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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimclimdm | Structured version Visualization version GIF version |
Description: A sequence of extended reals that converges to a real w.r.t. the standard topology on the extended reals, also converges w.r.t. to the standard topology on the complex numbers. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
Ref | Expression |
---|---|
xlimclimdm.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
xlimclimdm.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
xlimclimdm.3 | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
xlimclimdm.4 | ⊢ (𝜑 → 𝐹~~>*𝐴) |
xlimclimdm.5 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
xlimclimdm | ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climrel 15431 | . 2 ⊢ Rel ⇝ | |
2 | xlimclimdm.4 | . . 3 ⊢ (𝜑 → 𝐹~~>*𝐴) | |
3 | xlimclimdm.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
4 | xlimclimdm.2 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | xlimclimdm.3 | . . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
6 | xlimclimdm.5 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
7 | 3, 4, 5, 6 | xlimclim2 44490 | . . 3 ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) |
8 | 2, 7 | mpbid 231 | . 2 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
9 | releldm 5940 | . 2 ⊢ ((Rel ⇝ ∧ 𝐹 ⇝ 𝐴) → 𝐹 ∈ dom ⇝ ) | |
10 | 1, 8, 9 | sylancr 588 | 1 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 class class class wbr 5146 dom cdm 5674 Rel wrel 5679 ⟶wf 6535 ‘cfv 6539 ℝcr 11104 ℝ*cxr 11242 ℤcz 12553 ℤ≥cuz 12817 ⇝ cli 15423 ~~>*clsxlim 44468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4907 df-int 4949 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-er 8698 df-map 8817 df-pm 8818 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fi 9401 df-sup 9432 df-inf 9433 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-div 11867 df-nn 12208 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12468 df-z 12554 df-dec 12673 df-uz 12818 df-q 12928 df-rp 12970 df-xneg 13087 df-xadd 13088 df-xmul 13089 df-ioo 13323 df-ioc 13324 df-ico 13325 df-icc 13326 df-fz 13480 df-fl 13752 df-seq 13962 df-exp 14023 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15427 df-rlim 15428 df-struct 17075 df-slot 17110 df-ndx 17122 df-base 17140 df-plusg 17205 df-mulr 17206 df-starv 17207 df-tset 17211 df-ple 17212 df-ds 17214 df-unif 17215 df-rest 17363 df-topn 17364 df-topgen 17384 df-ordt 17442 df-ps 18514 df-tsr 18515 df-psmet 20920 df-xmet 20921 df-met 20922 df-bl 20923 df-mopn 20924 df-cnfld 20929 df-top 22377 df-topon 22394 df-topsp 22416 df-bases 22430 df-lm 22714 df-xms 23807 df-ms 23808 df-xlim 44469 |
This theorem is referenced by: xlimliminflimsup 44512 |
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