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Theorem rlimcnp2 24493

Description: Relate a limit of a real-valued sequence at infinity to the continuity of the function 𝑆(𝑦) = 𝑅(1 / 𝑦) at zero. (Contributed by Mario Carneiro, 1-Mar-2015.)

**Hypotheses**
Ref	Expression
rlimcnp2.a	⊢ (𝜑 → 𝐴 ⊆ (0[,)+∞))
rlimcnp2.0	⊢ (𝜑 → 0 ∈ 𝐴)
rlimcnp2.b	⊢ (𝜑 → 𝐵 ⊆ ℝ)
rlimcnp2.c	⊢ (𝜑 → 𝐶 ∈ ℂ)
rlimcnp2.r	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑆 ∈ ℂ)
rlimcnp2.d	⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝑦 ∈ 𝐵 ↔ (1 / 𝑦) ∈ 𝐴))
rlimcnp2.s	⊢ (𝑦 = (1 / 𝑥) → 𝑆 = 𝑅)
rlimcnp2.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)
rlimcnp2.k	⊢ 𝐾 = (𝐽 ↾_t 𝐴)

**Assertion**
Ref	Expression
rlimcnp2	⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ⇝_𝑟 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ if(𝑥 = 0, 𝐶, 𝑅)) ∈ ((𝐾 CnP 𝐽)‘0)))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐵,𝑦 𝑥,𝐶,𝑦 𝜑,𝑥,𝑦 𝑦,𝑅 𝑥,𝑆 Allowed substitution hints: 𝑅(𝑥) 𝑆(𝑦) 𝐽(𝑥,𝑦) 𝐾(𝑥,𝑦)

Proof of Theorem rlimcnp2 Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 3795	. . . . . . . 8 ⊢ (𝐵 ∩ (1[,)+∞)) ⊆ 𝐵
2		resmpt 5369	. . . . . . . 8 ⊢ ((𝐵 ∩ (1[,)+∞)) ⊆ 𝐵 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞))) = (𝑦 ∈ (𝐵 ∩ (1[,)+∞)) ↦ 𝑆))
3	1, 2	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞))) = (𝑦 ∈ (𝐵 ∩ (1[,)+∞)) ↦ 𝑆))
4		0xr 9965	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
5		0lt1 10429	. . . . . . . . . . 11 ⊢ 0 < 1
6		df-ioo 12050	. . . . . . . . . . . 12 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
7		df-ico 12052	. . . . . . . . . . . 12 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
8		xrltletr 11864	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((0 < 1 ∧ 1 ≤ 𝑤) → 0 < 𝑤))
9	6, 7, 8	ixxss1 12064	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ^* ∧ 0 < 1) → (1[,)+∞) ⊆ (0(,)+∞))
10	4, 5, 9	mp2an 704	. . . . . . . . . 10 ⊢ (1[,)+∞) ⊆ (0(,)+∞)
11		ioorp 12122	. . . . . . . . . 10 ⊢ (0(,)+∞) = ℝ⁺
12	10, 11	sseqtri 3600	. . . . . . . . 9 ⊢ (1[,)+∞) ⊆ ℝ⁺
13		sslin 3801	. . . . . . . . 9 ⊢ ((1[,)+∞) ⊆ ℝ⁺ → (𝐵 ∩ (1[,)+∞)) ⊆ (𝐵 ∩ ℝ⁺))
14	12, 13	ax-mp 5	. . . . . . . 8 ⊢ (𝐵 ∩ (1[,)+∞)) ⊆ (𝐵 ∩ ℝ⁺)
15		resmpt 5369	. . . . . . . 8 ⊢ ((𝐵 ∩ (1[,)+∞)) ⊆ (𝐵 ∩ ℝ⁺) → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞))) = (𝑦 ∈ (𝐵 ∩ (1[,)+∞)) ↦ 𝑆))
16	14, 15	mp1i 13	. . . . . . 7 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞))) = (𝑦 ∈ (𝐵 ∩ (1[,)+∞)) ↦ 𝑆))
17	3, 16	eqtr4d 2647	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞))) = ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞))))
18		resres 5329	. . . . . 6 ⊢ (((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)) = ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞)))
19		resres 5329	. . . . . 6 ⊢ (((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)) = ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞)))
20	17, 18, 19	3eqtr4g 2669	. . . . 5 ⊢ (𝜑 → (((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)) = (((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)))
21		rlimcnp2.r	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑆 ∈ ℂ)
22		eqid 2610	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑦 ∈ 𝐵 ↦ 𝑆)
23	21, 22	fmptd 6292	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝑆):𝐵⟶ℂ)
24		ffn 5958	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ↦ 𝑆):𝐵⟶ℂ → (𝑦 ∈ 𝐵 ↦ 𝑆) Fn 𝐵)
