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Theorem rlim0 14458

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 10245	. . 3 ⊢ (𝜑 → 0 ∈ ℂ)
4	1, 2, 3	rlim2 14446	. 2 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝_𝑟 0 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 0)) < 𝑥)))
5		subid1 10513	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (𝐵 − 0) = 𝐵)
6	5	fveq2d 6357	. . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (abs‘(𝐵 − 0)) = (abs‘𝐵))
7	6	breq1d 4814	. . . . . . 7 ⊢ (𝐵 ∈ ℂ → ((abs‘(𝐵 − 0)) < 𝑥 ↔ (abs‘𝐵) < 𝑥))
8	7	imbi2d 329	. . . . . 6 ⊢ (𝐵 ∈ ℂ → ((𝑦 ≤ 𝑧 → (abs‘(𝐵 − 0)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘𝐵) < 𝑥)))
9	8	ralimi 3090	. . . . 5 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ → ∀𝑧 ∈ 𝐴 ((𝑦 ≤ 𝑧 → (abs‘(𝐵 − 0)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘𝐵) < 𝑥)))
10		ralbi 3206	. . . . 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 3190	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 0)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘𝐵) < 𝑥)))
13	12	ralbidv 3124	. 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 268	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 196 ∈ wcel 2139 ∀wral 3050 ∃wrex 3051 ⊆ wss 3715 class class class wbr 4804 ↦ cmpt 4881 ‘cfv 6049 (class class class)co 6814 ℂcc 10146 ℝcr 10147 0cc0 10148 < clt 10286 ≤ cle 10287 − cmin 10478 ℝ⁺crp 12045 abscabs 14193 ⇝_𝑟 crli 14435

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-er 7913 df-pm 8028 df-en 8124 df-dom 8125 df-sdom 8126 df-pnf 10288 df-mnf 10289 df-ltxr 10291 df-sub 10480 df-rlim 14439

This theorem is referenced by: o1rlimmul 14568 dvfsumrlim 24013 rlimcxp 24920

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