Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimmnf | Structured version Visualization version GIF version |
Description: A function converges to minus infinity if it eventually becomes (and stays) smaller than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xlimmnf.k | ⊢ Ⅎ𝑘𝐹 |
xlimmnf.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
xlimmnf.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
xlimmnf.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
Ref | Expression |
---|---|
xlimmnf | ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xlimmnf.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | xlimmnf.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | xlimmnf.f | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
4 | 1, 2, 3 | xlimmnfv 43050 | . 2 ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦)) |
5 | breq2 5057 | . . . . 5 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑙) ≤ 𝑥)) | |
6 | 5 | rexralbidv 3220 | . . . 4 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥)) |
7 | fveq2 6717 | . . . . . . 7 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗)) | |
8 | 7 | raleqdv 3325 | . . . . . 6 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑙 ∈ (ℤ≥‘𝑗)(𝐹‘𝑙) ≤ 𝑥)) |
9 | xlimmnf.k | . . . . . . . . 9 ⊢ Ⅎ𝑘𝐹 | |
10 | nfcv 2904 | . . . . . . . . 9 ⊢ Ⅎ𝑘𝑙 | |
11 | 9, 10 | nffv 6727 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘𝑙) |
12 | nfcv 2904 | . . . . . . . 8 ⊢ Ⅎ𝑘 ≤ | |
13 | nfcv 2904 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑥 | |
14 | 11, 12, 13 | nfbr 5100 | . . . . . . 7 ⊢ Ⅎ𝑘(𝐹‘𝑙) ≤ 𝑥 |
15 | nfv 1922 | . . . . . . 7 ⊢ Ⅎ𝑙(𝐹‘𝑘) ≤ 𝑥 | |
16 | fveq2 6717 | . . . . . . . 8 ⊢ (𝑙 = 𝑘 → (𝐹‘𝑙) = (𝐹‘𝑘)) | |
17 | 16 | breq1d 5063 | . . . . . . 7 ⊢ (𝑙 = 𝑘 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑘) ≤ 𝑥)) |
18 | 14, 15, 17 | cbvralw 3349 | . . . . . 6 ⊢ (∀𝑙 ∈ (ℤ≥‘𝑗)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) |
19 | 8, 18 | bitrdi 290 | . . . . 5 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)) |
20 | 19 | cbvrexvw 3359 | . . . 4 ⊢ (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) |
21 | 6, 20 | bitrdi 290 | . . 3 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)) |
22 | 21 | cbvralvw 3358 | . 2 ⊢ (∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) |
23 | 4, 22 | bitrdi 290 | 1 ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 Ⅎwnfc 2884 ∀wral 3061 ∃wrex 3062 class class class wbr 5053 ⟶wf 6376 ‘cfv 6380 ℝcr 10728 -∞cmnf 10865 ℝ*cxr 10866 ≤ cle 10868 ℤcz 12176 ℤ≥cuz 12438 ~~>*clsxlim 43034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-1o 8202 df-er 8391 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fi 9027 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-z 12177 df-uz 12439 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-topgen 16948 df-ordt 17006 df-ps 18072 df-tsr 18073 df-top 21791 df-topon 21808 df-bases 21843 df-lm 22126 df-xlim 43035 |
This theorem is referenced by: xlimmnfmpt 43059 dfxlim2v 43063 xlimpnfxnegmnf2 43074 |
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