25	23, 24	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝑆) Fn 𝐵)
26		fnresdm 5914	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ↦ 𝑆) Fn 𝐵 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ 𝑆))
27	25, 26	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ 𝑆))
28	27	reseq1d 5316	. . . . 5 ⊢ (𝜑 → (((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)) = ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (1[,)+∞)))
29		inss1 3795	. . . . . . . . . . 11 ⊢ (𝐵 ∩ ℝ⁺) ⊆ 𝐵
30	29	sseli 3564	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∩ ℝ⁺) → 𝑦 ∈ 𝐵)
31	30, 21	sylan2 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → 𝑆 ∈ ℂ)
32		eqid 2610	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) = (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆)
33	31, 32	fmptd 6292	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆):(𝐵 ∩ ℝ⁺)⟶ℂ)
34		frel 5963	. . . . . . . 8 ⊢ ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆):(𝐵 ∩ ℝ⁺)⟶ℂ → Rel (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆))
35	33, 34	syl 17	. . . . . . 7 ⊢ (𝜑 → Rel (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆))
36	32, 31	dmmptd 5937	. . . . . . . 8 ⊢ (𝜑 → dom (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) = (𝐵 ∩ ℝ⁺))
37	36, 29	syl6eqss 3618	. . . . . . 7 ⊢ (𝜑 → dom (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ⊆ 𝐵)
38		relssres 5357	. . . . . . 7 ⊢ ((Rel (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ∧ dom (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ⊆ 𝐵) → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ 𝐵) = (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆))
39	35, 37, 38	syl2anc 691	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ 𝐵) = (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆))
40	39	reseq1d 5316	. . . . 5 ⊢ (𝜑 → (((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)) = ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (1[,)+∞)))
41	20, 28, 40	3eqtr3d 2652	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (1[,)+∞)) = ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (1[,)+∞)))
42	41	breq1d 4593	. . 3 ⊢ (𝜑 → (((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (1[,)+∞)) ⇝_𝑟 𝐶 ↔ ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (1[,)+∞)) ⇝_𝑟 𝐶))
43		rlimcnp2.b	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
44		1red 9934	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
45	23, 43, 44	rlimresb 14144	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ⇝_𝑟 𝐶 ↔ ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (1[,)+∞)) ⇝_𝑟 𝐶))
46	29, 43	syl5ss 3579	. . . 4 ⊢ (𝜑 → (𝐵 ∩ ℝ⁺) ⊆ ℝ)
47	33, 46, 44	rlimresb 14144	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ⇝_𝑟 𝐶 ↔ ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (1[,)+∞)) ⇝_𝑟 𝐶))
48	42, 45, 47	3bitr4d 299	. 2 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ⇝_𝑟 𝐶 ↔ (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ⇝_𝑟 𝐶))
49		inss2 3796	. . . . . . . . . . 11 ⊢ (𝐵 ∩ ℝ⁺) ⊆ ℝ⁺
50	49	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ ℝ⁺) ⊆ ℝ⁺)
51	50	sselda 3568	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → 𝑦 ∈ ℝ⁺)
52	51	rpreccld 11758	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → (1 / 𝑦) ∈ ℝ⁺)
53	52	rpne0d 11753	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → (1 / 𝑦) ≠ 0)
54	53	neneqd 2787	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → ¬ (1 / 𝑦) = 0)
