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| Mirrors > Home > MPE Home > Th. List > Mathboxes > climxlim2 | Structured version Visualization version GIF version | ||
| Description: A sequence of extended reals, converging w.r.t. the standard topology on the complex numbers is a converging sequence w.r.t. the standard topology on the extended reals. This is non-trivial, because +∞ and -∞ could, in principle, be complex numbers. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| Ref | Expression |
|---|---|
| climxlim2.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
| climxlim2.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
| climxlim2.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
| climxlim2.a | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
| Ref | Expression |
|---|---|
| climxlim2 | ⊢ (𝜑 → 𝐹~~>*𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | climxlim2.z | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
| 2 | 1 | eluzelz2 45414 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ) |
| 3 | 2 | ad2antlr 727 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → 𝑗 ∈ ℤ) |
| 4 | eqid 2737 | . . . 4 ⊢ (ℤ≥‘𝑗) = (ℤ≥‘𝑗) | |
| 5 | climxlim2.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
| 6 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹:𝑍⟶ℝ*) |
| 7 | 1 | uzssd3 45437 | . . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (ℤ≥‘𝑗) ⊆ 𝑍) |
| 8 | 7 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (ℤ≥‘𝑗) ⊆ 𝑍) |
| 9 | 6, 8 | fssresd 6775 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ*) |
| 10 | 9 | adantr 480 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ*) |
| 11 | simpr 484 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) | |
| 12 | climxlim2.a | . . . . . . 7 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
| 13 | 12 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 ⇝ 𝐴) |
| 14 | 1 | fvexi 6920 | . . . . . . . . 9 ⊢ 𝑍 ∈ V |
| 15 | 14 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ V) |
| 16 | 5, 15 | fexd 7247 | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ V) |
| 17 | climres 15611 | . . . . . . 7 ⊢ ((𝑗 ∈ ℤ ∧ 𝐹 ∈ V) → ((𝐹 ↾ (ℤ≥‘𝑗)) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴)) | |
| 18 | 2, 16, 17 | syl2anr 597 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝐹 ↾ (ℤ≥‘𝑗)) ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴)) |
| 19 | 13, 18 | mpbird 257 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑗)) ⇝ 𝐴) |
| 20 | 19 | adantr 480 | . . . 4 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → (𝐹 ↾ (ℤ≥‘𝑗)) ⇝ 𝐴) |
| 21 | 3, 4, 10, 11, 20 | climxlim2lem 45860 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → (𝐹 ↾ (ℤ≥‘𝑗))~~>*𝐴) |
| 22 | 1, 5 | fuzxrpmcn 45843 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
| 23 | 22 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 ∈ (ℝ* ↑pm ℂ)) |
| 24 | 2 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ) |
| 25 | 23, 24 | xlimres 45836 | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑗))~~>*𝐴)) |
| 26 | 25 | adantr 480 | . . 3 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → (𝐹~~>*𝐴 ↔ (𝐹 ↾ (ℤ≥‘𝑗))~~>*𝐴)) |
| 27 | 21, 26 | mpbird 257 | . 2 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) → 𝐹~~>*𝐴) |
| 28 | climxlim2.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 29 | 5 | ffnd 6737 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝑍) |
| 30 | climcl 15535 | . . . . 5 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ) | |
| 31 | 12, 30 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 32 | breldmg 5920 | . . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐴 ∈ ℂ ∧ 𝐹 ⇝ 𝐴) → 𝐹 ∈ dom ⇝ ) | |
| 33 | 16, 31, 12, 32 | syl3anc 1373 | . . 3 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) |
| 34 | 28, 1, 29, 33 | climrescn 45763 | . 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℂ) |
| 35 | 27, 34 | r19.29a 3162 | 1 ⊢ (𝜑 → 𝐹~~>*𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 dom cdm 5685 ↾ cres 5687 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑pm cpm 8867 ℂcc 11153 ℝ*cxr 11294 ℤcz 12613 ℤ≥cuz 12878 ⇝ cli 15520 ~~>*clsxlim 45833 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fl 13832 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17467 df-topn 17468 df-topgen 17488 df-ordt 17546 df-ps 18611 df-tsr 18612 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-lm 23237 df-xms 24330 df-ms 24331 df-xlim 45834 |
| This theorem is referenced by: dfxlim2v 45862 meaiuninc3v 46499 |
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