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Theorem rlim0 15461

Description: Express the predicate 𝐵(𝑧) converges to 0. (Contributed by Mario Carneiro, 16-Sep-2014.) (Revised by Mario Carneiro, 28-Feb-2015.)

**Hypotheses**
Ref	Expression
rlim0.1	⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ)
rlim0.2	⊢ (𝜑 → 𝐴 ⊆ ℝ)

**Assertion**
Ref	Expression
rlim0	⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝_𝑟 0 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘𝐵) < 𝑥)))

Distinct variable groups: 𝑥,𝑦,𝑧,𝐴 𝑥,𝐵,𝑦 𝜑,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑧) 𝐵(𝑧)

**Proof of Theorem rlim0**
Step	Hyp	Ref	Expression
1		rlim0.1	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ)
2		rlim0.2	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
3		0cnd 11128	. . 3 ⊢ (𝜑 → 0 ∈ ℂ)
4	1, 2, 3	rlim2 15449	. 2 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝_𝑟 0 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 0)) < 𝑥)))
5		subid1 11405	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (𝐵 − 0) = 𝐵)
6	5	fveq2d 6838	. . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (abs‘(𝐵 − 0)) = (abs‘𝐵))
7	6	breq1d 5096	. . . . . . 7 ⊢ (𝐵 ∈ ℂ → ((abs‘(𝐵 − 0)) < 𝑥 ↔ (abs‘𝐵) < 𝑥))
8	7	imbi2d 340	. . . . . 6 ⊢ (𝐵 ∈ ℂ → ((𝑦 ≤ 𝑧 → (abs‘(𝐵 − 0)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘𝐵) < 𝑥)))
9	8	ralimi 3075	. . . . 5 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ → ∀𝑧 ∈ 𝐴 ((𝑦 ≤ 𝑧 → (abs‘(𝐵 − 0)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘𝐵) < 𝑥)))
10		ralbi 3093	. . . . 5 ⊢ (∀𝑧 ∈ 𝐴 ((𝑦 ≤ 𝑧 → (abs‘(𝐵 − 0)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘𝐵) < 𝑥)) → (∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 0)) < 𝑥) ↔ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘𝐵) < 𝑥)))
11	1, 9, 10	3syl 18	. . . 4 ⊢ (𝜑 → (∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 0)) < 𝑥) ↔ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘𝐵) < 𝑥)))
12	11	rexbidv 3162	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 0)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘𝐵) < 𝑥)))
13	12	ralbidv 3161	. 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 0)) < 𝑥) ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘𝐵) < 𝑥)))
14	4, 13	bitrd 279	1 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝_𝑟 0 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘𝐵) < 𝑥)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 ⊆ wss 3890 class class class wbr 5086 ↦ cmpt 5167 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 ℝcr 11028 0cc0 11029 < clt 11170 ≤ cle 11171 − cmin 11368 ℝ⁺crp 12933 abscabs 15187 ⇝_𝑟 crli 15438

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 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 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-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-er 8636 df-pm 8769 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-ltxr 11175 df-sub 11370 df-rlim 15442

This theorem is referenced by: o1rlimmul 15572 dvfsumrlim 26008 rlimcxp 26951

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