rlimcnp2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > rlimcnp2		Structured version Visualization version GIF version

Theorem rlimcnp2 24693

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 3833	. . . . . . . 8 ⊢ (𝐵 ∩ (1[,)+∞)) ⊆ 𝐵
2		resmpt 5449	. . . . . . . 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 10086	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ^*
5		0lt1 10550	. . . . . . . . . . 11 ⊢ 0 < 1
6		df-ioo 12179	. . . . . . . . . . . 12 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
7		df-ico 12181	. . . . . . . . . . . 12 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
8		xrltletr 11988	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((0 < 1 ∧ 1 ≤ 𝑤) → 0 < 𝑤))
9	6, 7, 8	ixxss1 12193	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ^* ∧ 0 < 1) → (1[,)+∞) ⊆ (0(,)+∞))
10	4, 5, 9	mp2an 708	. . . . . . . . . 10 ⊢ (1[,)+∞) ⊆ (0(,)+∞)
11		ioorp 12251	. . . . . . . . . 10 ⊢ (0(,)+∞) = ℝ⁺
12	10, 11	sseqtri 3637	. . . . . . . . 9 ⊢ (1[,)+∞) ⊆ ℝ⁺
13		sslin 3839	. . . . . . . . 9 ⊢ ((1[,)+∞) ⊆ ℝ⁺ → (𝐵 ∩ (1[,)+∞)) ⊆ (𝐵 ∩ ℝ⁺))
14	12, 13	ax-mp 5	. . . . . . . 8 ⊢ (𝐵 ∩ (1[,)+∞)) ⊆ (𝐵 ∩ ℝ⁺)
15		resmpt 5449	. . . . . . . 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 2659	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞))) = ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞))))
18		resres 5409	. . . . . 6 ⊢ (((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)) = ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞)))
19		resres 5409	. . . . . 6 ⊢ (((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)) = ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (𝐵 ∩ (1[,)+∞)))
20	17, 18, 19	3eqtr4g 2681	. . . . 5 ⊢ (𝜑 → (((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)) = (((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)))
21		rlimcnp2.r	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝑆 ∈ ℂ)
22		eqid 2622	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 ↦ 𝑆) = (𝑦 ∈ 𝐵 ↦ 𝑆)
23	21, 22	fmptd 6385	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ 𝑆):𝐵⟶ℂ)
24		ffn 6045	. . . . . . . 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 6000	. . . . . . 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 5395	. . . . 5 ⊢ (𝜑 → (((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)) = ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (1[,)+∞)))
29		inss1 3833	. . . . . . . . . . 11 ⊢ (𝐵 ∩ ℝ⁺) ⊆ 𝐵
30	29	sseli 3599	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∩ ℝ⁺) → 𝑦 ∈ 𝐵)
31	30, 21	sylan2 491	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → 𝑆 ∈ ℂ)
32		eqid 2622	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) = (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆)
33	31, 32	fmptd 6385	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆):(𝐵 ∩ ℝ⁺)⟶ℂ)
34		frel 6050	. . . . . . . 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 6024	. . . . . . . 8 ⊢ (𝜑 → dom (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) = (𝐵 ∩ ℝ⁺))
37	36, 29	syl6eqss 3655	. . . . . . 7 ⊢ (𝜑 → dom (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ⊆ 𝐵)
38		relssres 5437	. . . . . . 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 693	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ 𝐵) = (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆))
40	39	reseq1d 5395	. . . . 5 ⊢ (𝜑 → (((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ 𝐵) ↾ (1[,)+∞)) = ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (1[,)+∞)))
