreccl 12265	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
46	45	ad2antlr 723	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝑥) ∈ ℝ⁺)
47	46	rpred 12281	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝑥) ∈ ℝ)
48	46	rpge0d 12285	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 0 ≤ (1 / 𝑥))
49	16	rpred 12281	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐𝐴) ∈ ℝ)
50	16	rpge0d 12285	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 0 ≤ (𝑛↑_𝑐𝐴))
51	47, 48, 49, 50, 24	cxplt2d 24990	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / 𝑥) < (𝑛↑_𝑐𝐴) ↔ ((1 / 𝑥)↑_𝑐(1 / 𝐴)) < ((𝑛↑_𝑐𝐴)↑_𝑐(1 / 𝐴))))
52	44, 51	mpbird 258	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝑥) < (𝑛↑_𝑐𝐴))
53	21, 16, 52	ltrec1d 12301	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / (𝑛↑_𝑐𝐴)) < 𝑥)
54	20, 53	eqbrtrd 4986	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥)
55	54	expr 457	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ℝ⁺) → ((𝑥↑_𝑐(1 / -𝐴)) < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
56	55	ralrimiva 3148	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → ∀𝑛 ∈ ℝ⁺ ((𝑥↑_𝑐(1 / -𝐴)) < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
57		breq1 4967	. . . . 5 ⊢ (𝑦 = (𝑥↑_𝑐(1 / -𝐴)) → (𝑦 < 𝑛 ↔ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛))
58	57	rspceaimv 3565	. . . 4 ⊢ (((𝑥↑_𝑐(1 / -𝐴)) ∈ ℝ ∧ ∀𝑛 ∈ ℝ⁺ ((𝑥↑_𝑐(1 / -𝐴)) < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
59	13, 56, 58	syl2anc 584	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
60	59	ralrimiva 3148	. 2 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
61		id 22	. . . . . . 7 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℝ⁺)
62		rpcxpcl 24940	. . . . . . 7 ⊢ ((𝑛 ∈ ℝ⁺ ∧ 𝐴 ∈ ℝ) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
63	61, 5, 62	syl2anr 596	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
64	63	rpreccld 12291	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (1 / (𝑛↑_𝑐𝐴)) ∈ ℝ⁺)
65	64	rpcnd 12283	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (1 / (𝑛↑_𝑐𝐴)) ∈ ℂ)
66	65	ralrimiva 3148	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑛 ∈ ℝ⁺ (1 / (𝑛↑_𝑐𝐴)) ∈ ℂ)
67		rpssre 12246	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
68	67	a1i 11	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ℝ⁺ ⊆ ℝ)
69	66, 68	rlim0lt 14700	. 2 ⊢ (𝐴 ∈ ℝ⁺ → ((𝑛 ∈ ℝ⁺ ↦ (1 / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥)))
70	60, 69	mpbird 258	1 ⊢ (𝐴 ∈ ℝ⁺ → (𝑛 ∈ ℝ⁺ ↦ (1 / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2080 ≠ wne 2983 ∀wral 3104 ∃wrex 3105 ⊆ wss 3861 class class class wbr 4964 ↦ cmpt 5043 ‘cfv 6228 (class class class)co 7019 ℂcc 10384 ℝcr 10385 0cc0 10386 1c1 10387 · cmul 10391 < clt 10524 ≤ cle 10525 -cneg 10720 / cdiv 11147 ℝ⁺crp 12239 abscabs 14427 ⇝_𝑟 crli 14676 ↑_𝑐ccxp 24820

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-rep 5084 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-inf2 8953 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 ax-pre-sup 10464 ax-addf 10465 ax-mulf 10466

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-int 4785 df-iun 4829 df-iin 4830 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-se 5406 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-isom 6237 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-of 7270 df-om 7440 df-1st 7548 df-2nd 7549 df-supp 7685 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-1o 7956 df-2o 7957 df-oadd 7960 df-er 8142 df-map 8261 df-pm 8262 df-ixp 8314 df-en 8361 df-dom 8362 df-sdom 8363 df-fin 8364 df-fsupp 8683 df-fi 8724 df-sup 8755 df-inf 8756 df-oi 8823 df-card 9217 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-div 11148 df-nn 11489 df-2 11550 df-3 11551 df-4 11552 df-5 11553 df-6 11554 df-7 11555 df-8 11556 df-9 11557 df-n0 11748 df-z 11832 df-dec 11949 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-ioo 12592 df-ioc 12593 df-ico 12594 df-icc 12595 df-fz 12743 df-fzo 12884 df-fl 13012 df-mod 13088 df-seq 13220 df-exp 13280 df-fac 13484 df-bc 13513 df-hash 13541 df-shft 14260 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-limsup 14662 df-clim 14679 df-rlim 14680 df-sum 14877 df-ef 15254 df-sin 15256 df-cos 15257 df-pi 15259 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-sca 16410 df-vsca 16411 df-ip 16412 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-hom 16418 df-cco 16419 df-rest 16525 df-topn 16526 df-0g 16544 df-gsum 16545 df-topgen 16546 df-pt 16547 df-prds 16550 df-xrs 16604 df-qtop 16609 df-imas 16610 df-xps 16612 df-mre 16686 df-mrc 16687 df-acs 16689 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-mulg 17982 df-cntz 18188 df-cmn 18635 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-fbas 20224 df-fg 20225 df-cnfld 20228 df-top 21186 df-topon 21203 df-topsp 21225 df-bases 21238 df-cld 21311 df-ntr 21312 df-cls 21313 df-nei 21390 df-lp 21428 df-perf 21429 df-cn 21519 df-cnp 21520 df-haus 21607 df-tx 21854 df-hmeo 22047 df-fil 22138 df-fm 22230 df-flim 22231 df-flf 22232 df-xms 22613 df-ms 22614 df-tms 22615 df-cncf 23169 df-limc 24147 df-dv 24148 df-log 24821 df-cxp 24822

This theorem is referenced by: sqrtlim 25232 signsplypnf 31429

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