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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 41580 | . 2 ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦)) |
5 | breq2 4929 | . . . . 5 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑙) ≤ 𝑥)) | |
6 | 5 | rexralbidv 3239 | . . . 4 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥)) |
7 | fveq2 6496 | . . . . . . 7 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗)) | |
8 | 7 | raleqdv 3348 | . . . . . 6 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑙 ∈ (ℤ≥‘𝑗)(𝐹‘𝑙) ≤ 𝑥)) |
9 | xlimmnf.k | . . . . . . . . 9 ⊢ Ⅎ𝑘𝐹 | |
10 | nfcv 2925 | . . . . . . . . 9 ⊢ Ⅎ𝑘𝑙 | |
11 | 9, 10 | nffv 6506 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘𝑙) |
12 | nfcv 2925 | . . . . . . . 8 ⊢ Ⅎ𝑘 ≤ | |
13 | nfcv 2925 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑥 | |
14 | 11, 12, 13 | nfbr 4972 | . . . . . . 7 ⊢ Ⅎ𝑘(𝐹‘𝑙) ≤ 𝑥 |
15 | nfv 1874 | . . . . . . 7 ⊢ Ⅎ𝑙(𝐹‘𝑘) ≤ 𝑥 | |
16 | fveq2 6496 | . . . . . . . 8 ⊢ (𝑙 = 𝑘 → (𝐹‘𝑙) = (𝐹‘𝑘)) | |
17 | 16 | breq1d 4935 | . . . . . . 7 ⊢ (𝑙 = 𝑘 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑘) ≤ 𝑥)) |
18 | 14, 15, 17 | cbvral 3372 | . . . . . 6 ⊢ (∀𝑙 ∈ (ℤ≥‘𝑗)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) |
19 | 8, 18 | syl6bb 279 | . . . . 5 ⊢ (𝑖 = 𝑗 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)) |
20 | 19 | cbvrexv 3377 | . . . 4 ⊢ (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) |
21 | 6, 20 | syl6bb 279 | . . 3 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)) |
22 | 21 | cbvralv 3376 | . 2 ⊢ (∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥) |
23 | 4, 22 | syl6bb 279 | 1 ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐹‘𝑘) ≤ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1508 ∈ wcel 2051 Ⅎwnfc 2909 ∀wral 3081 ∃wrex 3082 class class class wbr 4925 ⟶wf 6181 ‘cfv 6185 ℝcr 10332 -∞cmnf 10470 ℝ*cxr 10471 ≤ cle 10473 ℤcz 11791 ℤ≥cuz 12056 ~~>*clsxlim 41564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-pm 8207 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fi 8668 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-z 11792 df-uz 12057 df-ioo 12556 df-ioc 12557 df-ico 12558 df-icc 12559 df-topgen 16571 df-ordt 16628 df-ps 17680 df-tsr 17681 df-top 21221 df-topon 21238 df-bases 21273 df-lm 21556 df-xlim 41565 |
This theorem is referenced by: xlimmnfmpt 41589 dfxlim2v 41593 xlimpnfxnegmnf2 41604 |
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