41	20, 28, 40	3eqtr3d 2664	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (1[,)+∞)) = ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (1[,)+∞)))
42	41	breq1d 4663	. . 3 ⊢ (𝜑 → (((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (1[,)+∞)) ⇝_𝑟 𝐶 ↔ ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (1[,)+∞)) ⇝_𝑟 𝐶))
43		rlimcnp2.b	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℝ)
44		1red 10055	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
45	23, 43, 44	rlimresb 14296	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ⇝_𝑟 𝐶 ↔ ((𝑦 ∈ 𝐵 ↦ 𝑆) ↾ (1[,)+∞)) ⇝_𝑟 𝐶))
46	29, 43	syl5ss 3614	. . . 4 ⊢ (𝜑 → (𝐵 ∩ ℝ⁺) ⊆ ℝ)
47	33, 46, 44	rlimresb 14296	. . 3 ⊢ (𝜑 → ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ⇝_𝑟 𝐶 ↔ ((𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ↾ (1[,)+∞)) ⇝_𝑟 𝐶))
48	42, 45, 47	3bitr4d 300	. 2 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝑆) ⇝_𝑟 𝐶 ↔ (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆) ⇝_𝑟 𝐶))
49		inss2 3834	. . . . . . . . . . 11 ⊢ (𝐵 ∩ ℝ⁺) ⊆ ℝ⁺
50	49	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∩ ℝ⁺) ⊆ ℝ⁺)
51	50	sselda 3603	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → 𝑦 ∈ ℝ⁺)
52	51	rpreccld 11882	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → (1 / 𝑦) ∈ ℝ⁺)
53	52	rpne0d 11877	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → (1 / 𝑦) ≠ 0)
54	53	neneqd 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → ¬ (1 / 𝑦) = 0)
55	54	iffalsed 4097	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → if((1 / 𝑦) = 0, 𝐶, ⦋(1 / 𝑦) / 𝑥⦌𝑅) = ⦋(1 / 𝑦) / 𝑥⦌𝑅)
56		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑥 = (1 / 𝑦) → (1 / 𝑥) = (1 / (1 / 𝑦)))
57		rpcnne0 11850	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℝ⁺ → (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0))
58		recrec 10722	. . . . . . . . . . 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 2678	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) ∧ 𝑥 = (1 / 𝑦)) → (1 / 𝑥) = 𝑦)
61	60	eqcomd 2628	. . . . . . . 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 2628	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) ∧ 𝑥 = (1 / 𝑦)) → 𝑅 = 𝑆)
65	52, 64	csbied 3560	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → ⦋(1 / 𝑦) / 𝑥⦌𝑅 = 𝑆)
66	55, 65	eqtrd 2656	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → if((1 / 𝑦) = 0, 𝐶, ⦋(1 / 𝑦) / 𝑥⦌𝑅) = 𝑆)
67	66	mpteq2dva 4744	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ if((1 / 𝑦) = 0, 𝐶, ⦋(1 / 𝑦) / 𝑥⦌𝑅)) = (𝑦 ∈ (𝐵 ∩ ℝ⁺) ↦ 𝑆))
68	67	breq1d 4663	. 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 762	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ 𝑤 = 0) → 𝐶 ∈ ℂ)
73	69	sselda 3603	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ (0[,)+∞))
74		0re 10040	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
75		pnfxr 10092	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ^*
76		elico2 12237	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ^*) → (𝑤 ∈ (0[,)+∞) ↔ (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 < +∞)))
77	74, 75, 76	mp2an 708	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0[,)+∞) ↔ (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 < +∞))
78	73, 77	sylib 208	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 < +∞))
79	78	simp1d 1073	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ℝ)
