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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 45794 | . 2 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙))) |
5 | breq1 5151 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑙))) | |
6 | 5 | rexralbidv 3221 | . . . 4 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙))) |
7 | fveq2 6907 | . . . . . . 7 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗)) | |
8 | 7 | raleqdv 3324 | . . . . . 6 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑙 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑙))) |
9 | nfcv 2903 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑥 | |
10 | nfcv 2903 | . . . . . . . 8 ⊢ Ⅎ𝑘 ≤ | |
11 | xlimpnf.k | . . . . . . . . 9 ⊢ Ⅎ𝑘𝐹 | |
12 | nfcv 2903 | . . . . . . . . 9 ⊢ Ⅎ𝑘𝑙 | |
13 | 11, 12 | nffv 6917 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘𝑙) |
14 | 9, 10, 13 | nfbr 5195 | . . . . . . 7 ⊢ Ⅎ𝑘 𝑥 ≤ (𝐹‘𝑙) |
15 | nfv 1912 | . . . . . . 7 ⊢ Ⅎ𝑙 𝑥 ≤ (𝐹‘𝑘) | |
16 | fveq2 6907 | . . . . . . . 8 ⊢ (𝑙 = 𝑘 → (𝐹‘𝑙) = (𝐹‘𝑘)) | |
17 | 16 | breq2d 5160 | . . . . . . 7 ⊢ (𝑙 = 𝑘 → (𝑥 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑘))) |
18 | 14, 15, 17 | cbvralw 3304 | . . . . . 6 ⊢ (∀𝑙 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘)) |
19 | 8, 18 | bitrdi 287 | . . . . 5 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))) |
20 | 19 | cbvrexvw 3236 | . . . 4 ⊢ (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘)) |
21 | 6, 20 | bitrdi 287 | . . 3 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))) |
22 | 21 | cbvralvw 3235 | . 2 ⊢ (∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘)) |
23 | 4, 22 | bitrdi 287 | 1 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ (𝐹‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 Ⅎwnfc 2888 ∀wral 3059 ∃wrex 3068 class class class wbr 5148 ⟶wf 6559 ‘cfv 6563 ℝcr 11152 +∞cpnf 11290 ℝ*cxr 11292 ≤ cle 11294 ℤcz 12611 ℤ≥cuz 12876 ~~>*clsxlim 45774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-1o 8505 df-2o 8506 df-er 8744 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-z 12612 df-uz 12877 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-topgen 17490 df-ordt 17548 df-ps 18624 df-tsr 18625 df-top 22916 df-topon 22933 df-bases 22969 df-lm 23253 df-xlim 45775 |
This theorem is referenced by: xlimpnfmpt 45800 dfxlim2v 45803 xlimpnfxnegmnf2 45814 meaiuninc3v 46440 |
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