Theorem rlim2 15431

Description: Rewrite rlim 15430 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 2737	. . . . 5 ⊢ (𝑧 ∈ 𝐴 ↦ 𝐵) = (𝑧 ∈ 𝐴 ↦ 𝐵)
3	2	fmpt 7064	. . . 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 2738	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) = ((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤))
7	4, 5, 6	rlim 15430	. 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 1916	. . . . . . 7 ⊢ Ⅎ𝑧 𝑦 ≤ 𝑤
11		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑧abs
12		nffvmpt1 6853	. . . . . . . . . 10 ⊢ Ⅎ𝑧((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤)
13		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑧 −
14		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑧𝐶
15	12, 13, 14	nfov 7398	. . . . . . . . 9 ⊢ Ⅎ𝑧(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)
16	11, 15	nffv 6852	. . . . . . . 8 ⊢ Ⅎ𝑧(abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶))
17		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑧 <
18		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑧𝑥
19	16, 17, 18	nfbr 5147	. . . . . . 7 ⊢ Ⅎ𝑧(abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)) < 𝑥
20	10, 19	nfim 1898	. . . . . 6 ⊢ Ⅎ𝑧(𝑦 ≤ 𝑤 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑤) − 𝐶)) < 𝑥)
21		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑤(𝑦 ≤ 𝑧 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) < 𝑥)
22		breq2 5104	. . . . . . 7 ⊢ (𝑤 = 𝑧 → (𝑦 ≤ 𝑤 ↔ 𝑦 ≤ 𝑧))
23	22	imbrov2fvoveq 7393	. . . . . 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 3280	. . . . 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 6961	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → ((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) = 𝐵)
26	25	fvoveq1d 7390	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝐵 ∈ ℂ) → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) = (abs‘(𝐵 − 𝐶)))
27	26	breq1d 5110	. . . . . . . 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 3074	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ → ∀𝑧 ∈ 𝐴 ((𝑦 ≤ 𝑧 → (abs‘(((𝑧 ∈ 𝐴 ↦ 𝐵)‘𝑧) − 𝐶)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 𝐶)) < 𝑥)))
30		ralbi 3093	. . . . . 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 3162	. . 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 3161	. 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 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3903   class class class wbr 5100   ↦ cmpt 5181  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  ℂcc 11036  ℝcr 11037   < clt 11178   ≤ cle 11179   − cmin 11376  ℝ⁺crp 12917  abscabs 15169   ⇝_𝑟 crli 15420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-pm 8778  df-rlim 15424

This theorem is referenced by:  rlim2lt  15432  rlim3  15433  rlim0  15443  rlimi  15448  rlimconst  15479  climrlim2  15482  rlimcn1  15523  rlimcn3  15525  chtppilim  27454  pntlem3  27588