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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimmnfmpt | 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 |
---|---|
xlimmnfmpt.k | ⊢ Ⅎ𝑘𝜑 |
xlimmnfmpt.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
xlimmnfmpt.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
xlimmnfmpt.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*) |
xlimmnfmpt.f | ⊢ 𝐹 = (𝑘 ∈ 𝑍 ↦ 𝐵) |
Ref | Expression |
---|---|
xlimmnfmpt | ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐵 ≤ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xlimmnfmpt.f | . . . 4 ⊢ 𝐹 = (𝑘 ∈ 𝑍 ↦ 𝐵) | |
2 | nfmpt1 5257 | . . . 4 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑍 ↦ 𝐵) | |
3 | 1, 2 | nfcxfr 2889 | . . 3 ⊢ Ⅎ𝑘𝐹 |
4 | xlimmnfmpt.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
5 | xlimmnfmpt.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | xlimmnfmpt.k | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
7 | xlimmnfmpt.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*) | |
8 | 6, 7, 1 | fmptdf 7126 | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
9 | 3, 4, 5, 8 | xlimmnf 45367 | . 2 ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)(𝐹‘𝑘) ≤ 𝑦)) |
10 | nfv 1909 | . . . . . 6 ⊢ Ⅎ𝑘 𝑖 ∈ 𝑍 | |
11 | 6, 10 | nfan 1894 | . . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑖 ∈ 𝑍) |
12 | 5 | uztrn2 12874 | . . . . . . . 8 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑖)) → 𝑘 ∈ 𝑍) |
13 | 12 | adantll 712 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑖)) → 𝑘 ∈ 𝑍) |
14 | simpll 765 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑖)) → 𝜑) | |
15 | 14, 13, 7 | syl2anc 582 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑖)) → 𝐵 ∈ ℝ*) |
16 | 1 | fvmpt2 7015 | . . . . . . 7 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝐵 ∈ ℝ*) → (𝐹‘𝑘) = 𝐵) |
17 | 13, 15, 16 | syl2anc 582 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑘) = 𝐵) |
18 | 17 | breq1d 5159 | . . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑖)) → ((𝐹‘𝑘) ≤ 𝑦 ↔ 𝐵 ≤ 𝑦)) |
19 | 11, 18 | ralbida 3257 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑖)(𝐹‘𝑘) ≤ 𝑦 ↔ ∀𝑘 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦)) |
20 | 19 | rexbidva 3166 | . . 3 ⊢ (𝜑 → (∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)(𝐹‘𝑘) ≤ 𝑦 ↔ ∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦)) |
21 | 20 | ralbidv 3167 | . 2 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)(𝐹‘𝑘) ≤ 𝑦 ↔ ∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦)) |
22 | breq2 5153 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝐵 ≤ 𝑦 ↔ 𝐵 ≤ 𝑥)) | |
23 | 22 | rexralbidv 3210 | . . . . 5 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦 ↔ ∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥)) |
24 | fveq2 6896 | . . . . . . 7 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗)) | |
25 | 24 | raleqdv 3314 | . . . . . 6 ⊢ (𝑖 = 𝑗 → (∀𝑘 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝐵 ≤ 𝑥)) |
26 | 25 | cbvrexvw 3225 | . . . . 5 ⊢ (∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐵 ≤ 𝑥) |
27 | 23, 26 | bitrdi 286 | . . . 4 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐵 ≤ 𝑥)) |
28 | 27 | cbvralvw 3224 | . . 3 ⊢ (∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦 ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐵 ≤ 𝑥) |
29 | 28 | a1i 11 | . 2 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝐵 ≤ 𝑦 ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐵 ≤ 𝑥)) |
30 | 9, 21, 29 | 3bitrd 304 | 1 ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝐵 ≤ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 ∀wral 3050 ∃wrex 3059 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6549 ℝcr 11139 -∞cmnf 11278 ℝ*cxr 11279 ≤ cle 11281 ℤcz 12591 ℤ≥cuz 12855 ~~>*clsxlim 45344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-1o 8487 df-er 8725 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9436 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-z 12592 df-uz 12856 df-ioo 13363 df-ioc 13364 df-ico 13365 df-icc 13366 df-topgen 17428 df-ordt 17486 df-ps 18561 df-tsr 18562 df-top 22840 df-topon 22857 df-bases 22893 df-lm 23177 df-xlim 45345 |
This theorem is referenced by: (None) |
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