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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfxlim2 | Structured version Visualization version GIF version |
Description: An alternative definition for the convergence relation in the extended real numbers. This resembles what's found in most textbooks: three distinct definitions for the same symbol (limit of a sequence). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
dfxlim2.k | ⊢ Ⅎ𝑘𝐹 |
dfxlim2.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
dfxlim2.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
dfxlim2.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
Ref | Expression |
---|---|
dfxlim2 | ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹 ⇝ 𝐴 ∨ (𝐴 = -∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) ∨ (𝐴 = +∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfxlim2.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | dfxlim2.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | dfxlim2.f | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
4 | 1, 2, 3 | dfxlim2v 45022 | . 2 ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹 ⇝ 𝐴 ∨ (𝐴 = -∞ ∧ ∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦) ∨ (𝐴 = +∞ ∧ ∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙))))) |
5 | biid 261 | . . 3 ⊢ (𝐹 ⇝ 𝐴 ↔ 𝐹 ⇝ 𝐴) | |
6 | breq2 5152 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑙) ≤ 𝑥)) | |
7 | 6 | rexralbidv 3219 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥)) |
8 | fveq2 6891 | . . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗)) | |
9 | 8 | raleqdv 3324 | . . . . . . . 8 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑙 ∈ (ℤ≥‘𝑗)(𝐹‘𝑙) ≤ 𝑥)) |
10 | dfxlim2.k | . . . . . . . . . . 11 ⊢ Ⅎ𝑘𝐹 | |
11 | nfcv 2902 | . . . . . . . . . . 11 ⊢ Ⅎ𝑘𝑙 | |
12 | 10, 11 | nffv 6901 | . . . . . . . . . 10 ⊢ Ⅎ𝑘(𝐹‘𝑙) |
13 | nfcv 2902 | . . . . . . . . . 10 ⊢ Ⅎ𝑘 ≤ | |
14 | nfcv 2902 | . . . . . . . . . 10 ⊢ Ⅎ𝑘𝑥 | |
15 | 12, 13, 14 | nfbr 5195 | . . . . . . . . 9 ⊢ Ⅎ𝑘(𝐹‘𝑙) ≤ 𝑥 |
16 | nfv 1916 | . . . . . . . . 9 ⊢ Ⅎ𝑙(𝐹‘𝑘) ≤ 𝑥 | |
17 | fveq2 6891 | . . . . . . . . . 10 ⊢ (𝑙 = 𝑘 → (𝐹‘𝑙) = (𝐹‘𝑘)) | |
18 | 17 | breq1d 5158 | . . . . . . . . 9 ⊢ (𝑙 = 𝑘 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑘) ≤ 𝑥)) |
19 | 15, 16, 18 | cbvralw 3302 | . . . . . . . 8 ⊢ (∀𝑙 ∈ (ℤ≥‘𝑗)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) |
20 | 9, 19 | bitrdi 287 | . . . . . . 7 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)) |
21 | 20 | cbvrexvw 3234 | . . . . . 6 ⊢ (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) |
22 | 7, 21 | bitrdi 287 | . . . . 5 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)) |
23 | 22 | cbvralvw 3233 | . . . 4 ⊢ (∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) |
24 | 23 | anbi2i 622 | . . 3 ⊢ ((𝐴 = -∞ ∧ ∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦) ↔ (𝐴 = -∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)) |
25 | breq1 5151 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑙))) | |
26 | 25 | rexralbidv 3219 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙))) |
27 | 8 | raleqdv 3324 | . . . . . . . 8 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑙 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑙))) |
28 | 14, 13, 12 | nfbr 5195 | . . . . . . . . 9 ⊢ Ⅎ𝑘 𝑥 ≤ (𝐹‘𝑙) |
29 | nfv 1916 | . . . . . . . . 9 ⊢ Ⅎ𝑙 𝑥 ≤ (𝐹‘𝑘) | |
30 | 17 | breq2d 5160 | . . . . . . . . 9 ⊢ (𝑙 = 𝑘 → (𝑥 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑘))) |
31 | 28, 29, 30 | cbvralw 3302 | . . . . . . . 8 ⊢ (∀𝑙 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘)) |
32 | 27, 31 | bitrdi 287 | . . . . . . 7 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))) |
33 | 32 | cbvrexvw 3234 | . . . . . 6 ⊢ (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘)) |
34 | 26, 33 | bitrdi 287 | . . . . 5 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))) |
35 | 34 | cbvralvw 3233 | . . . 4 ⊢ (∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘)) |
36 | 35 | anbi2i 622 | . . 3 ⊢ ((𝐴 = +∞ ∧ ∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙)) ↔ (𝐴 = +∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))) |
37 | 5, 24, 36 | 3orbi123i 1155 | . 2 ⊢ ((𝐹 ⇝ 𝐴 ∨ (𝐴 = -∞ ∧ ∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦) ∨ (𝐴 = +∞ ∧ ∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙))) ↔ (𝐹 ⇝ 𝐴 ∨ (𝐴 = -∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) ∨ (𝐴 = +∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘)))) |
38 | 4, 37 | bitrdi 287 | 1 ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ (𝐹 ⇝ 𝐴 ∨ (𝐴 = -∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) ∨ (𝐴 = +∞ ∧ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ w3o 1085 = wceq 1540 ∈ wcel 2105 Ⅎwnfc 2882 ∀wral 3060 ∃wrex 3069 class class class wbr 5148 ⟶wf 6539 ‘cfv 6543 ℝcr 11115 +∞cpnf 11252 -∞cmnf 11253 ℝ*cxr 11254 ≤ cle 11256 ℤcz 12565 ℤ≥cuz 12829 ⇝ cli 15435 ~~>*clsxlim 44993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fi 9412 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ioo 13335 df-ioc 13336 df-ico 13337 df-icc 13338 df-fz 13492 df-fl 13764 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-rlim 15440 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-rest 17375 df-topn 17376 df-topgen 17396 df-ordt 17454 df-ps 18529 df-tsr 18530 df-psmet 21225 df-xmet 21226 df-met 21227 df-bl 21228 df-mopn 21229 df-cnfld 21234 df-top 22716 df-topon 22733 df-topsp 22755 df-bases 22769 df-lm 23053 df-xms 24146 df-ms 24147 df-xlim 44994 |
This theorem is referenced by: (None) |
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