cxplim - Metamath Proof Explorer

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Theorem cxplim 27305

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 13133	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
2	1	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
3		rpge0 13138	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ 𝑥)
4	3	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → 0 ≤ 𝑥)
5		rpre 13133	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
6	5	renegcld 11747	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → -𝐴 ∈ ℝ)
7	6	adantr 486	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → -𝐴 ∈ ℝ)
8		rpcn 13135	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
9		rpne0 13141	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≠ 0)
10	8, 9	negne0d 11671	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → -𝐴 ≠ 0)
11	10	adantr 486	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → -𝐴 ≠ 0)
12	7, 11	rereccld 12148	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (1 / -𝐴) ∈ ℝ)
13	2, 4, 12	recxpcld 27057	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(1 / -𝐴)) ∈ ℝ)
14		simprl 783	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑛 ∈ ℝ⁺)
15	5	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝐴 ∈ ℝ)
16	14, 15	rpcxpcld 27067	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
17	16	rpreccld 13178	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / (𝑛↑_𝑐𝐴)) ∈ ℝ⁺)
18	17	rprege0d 13175	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / (𝑛↑_𝑐𝐴)) ∈ ℝ ∧ 0 ≤ (1 / (𝑛↑_𝑐𝐴))))
19		absid 15469	. . . . . . . 8 ⊢ (((1 / (𝑛↑_𝑐𝐴)) ∈ ℝ ∧ 0 ≤ (1 / (𝑛↑_𝑐𝐴))) → (abs‘(1 / (𝑛↑_𝑐𝐴))) = (1 / (𝑛↑_𝑐𝐴)))
20	18, 19	syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (abs‘(1 / (𝑛↑_𝑐𝐴))) = (1 / (𝑛↑_𝑐𝐴)))
21		simplr 781	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑥 ∈ ℝ⁺)
22		simprr 785	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)
23		rpreccl 13152	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℝ⁺)
24	23	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝐴) ∈ ℝ⁺)
25	24	rpcnd 13170	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝐴) ∈ ℂ)
26	21, 25	cxprecd 27066	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / 𝑥)↑_𝑐(1 / 𝐴)) = (1 / (𝑥↑_𝑐(1 / 𝐴))))
27		rpcn 13135	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
28	27	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑥 ∈ ℂ)
29		rpne0 13141	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
30	29	ad2antlr 740	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑥 ≠ 0)
31	28, 30, 25	cxpnegd 27049	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑥↑_𝑐 -(1 / 𝐴)) = (1 / (𝑥↑_𝑐(1 / 𝐴))))
32		1cnd 11306	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 1 ∈ ℂ)
33	8	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝐴 ∈ ℂ)
34	9	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝐴 ≠ 0)
35	32, 33, 34	divneg2d 12111	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → -(1 / 𝐴) = (1 / -𝐴))
36	35	oveq2d 7437	. . . . . . . . . . 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 2801	. . . . . . . . . 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 12092	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝐴 · (1 / 𝐴)) = 1)
39	38	oveq2d 7437	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐(𝐴 · (1 / 𝐴))) = (𝑛↑_𝑐1))
40	14, 15, 25	cxpmuld 27071	. . . . . . . . . . 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 13170	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 𝑛 ∈ ℂ)
42	41	cxp1d 27040	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐1) = 𝑛)
43	39, 40, 42	3eqtr3d 2803	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((𝑛↑_𝑐𝐴)↑_𝑐(1 / 𝐴)) = 𝑛)
44	22, 37, 43	3brtr4d 5136	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → ((1 / 𝑥)↑_𝑐(1 / 𝐴)) < ((𝑛↑_𝑐𝐴)↑_𝑐(1 / 𝐴)))
45		rpreccl 13152	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
46	45	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝑥) ∈ ℝ⁺)
47	46	rpred 13168	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝑥) ∈ ℝ)
48	46	rpge0d 13172	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 0 ≤ (1 / 𝑥))
49	16	rpred 13168	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (𝑛↑_𝑐𝐴) ∈ ℝ)
50	16	rpge0d 13172	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → 0 ≤ (𝑛↑_𝑐𝐴))
51	47, 48, 49, 50, 24	cxplt2d 27060	. . . . . . . . 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 260	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / 𝑥) < (𝑛↑_𝑐𝐴))
53	21, 16, 52	ltrec1d 13188	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (1 / (𝑛↑_𝑐𝐴)) < 𝑥)
54	20, 53	eqbrtrd 5126	. . . . . 6 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑛 ∈ ℝ⁺ ∧ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛)) → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥)
55	54	expr 462	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ ℝ⁺) → ((𝑥↑_𝑐(1 / -𝐴)) < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
56	55	ralrimiva 3154	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → ∀𝑛 ∈ ℝ⁺ ((𝑥↑_𝑐(1 / -𝐴)) < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
57		breq1 5105	. . . . 5 ⊢ (𝑦 = (𝑥↑_𝑐(1 / -𝐴)) → (𝑦 < 𝑛 ↔ (𝑥↑_𝑐(1 / -𝐴)) < 𝑛))
58	57	rspceaimv 3582	. . . 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 596	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
60	59	ralrimiva 3154	. 2 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘(1 / (𝑛↑_𝑐𝐴))) < 𝑥))
61		id 23	. . . . . . 7 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℝ⁺)
62		rpcxpcl 27010	. . . . . . 7 ⊢ ((𝑛 ∈ ℝ⁺ ∧ 𝐴 ∈ ℝ) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
63	61, 5, 62	syl2anr 609	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
64	63	rpreccld 13178	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (1 / (𝑛↑_𝑐𝐴)) ∈ ℝ⁺)
65	64	rpcnd 13170	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (1 / (𝑛↑_𝑐𝐴)) ∈ ℂ)
66	65	ralrimiva 3154	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑛 ∈ ℝ⁺ (1 / (𝑛↑_𝑐𝐴)) ∈ ℂ)
67		rpssre 13132	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
68	67	a1i 11	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ℝ⁺ ⊆ ℝ)
69	66, 68	rlim0lt 15682	. 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 260	1 ⊢ (𝐴 ∈ ℝ⁺ → (𝑛 ∈ ℝ⁺ ↦ (1 / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 ⊆ wss 3895 class class class wbr 5102 ↦ cmpt 5185 ‘cfv 6539 (class class class)co 7421 ℂcc 11202 ℝcr 11203 0cc0 11204 1c1 11205 · cmul 11209 < clt 11347 ≤ cle 11348 -cneg 11546 / cdiv 11977 ℝ⁺crp 13124 abscabs 15407 ⇝_𝑟 crli 15658 ↑_𝑐ccxp 26889