80	79	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → 𝑤 ∈ ℝ)
81	78	simp2d 1074	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 0 ≤ 𝑤)
82		leloe 10124	. . . . . . . . . . . . . . 15 ⊢ ((0 ∈ ℝ ∧ 𝑤 ∈ ℝ) → (0 ≤ 𝑤 ↔ (0 < 𝑤 ∨ 0 = 𝑤)))
83	74, 79, 82	sylancr 695	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (0 ≤ 𝑤 ↔ (0 < 𝑤 ∨ 0 = 𝑤)))
84	81, 83	mpbid 222	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (0 < 𝑤 ∨ 0 = 𝑤))
85	84	ord 392	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (¬ 0 < 𝑤 → 0 = 𝑤))
86		eqcom 2629	. . . . . . . . . . . 12 ⊢ (0 = 𝑤 ↔ 𝑤 = 0)
87	85, 86	syl6ib 241	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (¬ 0 < 𝑤 → 𝑤 = 0))
88	87	con1d 139	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (¬ 𝑤 = 0 → 0 < 𝑤))
89	88	imp 445	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → 0 < 𝑤)
90	80, 89	elrpd 11869	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → 𝑤 ∈ ℝ⁺)
91		rpcnne0 11850	. . . . . . . . 9 ⊢ (𝑤 ∈ ℝ⁺ → (𝑤 ∈ ℂ ∧ 𝑤 ≠ 0))
92		recrec 10722	. . . . . . . . 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 3540	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → ⦋(1 / (1 / 𝑤)) / 𝑥⦌𝑅 = ⦋𝑤 / 𝑥⦌𝑅)
96		simplr 792	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → 𝑤 ∈ 𝐴)
97		simpll 790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → 𝜑)
98		rpreccl 11857	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ℝ⁺ → (1 / 𝑤) ∈ ℝ⁺)
99	98	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (1 / 𝑤) ∈ ℝ⁺)
100		rlimcnp2.d	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺) → (𝑦 ∈ 𝐵 ↔ (1 / 𝑦) ∈ 𝐴))
101	100	ralrimiva 2966	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑦 ∈ ℝ⁺ (𝑦 ∈ 𝐵 ↔ (1 / 𝑦) ∈ 𝐴))
102	101	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ∀𝑦 ∈ ℝ⁺ (𝑦 ∈ 𝐵 ↔ (1 / 𝑦) ∈ 𝐴))
103		eleq1 2689	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (1 / 𝑤) → (𝑦 ∈ 𝐵 ↔ (1 / 𝑤) ∈ 𝐵))
104		oveq2 6658	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (1 / 𝑤) → (1 / 𝑦) = (1 / (1 / 𝑤)))
105	104	eleq1d 2686	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (1 / 𝑤) → ((1 / 𝑦) ∈ 𝐴 ↔ (1 / (1 / 𝑤)) ∈ 𝐴))
106	103, 105	bibi12d 335	. . . . . . . . . . . . 13 ⊢ (𝑦 = (1 / 𝑤) → ((𝑦 ∈ 𝐵 ↔ (1 / 𝑦) ∈ 𝐴) ↔ ((1 / 𝑤) ∈ 𝐵 ↔ (1 / (1 / 𝑤)) ∈ 𝐴)))
107	106	rspcv 3305	. . . . . . . . . . . 12 ⊢ ((1 / 𝑤) ∈ ℝ⁺ → (∀𝑦 ∈ ℝ⁺ (𝑦 ∈ 𝐵 ↔ (1 / 𝑦) ∈ 𝐴) → ((1 / 𝑤) ∈ 𝐵 ↔ (1 / (1 / 𝑤)) ∈ 𝐴)))
108	99, 102, 107	sylc 65	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ((1 / 𝑤) ∈ 𝐵 ↔ (1 / (1 / 𝑤)) ∈ 𝐴))
109	93	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (1 / (1 / 𝑤)) = 𝑤)
110	109	eleq1d 2686	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ((1 / (1 / 𝑤)) ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
111	108, 110	bitr2d 269	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (𝑤 ∈ 𝐴 ↔ (1 / 𝑤) ∈ 𝐵))
112	97, 90, 111	syl2anc 693	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → (𝑤 ∈ 𝐴 ↔ (1 / 𝑤) ∈ 𝐵))
113	96, 112	mpbid 222	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → (1 / 𝑤) ∈ 𝐵)
114	90	rpreccld 11882	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → (1 / 𝑤) ∈ ℝ⁺)
115	113, 114	elind 3798	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → (1 / 𝑤) ∈ (𝐵 ∩ ℝ⁺))
116	65, 31	eqeltrd 2701	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐵 ∩ ℝ⁺)) → ⦋(1 / 𝑦) / 𝑥⦌𝑅 ∈ ℂ)
117	116	ralrimiva 2966	. . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ (𝐵 ∩ ℝ⁺)⦋(1 / 𝑦) / 𝑥⦌𝑅 ∈ ℂ)
