| Mathbox for Glauco Siliprandi |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimxrre | Structured version Visualization version GIF version | ||
| Description: If a sequence ranging over the extended reals converges w.r.t. the standard topology on the complex numbers, then there exists an upper set of the integers over which the function is real-valued (the weaker hypothesis 𝐹 ∈ dom ⇝ is probably not enough, since in principle we could have +∞ ∈ ℂ and -∞ ∈ ℂ). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
| Ref | Expression |
|---|---|
| xlimxrre.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
| xlimxrre.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
| xlimxrre.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
| xlimxrre.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| xlimxrre.c | ⊢ (𝜑 → 𝐹~~>*𝐴) |
| Ref | Expression |
|---|---|
| xlimxrre | ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elioore 13417 | . . . . . . 7 ⊢ ((𝐹‘𝑘) ∈ ((𝐴 − 1)(,)(𝐴 + 1)) → (𝐹‘𝑘) ∈ ℝ) | |
| 2 | 1 | anim2i 617 | . . . . . 6 ⊢ ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ((𝐴 − 1)(,)(𝐴 + 1))) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)) |
| 3 | 2 | ralimi 3083 | . . . . 5 ⊢ (∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ((𝐴 − 1)(,)(𝐴 + 1))) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)) |
| 4 | 3 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ((𝐴 − 1)(,)(𝐴 + 1)))) → ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ)) |
| 5 | xlimxrre.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
| 6 | 5 | ffund 6740 | . . . . . 6 ⊢ (𝜑 → Fun 𝐹) |
| 7 | ffvresb 7145 | . . . . . 6 ⊢ (Fun 𝐹 → ((𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))) | |
| 8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))) |
| 9 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ((𝐴 − 1)(,)(𝐴 + 1)))) → ((𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ℝ))) |
| 10 | 4, 9 | mpbird 257 | . . 3 ⊢ ((𝜑 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ((𝐴 − 1)(,)(𝐴 + 1)))) → (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) |
| 11 | 10 | adantrl 716 | . 2 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ((𝐴 − 1)(,)(𝐴 + 1))))) → (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) |
| 12 | xlimxrre.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 13 | peano2rem 11576 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ) | |
| 14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 − 1) ∈ ℝ) |
| 15 | 14 | rexrd 11311 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) ∈ ℝ*) |
| 16 | peano2re 11434 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ) | |
| 17 | 12, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ) |
| 18 | 17 | rexrd 11311 | . . . 4 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ*) |
| 19 | 12 | ltm1d 12200 | . . . 4 ⊢ (𝜑 → (𝐴 − 1) < 𝐴) |
| 20 | 12 | ltp1d 12198 | . . . 4 ⊢ (𝜑 → 𝐴 < (𝐴 + 1)) |
| 21 | 15, 18, 12, 19, 20 | eliood 45511 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ((𝐴 − 1)(,)(𝐴 + 1))) |
| 22 | iooordt 23225 | . . . 4 ⊢ ((𝐴 − 1)(,)(𝐴 + 1)) ∈ (ordTop‘ ≤ ) | |
| 23 | xlimxrre.c | . . . . . 6 ⊢ (𝜑 → 𝐹~~>*𝐴) | |
| 24 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑘𝐹 | |
| 25 | xlimxrre.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 26 | xlimxrre.z | . . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
| 27 | eqid 2737 | . . . . . . 7 ⊢ (ordTop‘ ≤ ) = (ordTop‘ ≤ ) | |
| 28 | 24, 25, 26, 5, 27 | xlimbr 45842 | . . . . . 6 ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ (𝐴 ∈ ℝ* ∧ ∀𝑢 ∈ (ordTop‘ ≤ )(𝐴 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))))) |
| 29 | 23, 28 | mpbid 232 | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ ∀𝑢 ∈ (ordTop‘ ≤ )(𝐴 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)))) |
| 30 | 29 | simprd 495 | . . . 4 ⊢ (𝜑 → ∀𝑢 ∈ (ordTop‘ ≤ )(𝐴 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) |
| 31 | eleq2 2830 | . . . . . 6 ⊢ (𝑢 = ((𝐴 − 1)(,)(𝐴 + 1)) → (𝐴 ∈ 𝑢 ↔ 𝐴 ∈ ((𝐴 − 1)(,)(𝐴 + 1)))) | |
| 32 | eleq2 2830 | . . . . . . . 8 ⊢ (𝑢 = ((𝐴 − 1)(,)(𝐴 + 1)) → ((𝐹‘𝑘) ∈ 𝑢 ↔ (𝐹‘𝑘) ∈ ((𝐴 − 1)(,)(𝐴 + 1)))) | |
| 33 | 32 | anbi2d 630 | . . . . . . 7 ⊢ (𝑢 = ((𝐴 − 1)(,)(𝐴 + 1)) → ((𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢) ↔ (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ((𝐴 − 1)(,)(𝐴 + 1))))) |
| 34 | 33 | rexralbidv 3223 | . . . . . 6 ⊢ (𝑢 = ((𝐴 − 1)(,)(𝐴 + 1)) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ((𝐴 − 1)(,)(𝐴 + 1))))) |
| 35 | 31, 34 | imbi12d 344 | . . . . 5 ⊢ (𝑢 = ((𝐴 − 1)(,)(𝐴 + 1)) → ((𝐴 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢)) ↔ (𝐴 ∈ ((𝐴 − 1)(,)(𝐴 + 1)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ((𝐴 − 1)(,)(𝐴 + 1)))))) |
| 36 | 35 | rspcva 3620 | . . . 4 ⊢ ((((𝐴 − 1)(,)(𝐴 + 1)) ∈ (ordTop‘ ≤ ) ∧ ∀𝑢 ∈ (ordTop‘ ≤ )(𝐴 ∈ 𝑢 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑢))) → (𝐴 ∈ ((𝐴 − 1)(,)(𝐴 + 1)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ((𝐴 − 1)(,)(𝐴 + 1))))) |
| 37 | 22, 30, 36 | sylancr 587 | . . 3 ⊢ (𝜑 → (𝐴 ∈ ((𝐴 − 1)(,)(𝐴 + 1)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ((𝐴 − 1)(,)(𝐴 + 1))))) |
| 38 | 21, 37 | mpd 15 | . 2 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ ((𝐴 − 1)(,)(𝐴 + 1)))) |
| 39 | 11, 38 | reximddv 3171 | 1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 class class class wbr 5143 dom cdm 5685 ↾ cres 5687 Fun wfun 6555 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 1c1 11156 + caddc 11158 ℝ*cxr 11294 ≤ cle 11296 − cmin 11492 ℤcz 12613 ℤ≥cuz 12878 (,)cioo 13387 ordTopcordt 17544 ~~>*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-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-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-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-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-topgen 17488 df-ordt 17546 df-ps 18611 df-tsr 18612 df-top 22900 df-topon 22917 df-bases 22953 df-lm 23237 df-xlim 45834 |
| This theorem is referenced by: xlimclim2 45855 xlimliminflimsup 45877 |
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