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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimpnf | Structured version Visualization version GIF version |
Description: A function converges to plus infinity if it eventually becomes (and stays) larger than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xlimpnf.k | ⊢ Ⅎ𝑘𝐹 |
xlimpnf.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
xlimpnf.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
xlimpnf.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
Ref | Expression |
---|---|
xlimpnf | ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xlimpnf.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | xlimpnf.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | xlimpnf.f | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
4 | 1, 2, 3 | xlimpnfv 40575 | . 2 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙))) |
5 | breq1 4789 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑙))) | |
6 | 5 | rexralbidv 3206 | . . . 4 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙))) |
7 | fveq2 6330 | . . . . . . 7 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗)) | |
8 | 7 | raleqdv 3293 | . . . . . 6 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑙 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑙))) |
9 | nfcv 2913 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑥 | |
10 | nfcv 2913 | . . . . . . . 8 ⊢ Ⅎ𝑘 ≤ | |
11 | xlimpnf.k | . . . . . . . . 9 ⊢ Ⅎ𝑘𝐹 | |
12 | nfcv 2913 | . . . . . . . . 9 ⊢ Ⅎ𝑘𝑙 | |
13 | 11, 12 | nffv 6337 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘𝑙) |
14 | 9, 10, 13 | nfbr 4833 | . . . . . . 7 ⊢ Ⅎ𝑘 𝑥 ≤ (𝐹‘𝑙) |
15 | nfv 1995 | . . . . . . 7 ⊢ Ⅎ𝑙 𝑥 ≤ (𝐹‘𝑘) | |
16 | fveq2 6330 | . . . . . . . 8 ⊢ (𝑙 = 𝑘 → (𝐹‘𝑙) = (𝐹‘𝑘)) | |
17 | 16 | breq2d 4798 | . . . . . . 7 ⊢ (𝑙 = 𝑘 → (𝑥 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑘))) |
18 | 14, 15, 17 | cbvral 3316 | . . . . . 6 ⊢ (∀𝑙 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘)) |
19 | 8, 18 | syl6bb 276 | . . . . 5 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))) |
20 | 19 | cbvrexv 3321 | . . . 4 ⊢ (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘)) |
21 | 6, 20 | syl6bb 276 | . . 3 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))) |
22 | 21 | cbvralv 3320 | . 2 ⊢ (∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘)) |
23 | 4, 22 | syl6bb 276 | 1 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1631 ∈ wcel 2145 Ⅎwnfc 2900 ∀wral 3061 ∃wrex 3062 class class class wbr 4786 ⟶wf 6025 ‘cfv 6029 ℝcr 10135 +∞cpnf 10271 ℝ*cxr 10273 ≤ cle 10275 ℤcz 11577 ℤ≥cuz 11886 ~~>*clsxlim 40555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-1st 7313 df-2nd 7314 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-oadd 7715 df-er 7894 df-pm 8010 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-fi 8471 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-z 11578 df-uz 11887 df-ioo 12377 df-ioc 12378 df-ico 12379 df-icc 12380 df-topgen 16305 df-ordt 16362 df-ps 17401 df-tsr 17402 df-top 20912 df-topon 20929 df-bases 20964 df-lm 21247 df-xlim 40556 |
This theorem is referenced by: xlimpnfmpt 40581 dfxlim2v 40584 meaiuninc3v 41211 |
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