![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > climge0 | Structured version Visualization version GIF version |
Description: A nonnegative sequence converges to a nonnegative number. (Contributed by NM, 11-Sep-2005.) (Proof shortened by Mario Carneiro, 10-May-2016.) |
Ref | Expression |
---|---|
climshft2.1 | β’ π = (β€β₯βπ) |
climshft2.2 | β’ (π β π β β€) |
climrecl.3 | β’ (π β πΉ β π΄) |
climrecl.4 | β’ ((π β§ π β π) β (πΉβπ) β β) |
climge0.5 | β’ ((π β§ π β π) β 0 β€ (πΉβπ)) |
Ref | Expression |
---|---|
climge0 | β’ (π β 0 β€ π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climshft2.2 | . . 3 β’ (π β π β β€) | |
2 | climshft2.1 | . . . 4 β’ π = (β€β₯βπ) | |
3 | 2 | uzsup 13868 | . . 3 β’ (π β β€ β sup(π, β*, < ) = +β) |
4 | 1, 3 | syl 17 | . 2 β’ (π β sup(π, β*, < ) = +β) |
5 | climrecl.3 | . . . 4 β’ (π β πΉ β π΄) | |
6 | climrel 15476 | . . . . . . 7 β’ Rel β | |
7 | 6 | brrelex1i 5738 | . . . . . 6 β’ (πΉ β π΄ β πΉ β V) |
8 | 5, 7 | syl 17 | . . . . 5 β’ (π β πΉ β V) |
9 | eqid 2728 | . . . . . 6 β’ (π β π β¦ (πΉβπ)) = (π β π β¦ (πΉβπ)) | |
10 | 2, 9 | climmpt 15555 | . . . . 5 β’ ((π β β€ β§ πΉ β V) β (πΉ β π΄ β (π β π β¦ (πΉβπ)) β π΄)) |
11 | 1, 8, 10 | syl2anc 582 | . . . 4 β’ (π β (πΉ β π΄ β (π β π β¦ (πΉβπ)) β π΄)) |
12 | 5, 11 | mpbid 231 | . . 3 β’ (π β (π β π β¦ (πΉβπ)) β π΄) |
13 | climrecl.4 | . . . . . 6 β’ ((π β§ π β π) β (πΉβπ) β β) | |
14 | 13 | recnd 11280 | . . . . 5 β’ ((π β§ π β π) β (πΉβπ) β β) |
15 | 14 | fmpttd 7130 | . . . 4 β’ (π β (π β π β¦ (πΉβπ)):πβΆβ) |
16 | 2, 1, 15 | rlimclim 15530 | . . 3 β’ (π β ((π β π β¦ (πΉβπ)) βπ π΄ β (π β π β¦ (πΉβπ)) β π΄)) |
17 | 12, 16 | mpbird 256 | . 2 β’ (π β (π β π β¦ (πΉβπ)) βπ π΄) |
18 | climge0.5 | . 2 β’ ((π β§ π β π) β 0 β€ (πΉβπ)) | |
19 | 4, 17, 13, 18 | rlimge0 15565 | 1 β’ (π β 0 β€ π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3473 class class class wbr 5152 β¦ cmpt 5235 βcfv 6553 supcsup 9471 βcc 11144 βcr 11145 0cc0 11146 +βcpnf 11283 β*cxr 11285 < clt 11286 β€ cle 11287 β€cz 12596 β€β₯cuz 12860 β cli 15468 βπ crli 15469 |
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-rep 5289 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 ax-pre-sup 11224 |
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-rmo 3374 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-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-pred 6310 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-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-fl 13797 df-seq 14007 df-exp 14067 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-rlim 15473 |
This theorem is referenced by: climle 15624 radcnvrat 43782 |
Copyright terms: Public domain | W3C validator |