Theorem rlim2 15513

Description: Rewrite rlim 15512 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 2761	. . . . 5 ⊢ (𝑧 ∈ 𝐴 ↦ 𝐵) = (𝑧 ∈ 𝐴 ↦ 𝐵)
3	2	fmpt 7085	. . . 4 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ ↔ (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
4	1, 3	sylib 220	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
5		rlim2.2	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
6		eqidd 2762	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) = ((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤))
7	4, 5, 6	rlim 15512	. 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 540	. 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 1933	. . . . . . 7 ⊢ Ⅎ𝑧 𝑦 ≤ 𝑤
11		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑧abs
12		nffvmpt1 6872	. . . . . . . . . 10 ⊢ Ⅎ𝑧((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤)
13		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑧 −
14		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑧𝐶
15	12, 13, 14	nfov 7420	. . . . . . . . 9 ⊢ Ⅎ𝑧(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)
16	11, 15	nffv 6871	. . . . . . . 8 ⊢ Ⅎ𝑧(abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶))
17		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑧 <
18		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑧𝑥
19	16, 17, 18	nfbr 5144	. . . . . . 7 ⊢ Ⅎ𝑧(abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)) < 𝑥
20	10, 19	nfim 1915	. . . . . 6 ⊢ Ⅎ𝑧(𝑦 ≤ 𝑤 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)) < 𝑥)
21		nfv 1933	. . . . . 6 ⊢ Ⅎ𝑤(𝑦 ≤ 𝑧 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) < 𝑥)
22		breq2 5101	. . . . . . 7 ⊢ (𝑤 = 𝑧 → (𝑦 ≤ 𝑤 ↔ 𝑦 ≤ 𝑧))
23	22	imbrov2fvoveq 7415	. . . . . 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 3303	. . . . 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 6981	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → ((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) = 𝐵)
26	25	fvoveq1d 7412	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) = (abs‘(𝐵 − 𝐶)))
27	26	breq1d 5107	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → ((abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) < 𝑥 ↔ (abs‘(𝐵 − 𝐶)) < 𝑥))
28	27	imbi2d 342	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → ((𝑦 ≤ 𝑧 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
29	28	ralimiaa 3097	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ → ∀𝑧 ∈ 𝐴 ((𝑦 ≤ 𝑧 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
30		ralbi 3116	. . . . . 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 285	. . . 4 ⊢ (𝜑 → (∀𝑤 ∈ 𝐴 (𝑦 ≤ 𝑤 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)) < 𝑥) ↔ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
33	32	rexbidv 3185	. . 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 3184	. 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 309	1 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085   ⊆ wss 3902   class class class wbr 5097   ↦ cmpt 5178  ⟶wf 6511  ‘cfv 6515  (class class class)co 7390  ℂcc 11064  ℝcr 11065   < clt 11209   ≤ cle 11210   − cmin 11407  ℝ⁺crp 12986  abscabs 15251   ⇝_𝑟 crli 15502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712  ax-cnex 11122  ax-resscn 11123

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-pm 8804  df-rlim 15506

This theorem is referenced by:  rlim2lt  15514  rlim3  15515  rlim0  15525  rlimi  15530  rlimconst  15561  climrlim2  15564  rlimcn1  15605  rlimcn3  15607  chtppilim  27526  pntlem3  27660