MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovolmge0 Structured version   Visualization version   Unicode version
Theorem ovolmge0 23245
Description: The set M is composed of nonnegative extended real numbers. (Contributed by Mario Carneiro, 16-Mar-2014.)
Hypothesis
Ref Expression
ovolval.1 |- M = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }
Assertion
Ref Expression
ovolmge0 |- ( B e. M -> 0 <_ B )
Distinct variable groups:   y, f, A   B, f, y
Allowed substitution hints:   M( y, f)
Proof of Theorem ovolmge0
StepHypRef Expression
1 ovolval.1 . . 3 |- M = { y e. RR* | E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ y = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) }
21elovolm 23243 . 2 |- ( B e. M <-> E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ B = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) )
3 reex 10027 . . . . . . . . 9 |- RR e. _V
43, 3xpex 6962 . . . . . . . 8 |- ( RR X. RR ) e. _V
54inex2 4800 . . . . . . 7 |- ( <_ i^i ( RR X. RR ) ) e. _V
6 nnex 11026 . . . . . . 7 |- NN e. _V
75, 6elmap 7886 . . . . . 6 |- ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) <-> f : NN --> ( <_ i^i ( RR X. RR ) ) )
8 eqid 2622 . . . . . . . . . 10 |- ( ( abs o. - ) o. f ) = ( ( abs o. - ) o. f )
9 eqid 2622 . . . . . . . . . 10 |- seq 1 ( + , ( ( abs o. - ) o. f ) ) = seq 1 ( + , ( ( abs o. - ) o. f ) )
108, 9ovolsf 23241 . . . . . . . . 9 |- ( f : NN --> ( <_ i^i ( RR X. RR ) ) -> seq 1 ( + , ( ( abs o. - ) o. f ) ) : NN --> ( 0 [,) +oo ) )
11 1nn 11031 . . . . . . . . 9 |- 1 e. NN
12 ffvelrn 6357 . . . . . . . . 9 |- ( ( seq 1 ( + , ( ( abs o. - ) o. f ) ) : NN --> ( 0 [,) +oo ) /\ 1 e. NN ) -> ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) e. ( 0 [,) +oo ) )
1310, 11, 12sylancl 694 . . . . . . . 8 |- ( f : NN --> ( <_ i^i ( RR X. RR ) ) -> ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) e. ( 0 [,) +oo ) )
14 elrege0 12278 . . . . . . . . 9 |- ( ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) e. ( 0 [,) +oo ) <-> ( ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) e. RR /\ 0 <_ ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) ) )
1514simprbi 480 . . . . . . . 8 |- ( ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) e. ( 0 [,) +oo ) -> 0 <_ ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) )
1613, 15syl 17 . . . . . . 7 |- ( f : NN --> ( <_ i^i ( RR X. RR ) ) -> 0 <_ ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) )
17 frn 6053 . . . . . . . . . 10 |- ( seq 1 ( + , ( ( abs o. - ) o. f ) ) : NN --> ( 0 [,) +oo ) -> ran seq 1 ( + , ( ( abs o. - ) o. f ) ) C_ ( 0 [,) +oo ) )
1810, 17syl 17 . . . . . . . . 9 |- ( f : NN --> ( <_ i^i ( RR X. RR ) ) -> ran seq 1 ( + , ( ( abs o. - ) o. f ) ) C_ ( 0 [,) +oo ) )
19 icossxr 12258 . . . . . . . . 9 |- ( 0 [,) +oo ) C_ RR*
2018, 19syl6ss 3615 . . . . . . . 8 |- ( f : NN --> ( <_ i^i ( RR X. RR ) ) -> ran seq 1 ( + , ( ( abs o. - ) o. f ) ) C_ RR* )
21 ffn 6045 . . . . . . . . . 10 |- ( seq 1 ( + , ( ( abs o. - ) o. f ) ) : NN --> ( 0 [,) +oo ) -> seq 1 ( + , ( ( abs o. - ) o. f ) ) Fn NN )
2210, 21syl 17 . . . . . . . . 9 |- ( f : NN --> ( <_ i^i ( RR X. RR ) ) -> seq 1 ( + , ( ( abs o. - ) o. f ) ) Fn NN )
23 fnfvelrn 6356 . . . . . . . . 9 |- ( ( seq 1 ( + , ( ( abs o. - ) o. f ) ) Fn NN /\ 1 e. NN ) -> ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) e. ran seq 1 ( + , ( ( abs o. - ) o. f ) ) )
2422, 11, 23sylancl 694 . . . . . . . 8 |- ( f : NN --> ( <_ i^i ( RR X. RR ) ) -> ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) e. ran seq 1 ( + , ( ( abs o. - ) o. f ) ) )
