Theorem rlim2 15403

Description: Rewrite rlim 15402 for a mapping operation. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)

Hypotheses

Ref	Expression
rlim2.1	⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ)
rlim2.2	⊢ (𝜑 → 𝐴 ⊆ ℝ)
rlim2.3	⊢ (𝜑 → 𝐶 ∈ ℂ)

Assertion

Ref	Expression
rlim2	⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦,𝑧   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑧)   𝐵(𝑧)

Proof of Theorem rlim2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rlim2.1	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ)
2		eqid 2731	. . . . 5 ⊢ (𝑧 ∈ 𝐴 ↦ 𝐵) = (𝑧 ∈ 𝐴 ↦ 𝐵)
3	2	fmpt 7043	. . . 4 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ ↔ (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
4	1, 3	sylib 218	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
5		rlim2.2	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
6		eqidd 2732	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) = ((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤))
7	4, 5, 6	rlim 15402	. 2 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝐴 (𝑦 ≤ 𝑤 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)) < 𝑥))))
8		rlim2.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ)
9	8	biantrurd 532	. 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝐴 (𝑦 ≤ 𝑤 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ (𝐶 ∈ ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝐴 (𝑦 ≤ 𝑤 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)) < 𝑥))))
10		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑧 𝑦 ≤ 𝑤
11		nfcv 2894	. . . . . . . . 9 ⊢ Ⅎ𝑧abs
12		nffvmpt1 6833	. . . . . . . . . 10 ⊢ Ⅎ𝑧((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤)
13		nfcv 2894	. . . . . . . . . 10 ⊢ Ⅎ𝑧 −
14		nfcv 2894	. . . . . . . . . 10 ⊢ Ⅎ𝑧𝐶
15	12, 13, 14	nfov 7376	. . . . . . . . 9 ⊢ Ⅎ𝑧(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)
16	11, 15	nffv 6832	. . . . . . . 8 ⊢ Ⅎ𝑧(abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶))
17		nfcv 2894	. . . . . . . 8 ⊢ Ⅎ𝑧 <
18		nfcv 2894	. . . . . . . 8 ⊢ Ⅎ𝑧𝑥
19	16, 17, 18	nfbr 5138	. . . . . . 7 ⊢ Ⅎ𝑧(abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)) < 𝑥
20	10, 19	nfim 1897	. . . . . 6 ⊢ Ⅎ𝑧(𝑦 ≤ 𝑤 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)) < 𝑥)
21		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑤(𝑦 ≤ 𝑧 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) < 𝑥)
22		breq2 5095	. . . . . . 7 ⊢ (𝑤 = 𝑧 → (𝑦 ≤ 𝑤 ↔ 𝑦 ≤ 𝑧))
23	22	imbrov2fvoveq 7371	. . . . . 6 ⊢ (𝑤 = 𝑧 → ((𝑦 ≤ 𝑤 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) < 𝑥)))
24	20, 21, 23	cbvralw 3274	. . . . 5 ⊢ (∀𝑤 ∈ 𝐴 (𝑦 ≤ 𝑤 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) < 𝑥))
25	2	fvmpt2 6940	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → ((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) = 𝐵)
26	25	fvoveq1d 7368	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) = (abs‘(𝐵 − 𝐶)))
27	26	breq1d 5101	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → ((abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) < 𝑥 ↔ (abs‘(𝐵 − 𝐶)) < 𝑥))
28	27	imbi2d 340	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → ((𝑦 ≤ 𝑧 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
29	28	ralimiaa 3068	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ → ∀𝑧 ∈ 𝐴 ((𝑦 ≤ 𝑧 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
30		ralbi 3087	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐴 ((𝑦 ≤ 𝑧 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)) → (∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
31	1, 29, 30	3syl 18	. . . . 5 ⊢ (𝜑 → (∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
32	24, 31	bitrid 283	. . . 4 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 (𝑦 ≤ 𝑤 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
33	32	rexbidv 3156	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝐴 (𝑦 ≤ 𝑤 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
34	33	ralbidv 3155	. 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝐴 (𝑦 ≤ 𝑤 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
35	7, 9, 34	3bitr2d 307	1 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ⊆ wss 3902   class class class wbr 5091   ↦ cmpt 5172  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  ℂcc 11004  ℝcr 11005   < clt 11146   ≤ cle 11147   − cmin 11344  ℝ⁺crp 12890  abscabs 15141   ⇝_𝑟 crli 15392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11062  ax-resscn 11063

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-pm 8753  df-rlim 15396

This theorem is referenced by:  rlim2lt  15404  rlim3  15405  rlim0  15415  rlimi  15420  rlimconst  15451  climrlim2  15454  rlimcn1  15495  rlimcn3  15497  chtppilim  27414  pntlem3  27548