118	117	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → ∀𝑦 ∈ (𝐵 ∩ ℝ⁺)⦋(1 / 𝑦) / 𝑥⦌𝑅 ∈ ℂ)
119	104	csbeq1d 3540	. . . . . . . . 9 ⊢ (𝑦 = (1 / 𝑤) → ⦋(1 / 𝑦) / 𝑥⦌𝑅 = ⦋(1 / (1 / 𝑤)) / 𝑥⦌𝑅)
120	119	eleq1d 2686	. . . . . . . 8 ⊢ (𝑦 = (1 / 𝑤) → (⦋(1 / 𝑦) / 𝑥⦌𝑅 ∈ ℂ ↔ ⦋(1 / (1 / 𝑤)) / 𝑥⦌𝑅 ∈ ℂ))
121	120	rspcv 3305	. . . . . . 7 ⊢ ((1 / 𝑤) ∈ (𝐵 ∩ ℝ⁺) → (∀𝑦 ∈ (𝐵 ∩ ℝ⁺)⦋(1 / 𝑦) / 𝑥⦌𝑅 ∈ ℂ → ⦋(1 / (1 / 𝑤)) / 𝑥⦌𝑅 ∈ ℂ))
122	115, 118, 121	sylc 65	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → ⦋(1 / (1 / 𝑤)) / 𝑥⦌𝑅 ∈ ℂ)
123	95, 122	eqeltrrd 2702	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 = 0) → ⦋𝑤 / 𝑥⦌𝑅 ∈ ℂ)
124	72, 123	ifclda 4120	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅) ∈ ℂ)
125	99	biantrud 528	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → ((1 / 𝑤) ∈ 𝐵 ↔ ((1 / 𝑤) ∈ 𝐵 ∧ (1 / 𝑤) ∈ ℝ⁺)))
126	111, 125	bitrd 268	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (𝑤 ∈ 𝐴 ↔ ((1 / 𝑤) ∈ 𝐵 ∧ (1 / 𝑤) ∈ ℝ⁺)))
127		elin 3796	. . . . 5 ⊢ ((1 / 𝑤) ∈ (𝐵 ∩ ℝ⁺) ↔ ((1 / 𝑤) ∈ 𝐵 ∧ (1 / 𝑤) ∈ ℝ⁺))
128	126, 127	syl6bbr 278	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ⁺) → (𝑤 ∈ 𝐴 ↔ (1 / 𝑤) ∈ (𝐵 ∩ ℝ⁺)))
129		iftrue 4092	. . . 4 ⊢ (𝑤 = 0 → if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅) = 𝐶)
130		eqeq1 2626	. . . . 5 ⊢ (𝑤 = (1 / 𝑦) → (𝑤 = 0 ↔ (1 / 𝑦) = 0))
131		csbeq1 3536	. . . . 5 ⊢ (𝑤 = (1 / 𝑦) → ⦋𝑤 / 𝑥⦌𝑅 = ⦋(1 / 𝑦) / 𝑥⦌𝑅)
132	130, 131	ifbieq2d 4111	. . . 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 24692	. . 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 2764	. . . . 5 ⊢ Ⅎ𝑤if(𝑥 = 0, 𝐶, 𝑅)
137		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑥 𝑤 = 0
138		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑥𝐶
139		nfcsb1v 3549	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝑅
140	137, 138, 139	nfif 4115	. . . . 5 ⊢ Ⅎ𝑥if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅)
141		eqeq1 2626	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑥 = 0 ↔ 𝑤 = 0))
142		csbeq1a 3542	. . . . . 6 ⊢ (𝑥 = 𝑤 → 𝑅 = ⦋𝑤 / 𝑥⦌𝑅)
143	141, 142	ifbieq2d 4111	. . . . 5 ⊢ (𝑥 = 𝑤 → if(𝑥 = 0, 𝐶, 𝑅) = if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅))
144	136, 140, 143	cbvmpt 4749	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ if(𝑥 = 0, 𝐶, 𝑅)) = (𝑤 ∈ 𝐴 ↦ if(𝑤 = 0, 𝐶, ⦋𝑤 / 𝑥⦌𝑅))
145	144	eleq1i 2692	. . 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 278	. 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 296	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 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ⦋csb 3533 ∩ cin 3573 ⊆ wss 3574 ifcif 4086 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ↾ cres 5116 Rel wrel 5119 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 +∞cpnf 10071 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 / cdiv 10684 ℝ⁺crp 11832 (,)cioo 12175 [,)cico 12177 ⇝_𝑟 crli 14216 ↾_t crest 16081 TopOpenctopn 16082 ℂ_fldccnfld 19746 CnP ccnp 21029

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 ax-pre-sup 10014

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-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-fz 12327 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-rlim 14220 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-rest 16083 df-topn 16084 df-topgen 16104 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-bases 20750 df-cnp 21032

This theorem is referenced by: rlimcnp3 24694

W3C validator