MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniioombllem1 Structured version   Unicode version

Theorem uniioombllem1 21856
Description: Lemma for uniioombl 21864. (Contributed by Mario Carneiro, 25-Mar-2015.)
Hypotheses
Ref Expression
uniioombl.1  |-  ( ph  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
uniioombl.2  |-  ( ph  -> Disj  x  e.  NN  ( (,) `  ( F `  x ) ) )
uniioombl.3  |-  S  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
uniioombl.a  |-  A  = 
U. ran  ( (,)  o.  F )
uniioombl.e  |-  ( ph  ->  ( vol* `  E )  e.  RR )
uniioombl.c  |-  ( ph  ->  C  e.  RR+ )
uniioombl.g  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
uniioombl.s  |-  ( ph  ->  E  C_  U. ran  ( (,)  o.  G ) )
uniioombl.t  |-  T  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )
uniioombl.v  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol* `  E )  +  C
) )
Assertion
Ref Expression
uniioombllem1  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
Distinct variable groups:    x, F    x, G    x, A    x, C    ph, x    x, T
Allowed substitution hints:    S( x)    E( x)

Proof of Theorem uniioombllem1
StepHypRef Expression
1 uniioombl.g . . . . 5  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
2 eqid 2441 . . . . . 6  |-  ( ( abs  o.  -  )  o.  G )  =  ( ( abs  o.  -  )  o.  G )
3 uniioombl.t . . . . . 6  |-  T  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )
42, 3ovolsf 21750 . . . . 5  |-  ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  T : NN --> ( 0 [,) +oo ) )
51, 4syl 16 . . . 4  |-  ( ph  ->  T : NN --> ( 0 [,) +oo ) )
6 frn 5723 . . . 4  |-  ( T : NN --> ( 0 [,) +oo )  ->  ran  T  C_  ( 0 [,) +oo ) )
75, 6syl 16 . . 3  |-  ( ph  ->  ran  T  C_  (
0 [,) +oo )
)
8 rge0ssre 11632 . . 3  |-  ( 0 [,) +oo )  C_  RR
97, 8syl6ss 3498 . 2  |-  ( ph  ->  ran  T  C_  RR )
10 1nn 10548 . . . . 5  |-  1  e.  NN
11 fdm 5721 . . . . . 6  |-  ( T : NN --> ( 0 [,) +oo )  ->  dom  T  =  NN )
125, 11syl 16 . . . . 5  |-  ( ph  ->  dom  T  =  NN )
1310, 12syl5eleqr 2536 . . . 4  |-  ( ph  ->  1  e.  dom  T
)
14 ne0i 3773 . . . 4  |-  ( 1  e.  dom  T  ->  dom  T  =/=  (/) )
1513, 14syl 16 . . 3  |-  ( ph  ->  dom  T  =/=  (/) )
16 dm0rn0 5205 . . . 4  |-  ( dom 
T  =  (/)  <->  ran  T  =  (/) )
1716necon3bii 2709 . . 3  |-  ( dom 
T  =/=  (/)  <->  ran  T  =/=  (/) )
1815, 17sylib 196 . 2  |-  ( ph  ->  ran  T  =/=  (/) )
19 icossxr 11613 . . . . 5  |-  ( 0 [,) +oo )  C_  RR*
207, 19syl6ss 3498 . . . 4  |-  ( ph  ->  ran  T  C_  RR* )
21 supxrcl 11510 . . . 4  |-  ( ran 
T  C_  RR*  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
2220, 21syl 16 . . 3  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
23 uniioombl.e . . . . 5  |-  ( ph  ->  ( vol* `  E )  e.  RR )
24 uniioombl.c . . . . . 6  |-  ( ph  ->  C  e.  RR+ )
2524rpred 11260 . . . . 5  |-  ( ph  ->  C  e.  RR )
2623, 25readdcld 9621 . . . 4  |-  ( ph  ->  ( ( vol* `  E )  +  C
)  e.  RR )
2726rexrd 9641 . . 3  |-  ( ph  ->  ( ( vol* `  E )  +  C
)  e.  RR* )
28 pnfxr 11325 . . . 4  |- +oo  e.  RR*
2928a1i 11 . . 3  |-  ( ph  -> +oo  e.  RR* )
30 uniioombl.v . . 3  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol* `  E )  +  C
) )
31 ltpnf 11335 . . . 4  |-  ( ( ( vol* `  E )  +  C
)  e.  RR  ->  ( ( vol* `  E )  +  C
)  < +oo )
3226, 31syl 16 . . 3  |-  ( ph  ->  ( ( vol* `  E )  +  C
)  < +oo )
3322, 27, 29, 30, 32xrlelttrd 11367 . 2  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  < +oo )
34 supxrbnd 11524 . 2  |-  ( ( ran  T  C_  RR  /\ 
ran  T  =/=  (/)  /\  sup ( ran  T ,  RR* ,  <  )  < +oo )  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
359, 18, 33, 34syl3anc 1227 1  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1381    e. wcel 1802    =/= wne 2636    i^i cin 3457    C_ wss 3458   (/)c0 3767   U.cuni 4230  Disj wdisj 4403   class class class wbr 4433    X. cxp 4983   dom cdm 4985   ran crn 4986    o. ccom 4989   -->wf 5570   ` cfv 5574  (class class class)co 6277   supcsup 7898   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493   +oocpnf 9623   RR*cxr 9625    < clt 9626    <_ cle 9627    - cmin 9805   NNcn 10537   RR+crp 11224   (,)cioo 11533   [,)cico 11535    seqcseq 12081   abscabs 13041   vol*covol 21740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-sup 7899  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-ico 11539  df-fz 11677  df-seq 12082  df-exp 12141  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043
This theorem is referenced by:  uniioombllem3  21860  uniioombllem4  21861  uniioombllem5  21862  uniioombllem6  21863
  Copyright terms: Public domain W3C validator