55	54	iffalsed 4047	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → if((1 / 𝑦) = 0, 𝐶, ⦋(1 / 𝑦) / 𝑥⦌𝑅) = ⦋(1 / 𝑦) / 𝑥⦌𝑅)
56		oveq2 6557	. . . . . . . . . 10 ⊢ (𝑥 = (1 / 𝑦) → (1 / 𝑥) = (1 / (1 / 𝑦)))
57		rpcnne0 11726	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ⁺ → (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
58		recrec 10601	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℂ ∧ 𝑦 ≠ 0) → (1 / (1 / 𝑦)) = 𝑦)
59	51, 57, 58	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → (1 / (1 / 𝑦)) = 𝑦)
60	56, 59	sylan9eqr 2666	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) ∧ 𝑥 = (1 / 𝑦)) → (1 / 𝑥) = 𝑦)
61	60	eqcomd 2616	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) ∧ 𝑥 = (1 / 𝑦)) → 𝑦 = (1 / 𝑥))
62		rlimcnp2.s	. . . . . . . 8 ⊢ (𝑦 = (1 / 𝑥) → 𝑆 = 𝑅)
63	61, 62	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) ∧ 𝑥 = (1 / 𝑦)) → 𝑆 = 𝑅)
64	63	eqcomd 2616	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) ∧ 𝑥 = (1 / 𝑦)) → 𝑅 = 𝑆)
65	52, 64	csbied 3526	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → ⦋(1 / 𝑦) / 𝑥⦌𝑅 = 𝑆)
66	55, 65	eqtrd 2644	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → if((1 / 𝑦) = 0, 𝐶, ⦋(1 / 𝑦) / 𝑥⦌𝑅) = 𝑆)
67	66	mpteq2dva 4672	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ if((1 / 𝑦) = 0, 𝐶, ⦋(1 / 𝑦) / 𝑥⦌𝑅)) = (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆))
68	67	breq1d 4593	. 2 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ if((1 / 𝑦) = 0, 𝐶, ⦋(1 / 𝑦) / 𝑥⦌𝑅)) ⇝_𝑟 𝐶 ↔ (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ⇝_𝑟 𝐶))
69		rlimcnp2.a	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ (0[,)+∞))
70		rlimcnp2.0	. . . 4 ⊢ (𝜑 → 0 ∈ 𝐴)
71		rlimcnp2.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ)
72	71	ad2antrr 758	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ 𝑤 = 0) → 𝐶 ∈ ℂ)
73	69	sselda 3568	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ (0[,)+∞))
74		0re 9919	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
75		pnfxr 9971	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
76		elico2 12108	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ^*) → (𝑤 ∈ (0[,)+∞) ↔ (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 < +∞)))
77	74, 75, 76	mp2an 704	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0[,)+∞) ↔ (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 < +∞))
78	73, 77	sylib 207	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 < +∞))
79	78	simp1d 1066	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
80	79	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → 𝑤 ∈ ℝ)
81	78	simp2d 1067	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 0 ≤ 𝑤)
82		leloe 10003	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ 𝑤 ∈ ℝ) → (0 ≤ 𝑤 ↔ (0 < 𝑤 ∨ 0 = 𝑤)))
83	74, 79, 82	sylancr 694	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (0 ≤ 𝑤 ↔ (0 < 𝑤 ∨ 0 = 𝑤)))
84	81, 83	mpbid 221	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (0 < 𝑤 ∨ 0 = 𝑤))
85	84	ord 391	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (¬ 0 < 𝑤 → 0 = 𝑤))
86		eqcom 2617	. . . . . . . . . . . 12 ⊢ (0 = 𝑤 ↔ 𝑤 = 0)
87	85, 86	syl6ib 240	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (¬ 0 < 𝑤 → 𝑤 = 0))
88	87	con1d 138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (¬ 𝑤 = 0 → 0 < 𝑤))
89	88	imp 444	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → 0 < 𝑤)
