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Theorem cxplim 26121

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 12738	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
2	1	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
3		rpge0 12743	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ 𝑥)
4	3	adantl 482	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ 𝑥)
5		rpre 12738	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
6	5	renegcld 11402	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → -𝐴 ∈ ℝ)
7	6	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → -𝐴 ∈ ℝ)
8		rpcn 12740	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
9		rpne0 12746	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
10	8, 9	negne0d 11330	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → -𝐴 ≠ 0)
11	10	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → -𝐴 ≠ 0)
12	7, 11	rereccld 11802	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 / -𝐴) ∈ ℝ)
13	2, 4, 12	recxpcld 25878	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(1 / -𝐴)) ∈ ℝ)
14		simprl 768	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑛 ∈ ℝ⁺)
15	5	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝐴 ∈ ℝ)
16	14, 15	rpcxpcld 25887	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
17	16	rpreccld 12782	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / (𝑛↑_𝑐𝐴)) ∈ ℝ⁺)
18	17	rprege0d 12779	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / (𝑛↑_𝑐𝐴)) ∈ ℝ ∧ 0 ≤ (1 / (𝑛↑_𝑐𝐴))))
19		absid 15008	. . . . . . . 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 766	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑥 ∈ ℝ⁺)
22		simprr 770	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)
23		rpreccl 12756	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℝ⁺)
24	23	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝐴) ∈ ℝ⁺)
25	24	rpcnd 12774	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝐴) ∈ ℂ)
26	21, 25	cxprecd 25886	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / 𝑥)↑_𝑐(1 / 𝐴)) = (1 / (𝑥↑_𝑐(1 / 𝐴))))
27		rpcn 12740	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
28	27	ad2antlr 724	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑥 ∈ ℂ)
29		rpne0 12746	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
30	29	ad2antlr 724	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑥 ≠ 0)
31	28, 30, 25	cxpnegd 25870	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑥↑_𝑐-(1 / 𝐴)) = (1 / (𝑥↑_𝑐(1 / 𝐴))))
32		1cnd 10970	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 1 ∈ ℂ)
33	8	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝐴 ∈ ℂ)
34	9	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝐴 ≠ 0)
35	32, 33, 34	divneg2d 11765	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → -(1 / 𝐴) = (1 / -𝐴))
36	35	oveq2d 7291	. . . . . . . . . . 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 2784	. . . . . . . . . 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 11746	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝐴 · (1 / 𝐴)) = 1)
39	38	oveq2d 7291	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐(𝐴 · (1 / 𝐴))) = (𝑛↑_𝑐1))
40	14, 15, 25	cxpmuld 25891	. . . . . . . . . . 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 12774	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑛 ∈ ℂ)
42	41	cxp1d 25861	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐1) = 𝑛)
43	39, 40, 42	3eqtr3d 2786	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((𝑛↑_𝑐𝐴)↑_𝑐(1 / 𝐴)) = 𝑛)
44	22, 37, 43	3brtr4d 5106	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / 𝑥)↑_𝑐(1 / 𝐴)) < ((𝑛↑_𝑐𝐴)↑_𝑐(1 / 𝐴)))
45		rpreccl 12756	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
46	45	ad2antlr 724	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝑥) ∈ ℝ⁺)
47	46	rpred 12772	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝑥) ∈ ℝ)
48	46	rpge0d 12776	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 0 ≤ (1 / 𝑥))
49	16	rpred 12772	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐𝐴) ∈ ℝ)
50	16	rpge0d 12776	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 0 ≤ (𝑛↑_𝑐𝐴))
51	47, 48, 49, 50, 24	cxplt2d 25881	. . . . . . . . 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 256	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝑥) < (𝑛↑_𝑐𝐴))
53	21, 16, 52	ltrec1d 12792	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / (𝑛↑_𝑐𝐴)) < 𝑥)
54	20, 53	eqbrtrd 5096	. . . . . 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 3103	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → ∀𝑛 ∈ ℝ⁺ ((𝑥↑_𝑐(1 / -𝐴)) < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
57		breq1 5077	. . . . 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 3103	. 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 25831	. . . . . . 7 ⊢ ((𝑛 ∈ ℝ⁺ ∧ 𝐴 ∈ ℝ) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
63	61, 5, 62	syl2anr 597	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
64	63	rpreccld 12782	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (1 / (𝑛↑_𝑐𝐴)) ∈ ℝ⁺)
65	64	rpcnd 12774	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (1 / (𝑛↑_𝑐𝐴)) ∈ ℂ)
66	65	ralrimiva 3103	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑛 ∈ ℝ⁺ (1 / (𝑛↑_𝑐𝐴)) ∈ ℂ)
67		rpssre 12737	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
68	67	a1i 11	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ℝ⁺ ⊆ ℝ)
69	66, 68	rlim0lt 15218	. 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 256	1 ⊢ (𝐴 ∈ ℝ⁺ → (𝑛 ∈ ℝ⁺ ↦ (1 / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 · cmul 10876 < clt 11009 ≤ cle 11010 -cneg 11206 / cdiv 11632 ℝ⁺crp 12730 abscabs 14945 ⇝_𝑟 crli 15194 ↑_𝑐ccxp 25711

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-inf2 9399 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 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

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-rmo 3071 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-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 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-se 5545 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-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 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-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-log 25712 df-cxp 25713

This theorem is referenced by: sqrtlim 26122 signsplypnf 32529

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