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Mirrors > Home > MPE Home > Th. List > rlimmptrcl | Structured version Visualization version GIF version |
Description: Reverse closure for a real limit. (Contributed by Mario Carneiro, 10-May-2016.) |
Ref | Expression |
---|---|
rlimabs.1 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
rlimabs.2 | ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶) |
Ref | Expression |
---|---|
rlimmptrcl | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlimabs.2 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶) | |
2 | rlimf 14573 | . . . . 5 ⊢ ((𝑘 ∈ 𝐴 ↦ 𝐵) ⇝𝑟 𝐶 → (𝑘 ∈ 𝐴 ↦ 𝐵):dom (𝑘 ∈ 𝐴 ↦ 𝐵)⟶ℂ) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):dom (𝑘 ∈ 𝐴 ↦ 𝐵)⟶ℂ) |
4 | eqid 2799 | . . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
5 | rlimabs.1 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
6 | 4, 5 | dmmptd 6235 | . . . . 5 ⊢ (𝜑 → dom (𝑘 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
7 | 6 | feq2d 6242 | . . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵):dom (𝑘 ∈ 𝐴 ↦ 𝐵)⟶ℂ ↔ (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)) |
8 | 3, 7 | mpbid 224 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
9 | 4 | fmpt 6606 | . . 3 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ ↔ (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
10 | 8, 9 | sylibr 226 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ) |
11 | 10 | r19.21bi 3113 | 1 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 ∀wral 3089 class class class wbr 4843 ↦ cmpt 4922 dom cdm 5312 ⟶wf 6097 ℂcc 10222 ⇝𝑟 crli 14557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-pm 8098 df-rlim 14561 |
This theorem is referenced by: rlimabs 14680 rlimcj 14681 rlimre 14682 rlimim 14683 rlimadd 14714 rlimsub 14715 rlimmul 14716 rlimdiv 14717 rlimneg 14718 fsumrlim 14881 dvfsumrlim 24135 rlimcxp 25052 cxploglim2 25057 |
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