25 supxrub 12154 . . . . . . . 8 |- ( ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) C_ RR* /\ ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) e. ran seq 1 ( + , ( ( abs o. - ) o. f ) ) ) -> ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) <_ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) )
2620, 24, 25syl2anc 693 . . . . . . 7 |- ( f : NN --> ( <_ i^i ( RR X. RR ) ) -> ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) <_ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) )
2719, 13sseldi 3601 . . . . . . . 8 |- ( f : NN --> ( <_ i^i ( RR X. RR ) ) -> ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) e. RR* )
28 supxrcl 12145 . . . . . . . . 9 |- ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) C_ RR* -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) e. RR* )
2920, 28syl 17 . . . . . . . 8 |- ( f : NN --> ( <_ i^i ( RR X. RR ) ) -> sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) e. RR* )
30 0xr 10086 . . . . . . . . 9 |- 0 e. RR*
31 xrletr 11989 . . . . . . . . 9 |- ( ( 0 e. RR* /\ ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) e. RR* /\ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) e. RR* ) -> ( ( 0 <_ ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) /\ ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) <_ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) -> 0 <_ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) )
3230, 31mp3an1 1411 . . . . . . . 8 |- ( ( ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) e. RR* /\ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) e. RR* ) -> ( ( 0 <_ ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) /\ ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) <_ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) -> 0 <_ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) )
3327, 29, 32syl2anc 693 . . . . . . 7 |- ( f : NN --> ( <_ i^i ( RR X. RR ) ) -> ( ( 0 <_ ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) /\ ( seq 1 ( + , ( ( abs o. - ) o. f ) ) ` 1 ) <_ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) -> 0 <_ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) )
3416, 26, 33mp2and 715 . . . . . 6 |- ( f : NN --> ( <_ i^i ( RR X. RR ) ) -> 0 <_ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) )
357, 34sylbi 207 . . . . 5 |- ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) -> 0 <_ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) )
36 breq2 4657 . . . . 5 |- ( B = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) -> ( 0 <_ B <-> 0 <_ sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) )
3735, 36syl5ibrcom 237 . . . 4 |- ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) -> ( B = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) -> 0 <_ B ) )
3837adantld 483 . . 3 |- ( f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) -> ( ( A C_ U. ran ( (,) o. f ) /\ B = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) -> 0 <_ B ) )
3938rexlimiv 3027 . 2 |- ( E. f e. ( ( <_ i^i ( RR X. RR ) ) ^m NN ) ( A C_ U. ran ( (,) o. f ) /\ B = sup ( ran seq 1 ( + , ( ( abs o. - ) o. f ) ) , RR* , < ) ) -> 0 <_ B )
402, 39sylbi 207 1 |- ( B e. M -> 0 <_ B )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   i^i cin 3573   C_ wss 3574   U.cuni 4436   class class class wbr 4653   X. cxp 5112   ran crn 5115   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   supcsup 8346   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   (,)cioo 12175   [,)cico 12177   seqcseq 12801   abscabs 13974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976
This theorem is referenced by:  ovolge0  23249
  Copyright terms: Public domain W3C validator