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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 | β’ β²ππΉ |
xlimmnf.m | β’ (π β π β β€) |
xlimmnf.z | β’ π = (β€β₯βπ) |
xlimmnf.f | β’ (π β πΉ:πβΆβ*) |
Ref | Expression |
---|---|
xlimmnf | β’ (π β (πΉ~~>*-β β βπ₯ β β βπ β π βπ β (β€β₯βπ)(πΉβπ) β€ π₯)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xlimmnf.m | . . 3 β’ (π β π β β€) | |
2 | xlimmnf.z | . . 3 β’ π = (β€β₯βπ) | |
3 | xlimmnf.f | . . 3 β’ (π β πΉ:πβΆβ*) | |
4 | 1, 2, 3 | xlimmnfv 45251 | . 2 β’ (π β (πΉ~~>*-β β βπ¦ β β βπ β π βπ β (β€β₯βπ)(πΉβπ) β€ π¦)) |
5 | breq2 5156 | . . . . 5 β’ (π¦ = π₯ β ((πΉβπ) β€ π¦ β (πΉβπ) β€ π₯)) | |
6 | 5 | rexralbidv 3218 | . . . 4 β’ (π¦ = π₯ β (βπ β π βπ β (β€β₯βπ)(πΉβπ) β€ π¦ β βπ β π βπ β (β€β₯βπ)(πΉβπ) β€ π₯)) |
7 | fveq2 6902 | . . . . . . 7 β’ (π = π β (β€β₯βπ) = (β€β₯βπ)) | |
8 | 7 | raleqdv 3323 | . . . . . 6 β’ (π = π β (βπ β (β€β₯βπ)(πΉβπ) β€ π₯ β βπ β (β€β₯βπ)(πΉβπ) β€ π₯)) |
9 | xlimmnf.k | . . . . . . . . 9 β’ β²ππΉ | |
10 | nfcv 2899 | . . . . . . . . 9 β’ β²ππ | |
11 | 9, 10 | nffv 6912 | . . . . . . . 8 β’ β²π(πΉβπ) |
12 | nfcv 2899 | . . . . . . . 8 β’ β²π β€ | |
13 | nfcv 2899 | . . . . . . . 8 β’ β²ππ₯ | |
14 | 11, 12, 13 | nfbr 5199 | . . . . . . 7 β’ β²π(πΉβπ) β€ π₯ |
15 | nfv 1909 | . . . . . . 7 β’ β²π(πΉβπ) β€ π₯ | |
16 | fveq2 6902 | . . . . . . . 8 β’ (π = π β (πΉβπ) = (πΉβπ)) | |
17 | 16 | breq1d 5162 | . . . . . . 7 β’ (π = π β ((πΉβπ) β€ π₯ β (πΉβπ) β€ π₯)) |
18 | 14, 15, 17 | cbvralw 3301 | . . . . . 6 β’ (βπ β (β€β₯βπ)(πΉβπ) β€ π₯ β βπ β (β€β₯βπ)(πΉβπ) β€ π₯) |
19 | 8, 18 | bitrdi 286 | . . . . 5 β’ (π = π β (βπ β (β€β₯βπ)(πΉβπ) β€ π₯ β βπ β (β€β₯βπ)(πΉβπ) β€ π₯)) |
20 | 19 | cbvrexvw 3233 | . . . 4 β’ (βπ β π βπ β (β€β₯βπ)(πΉβπ) β€ π₯ β βπ β π βπ β (β€β₯βπ)(πΉβπ) β€ π₯) |
21 | 6, 20 | bitrdi 286 | . . 3 β’ (π¦ = π₯ β (βπ β π βπ β (β€β₯βπ)(πΉβπ) β€ π¦ β βπ β π βπ β (β€β₯βπ)(πΉβπ) β€ π₯)) |
22 | 21 | cbvralvw 3232 | . 2 β’ (βπ¦ β β βπ β π βπ β (β€β₯βπ)(πΉβπ) β€ π¦ β βπ₯ β β βπ β π βπ β (β€β₯βπ)(πΉβπ) β€ π₯) |
23 | 4, 22 | bitrdi 286 | 1 β’ (π β (πΉ~~>*-β β βπ₯ β β βπ β π βπ β (β€β₯βπ)(πΉβπ) β€ π₯)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 β²wnfc 2879 βwral 3058 βwrex 3067 class class class wbr 5152 βΆwf 6549 βcfv 6553 βcr 11145 -βcmnf 11284 β*cxr 11285 β€ cle 11287 β€cz 12596 β€β₯cuz 12860 ~~>*clsxlim 45235 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-1o 8493 df-er 8731 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fi 9442 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-z 12597 df-uz 12861 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-topgen 17432 df-ordt 17490 df-ps 18565 df-tsr 18566 df-top 22816 df-topon 22833 df-bases 22869 df-lm 23153 df-xlim 45236 |
This theorem is referenced by: xlimmnfmpt 45260 dfxlim2v 45264 xlimpnfxnegmnf2 45275 |
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