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Theorem rlim0 15525

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 11165	. . 3 ⊢ (𝜑 → 0 ∈ ℂ)
4	1, 2, 3	rlim2 15513	. 2 ⊢ (𝜑 → ((𝑧 ∈ 𝐴 ↦ 𝐵) ⇝_𝑟 0 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 0)) < 𝑥)))
5		subid1 11444	. . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (𝐵 − 0) = 𝐵)
6	5	fveq2d 6865	. . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (abs‘(𝐵 − 0)) = (abs‘𝐵))
7	6	breq1d 5107	. . . . . . 7 ⊢ (𝐵 ∈ ℂ → ((abs‘(𝐵 − 0)) < 𝑥 ↔ (abs‘𝐵) < 𝑥))
8	7	imbi2d 342	. . . . . 6 ⊢ (𝐵 ∈ ℂ → ((𝑦 ≤ 𝑧 → (abs‘(𝐵 − 0)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘𝐵) < 𝑥)))
9	8	ralimi 3098	. . . . 5 ⊢ (∀𝑧 ∈ 𝐴 𝐵 ∈ ℂ → ∀𝑧 ∈ 𝐴 ((𝑦 ≤ 𝑧 → (abs‘(𝐵 − 0)) < 𝑥) ↔ (𝑦 ≤ 𝑧 → (abs‘𝐵) < 𝑥)))
10		ralbi 3116	. . . . 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 3185	. . 3 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘(𝐵 − 0)) < 𝑥) ↔ ∃𝑦 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → (abs‘𝐵) < 𝑥)))
13	12	ralbidv 3184	. 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 281	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 208 ∈ wcel 2141 ∀wral 3075 ∃wrex 3085 ⊆ wss 3902 class class class wbr 5097 ↦ cmpt 5178 ‘cfv 6515 (class class class)co 7390 ℂcc 11064 ℝcr 11065 0cc0 11066 < 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 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 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-nel 3061 df-ral 3076 df-rex 3086 df-reu 3367 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-po 5551 df-so 5552 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-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-er 8671 df-pm 8804 df-en 8921 df-dom 8922 df-sdom 8923 df-pnf 11211 df-mnf 11212 df-ltxr 11214 df-sub 11409 df-rlim 15506

This theorem is referenced by: o1rlimmul 15636 dvfsumrlim 26080 rlimcxp 27025

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