90	80, 89	elrpd 11745	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → 𝑤 ∈ ℝ⁺)
91		rpcnne0 11726	. . . . . . . . 9 ⊢ (𝑤 ∈ ℝ⁺ → (𝑤 ∈ ℂ ∧ 𝑤 ≠ 0))
92		recrec 10601	. . . . . . . . 9 ⊢ ((𝑤 ∈ ℂ ∧ 𝑤 ≠ 0) → (1 / (1 / 𝑤)) = 𝑤)
93	91, 92	syl 17	. . . . . . . 8 ⊢ (𝑤 ∈ ℝ⁺ → (1 / (1 / 𝑤)) = 𝑤)
94	90, 93	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → (1 / (1 / 𝑤)) = 𝑤)
95	94	csbeq1d 3506	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → ⦋(1 / (1 / 𝑤)) / 𝑥⦌𝑅 = ⦋𝑤 / 𝑥⦌𝑅)
96		simplr 788	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → 𝑤 ∈ 𝐴)
97		simpll 786	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → 𝜑)
98		rpreccl 11733	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ℝ⁺ → (1 / 𝑤) ∈ ℝ⁺)
99	98	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (1 / 𝑤) ∈ ℝ⁺)
100		rlimcnp2.d	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝑦 ∈ 𝐵 ↔ (1 / 𝑦) ∈ 𝐴))
101	100	ralrimiva 2949	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ (𝑦 ∈ 𝐵 ↔ (1 / 𝑦) ∈ 𝐴))
102	101	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ∀𝑦 ∈ ℝ⁺ (𝑦 ∈ 𝐵 ↔ (1 / 𝑦) ∈ 𝐴))
103		eleq1 2676	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (1 / 𝑤) → (𝑦 ∈ 𝐵 ↔ (1 / 𝑤) ∈ 𝐵))
104		oveq2 6557	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (1 / 𝑤) → (1 / 𝑦) = (1 / (1 / 𝑤)))
105	104	eleq1d 2672	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (1 / 𝑤) → ((1 / 𝑦) ∈ 𝐴 ↔ (1 / (1 / 𝑤)) ∈ 𝐴))
106	103, 105	bibi12d 334	. . . . . . . . . . . . 13 ⊢ (𝑦 = (1 / 𝑤) → ((𝑦 ∈ 𝐵 ↔ (1 / 𝑦) ∈ 𝐴) ↔ ((1 / 𝑤) ∈ 𝐵 ↔ (1 / (1 / 𝑤)) ∈ 𝐴)))
107	106	rspcv 3278	. . . . . . . . . . . 12 ⊢ ((1 / 𝑤) ∈ ℝ⁺ → (∀𝑦 ∈ ℝ⁺ (𝑦 ∈ 𝐵 ↔ (1 / 𝑦) ∈ 𝐴) → ((1 / 𝑤) ∈ 𝐵 ↔ (1 / (1 / 𝑤)) ∈ 𝐴)))
108	99, 102, 107	sylc 63	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ((1 / 𝑤) ∈ 𝐵 ↔ (1 / (1 / 𝑤)) ∈ 𝐴))
109	93	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (1 / (1 / 𝑤)) = 𝑤)
110	109	eleq1d 2672	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ((1 / (1 / 𝑤)) ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
111	108, 110	bitr2d 268	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (𝑤 ∈ 𝐴 ↔ (1 / 𝑤) ∈ 𝐵))
112	97, 90, 111	syl2anc 691	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → (𝑤 ∈ 𝐴 ↔ (1 / 𝑤) ∈ 𝐵))
113	96, 112	mpbid 221	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → (1 / 𝑤) ∈ 𝐵)
114	90	rpreccld 11758	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → (1 / 𝑤) ∈ ℝ⁺)
115	113, 114	elind 3760	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → (1 / 𝑤) ∈ (𝐵 ∩ ℝ⁺))
116	65, 31	eqeltrd 2688	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → ⦋(1 / 𝑦) / 𝑥⦌𝑅 ∈ ℂ)
117	116	ralrimiva 2949	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ (𝐵 ∩ ℝ⁺)⦋(1 / 𝑦) / 𝑥⦌𝑅 ∈ ℂ)
118	117	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → ∀𝑦 ∈ (𝐵 ∩ ℝ⁺)⦋(1 / 𝑦) / 𝑥⦌𝑅 ∈ ℂ)
119	104	csbeq1d 3506	. . . . . . . . 9 ⊢ (𝑦 = (1 / 𝑤) → ⦋(1 / 𝑦) / 𝑥⦌𝑅 = ⦋(1 / (1 / 𝑤)) / 𝑥⦌𝑅)
120	119	eleq1d 2672	. . . . . . . 8 ⊢ (𝑦 = (1 / 𝑤) → (⦋(1 / 𝑦) / 𝑥⦌𝑅 ∈ ℂ ↔ ⦋(1 / (1 / 𝑤)) / 𝑥⦌𝑅 ∈ ℂ))
121	120	rspcv 3278	. . . . . . 7 ⊢ ((1 / 𝑤) ∈ (𝐵 ∩ ℝ⁺) → (∀𝑦 ∈ (𝐵 ∩ ℝ⁺)⦋(1 / 𝑦) / 𝑥⦌𝑅 ∈ ℂ → ⦋(1 / (1 / 𝑤)) / 𝑥⦌𝑅 ∈ ℂ))
122	115, 118, 121	sylc 63	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → ⦋(1 / (1 / 𝑤)) / 𝑥⦌𝑅 ∈ ℂ)
123	95, 122	eqeltrrd 2689	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → ⦋𝑤 / 𝑥⦌𝑅 ∈ ℂ)
124	72, 123	ifclda 4070	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅) ∈ ℂ)