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-inf2 9646 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-pre-sup 11282 ax-addf 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-of 7694 df-om 7879 df-1st 8002 df-2nd 8003 df-supp 8181 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-2o 8481 df-er 8721 df-map 8853 df-pm 8854 df-ixp 8930 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-fsupp 9358 df-fi 9407 df-sup 9438 df-inf 9439 df-oi 9508 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-z 12698 df-dec 12819 df-uz 12970 df-q 13080 df-rp 13125 df-xneg 13245 df-xadd 13246 df-xmul 13247 df-ioo 13484 df-ioc 13485 df-ico 13486 df-icc 13487 df-fz 13644 df-fzo 13793 df-fl 13936 df-mod 14014 df-seq 14149 df-exp 14209 df-fac 14422 df-bc 14451 df-hash 14479 df-shft 15226 df-cj 15272 df-re 15273 df-im 15274 df-sqrt 15408 df-abs 15409 df-limsup 15644 df-clim 15661 df-rlim 15662 df-sum 15860 df-ef 16239 df-sin 16241 df-cos 16242 df-pi 16244 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-starv 17449 df-sca 17450 df-vsca 17451 df-ip 17452 df-tset 17453 df-ple 17454 df-ds 17456 df-unif 17457 df-hom 17458 df-cco 17459 df-rest 17599 df-topn 17600 df-0g 17618 df-gsum 17619 df-topgen 17620 df-pt 17621 df-prds 17624 df-xrs 17680 df-qtop 17685 df-imas 17686 df-xps 17688 df-mre 17762 df-mrc 17763 df-acs 17765 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-submnd 18985 df-mulg 19284 df-cntz 19537 df-cmn 20002 df-psmet 21676 df-xmet 21677 df-met 21678 df-bl 21679 df-mopn 21680 df-fbas 21681 df-fg 21682 df-cnfld 21685 df-top 23218 df-topon 23235 df-topsp 23257 df-bases 23270 df-cld 23343 df-ntr 23344 df-cls 23345 df-nei 23422 df-lp 23460 df-perf 23461 df-cn 23551 df-cnp 23552 df-haus 23639 df-tx 23887 df-hmeo 24080 df-fil 24171 df-fm 24263 df-flim 24264 df-flf 24265 df-xms 24645 df-ms 24646 df-tms 24647 df-cncf 25205 df-limc 26192 df-dv 26193 df-log 26890 df-cxp 26891

This theorem is used by: sqrtlim 27306 signsplypnf 35226

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