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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 44161 | . 2 β’ (π β (πΉ~~>*-β β βπ¦ β β βπ β π βπ β (β€β₯βπ)(πΉβπ) β€ π¦)) |
5 | breq2 5110 | . . . . 5 β’ (π¦ = π₯ β ((πΉβπ) β€ π¦ β (πΉβπ) β€ π₯)) | |
6 | 5 | rexralbidv 3211 | . . . 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 6843 | . . . . . . 7 β’ (π = π β (β€β₯βπ) = (β€β₯βπ)) | |
8 | 7 | raleqdv 3312 | . . . . . 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 2904 | . . . . . . . . 9 β’ β²ππ | |
11 | 9, 10 | nffv 6853 | . . . . . . . 8 β’ β²π(πΉβπ) |
12 | nfcv 2904 | . . . . . . . 8 β’ β²π β€ | |
13 | nfcv 2904 | . . . . . . . 8 β’ β²ππ₯ | |
14 | 11, 12, 13 | nfbr 5153 | . . . . . . 7 β’ β²π(πΉβπ) β€ π₯ |
15 | nfv 1918 | . . . . . . 7 β’ β²π(πΉβπ) β€ π₯ | |
16 | fveq2 6843 | . . . . . . . 8 β’ (π = π β (πΉβπ) = (πΉβπ)) | |
17 | 16 | breq1d 5116 | . . . . . . 7 β’ (π = π β ((πΉβπ) β€ π₯ β (πΉβπ) β€ π₯)) |
18 | 14, 15, 17 | cbvralw 3288 | . . . . . 6 β’ (βπ β (β€β₯βπ)(πΉβπ) β€ π₯ β βπ β (β€β₯βπ)(πΉβπ) β€ π₯) |
19 | 8, 18 | bitrdi 287 | . . . . 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 3225 | . . . 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 287 | . . 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 3224 | . 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 287 | 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 1542 β wcel 2107 β²wnfc 2884 βwral 3061 βwrex 3070 class class class wbr 5106 βΆwf 6493 βcfv 6497 βcr 11055 -βcmnf 11192 β*cxr 11193 β€ cle 11195 β€cz 12504 β€β₯cuz 12768 ~~>*clsxlim 44145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-1o 8413 df-er 8651 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fi 9352 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-z 12505 df-uz 12769 df-ioo 13274 df-ioc 13275 df-ico 13276 df-icc 13277 df-topgen 17330 df-ordt 17388 df-ps 18460 df-tsr 18461 df-top 22259 df-topon 22276 df-bases 22312 df-lm 22596 df-xlim 44146 |
This theorem is referenced by: xlimmnfmpt 44170 dfxlim2v 44174 xlimpnfxnegmnf2 44185 |
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