125	99	biantrud 527	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ((1 / 𝑤) ∈ 𝐵 ↔ ((1 / 𝑤) ∈ 𝐵 ∧ (1 / 𝑤) ∈ ℝ⁺)))
126	111, 125	bitrd 267	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (𝑤 ∈ 𝐴 ↔ ((1 / 𝑤) ∈ 𝐵 ∧ (1 / 𝑤) ∈ ℝ⁺)))
127		elin 3758	. . . . 5 ⊢ ((1 / 𝑤) ∈ (𝐵 ∩ ℝ⁺) ↔ ((1 / 𝑤) ∈ 𝐵 ∧ (1 / 𝑤) ∈ ℝ⁺))
128	126, 127	syl6bbr 277	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (𝑤 ∈ 𝐴 ↔ (1 / 𝑤) ∈ (𝐵 ∩ ℝ⁺)))
129		iftrue 4042	. . . 4 ⊢ (𝑤 = 0 → if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅) = 𝐶)
130		eqeq1 2614	. . . . 5 ⊢ (𝑤 = (1 / 𝑦) → (𝑤 = 0 ↔ (1 / 𝑦) = 0))
131		csbeq1 3502	. . . . 5 ⊢ (𝑤 = (1 / 𝑦) → ⦋𝑤 / 𝑥⦌𝑅 = ⦋(1 / 𝑦) / 𝑥⦌𝑅)
132	130, 131	ifbieq2d 4061	. . . 4 ⊢ (𝑤 = (1 / 𝑦) → if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅) = if((1 / 𝑦) = 0, 𝐶, ⦋(1 / 𝑦) / 𝑥⦌𝑅))
133		rlimcnp2.j	. . . 4 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
134		rlimcnp2.k	. . . 4 ⊢ 𝐾 = (𝐽 ↾_t 𝐴)
135	69, 70, 50, 124, 128, 129, 132, 133, 134	rlimcnp 24492	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ if((1 / 𝑦) = 0, 𝐶, ⦋(1 / 𝑦) / 𝑥⦌𝑅)) ⇝_𝑟 𝐶 ↔ (𝑤 ∈ 𝐴 ↦ if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅)) ∈ ((𝐾 CnP 𝐽)‘0)))
136		nfcv 2751	. . . . 5 ⊢ Ⅎ𝑤if(𝑥 = 0, 𝐶, 𝑅)
137		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑥 𝑤 = 0
138		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑥𝐶
139		nfcsb1v 3515	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝑅
140	137, 138, 139	nfif 4065	. . . . 5 ⊢ Ⅎ𝑥if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅)
141		eqeq1 2614	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑥 = 0 ↔ 𝑤 = 0))
142		csbeq1a 3508	. . . . . 6 ⊢ (𝑥 = 𝑤 → 𝑅 = ⦋𝑤 / 𝑥⦌𝑅)
143	141, 142	ifbieq2d 4061	. . . . 5 ⊢ (𝑥 = 𝑤 → if(𝑥 = 0, 𝐶, 𝑅) = if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅))
144	136, 140, 143	cbvmpt 4677	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ if(𝑥 = 0, 𝐶, 𝑅)) = (𝑤 ∈ 𝐴 ↦ if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅))
145	144	eleq1i 2679	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ if(𝑥 = 0, 𝐶, 𝑅)) ∈ ((𝐾 CnP 𝐽)‘0) ↔ (𝑤 ∈ 𝐴 ↦ if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅)) ∈ ((𝐾 CnP 𝐽)‘0))
146	135, 145	syl6bbr 277	. 2 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ if((1 / 𝑦) = 0, 𝐶, ⦋(1 / 𝑦) / 𝑥⦌𝑅)) ⇝_𝑟 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ if(𝑥 = 0, 𝐶, 𝑅)) ∈ ((𝐾 CnP 𝐽)‘0)))
147	48, 68, 146	3bitr2d 295	1 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ⇝_𝑟 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ if(𝑥 = 0, 𝐶, 𝑅)) ∈ ((𝐾 CnP 𝐽)‘0)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ⦋csb 3499 ∩ cin 3539 ⊆ wss 3540 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ↾ cres 5040 Rel wrel 5043 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 +∞cpnf 9950 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 / cdiv 10563 ℝ⁺crp 11708 (,)cioo 12046 [,)cico 12048 ⇝_𝑟 crli 14064 ↾_t crest 15904 TopOpenctopn 15905 ℂ_fldccnfld 19567 CnP ccnp 20839

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 ax-pre-sup 9893

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-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-rlim 14068 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-rest 15906 df-topn 15907 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-cnp 20842

This theorem is referenced by: rlimcnp3 24494

W3C validator