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Theorem uniioombllem1 21060
Description: Lemma for uniioombl 21068. (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 2442 . . . . . 6  |-  ( ( abs  o.  -  )  o.  G )  =  ( ( abs  o.  -  )  o.  G )
3 uniioombl.t . . . . . 6  |-  T  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )
42, 3ovolsf 20955 . . . . 5  |-  ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  T : NN --> ( 0 [,) +oo ) )
51, 4syl 16 . . . 4  |-  ( ph  ->  T : NN --> ( 0 [,) +oo ) )
6 frn 5564 . . . 4  |-  ( T : NN --> ( 0 [,) +oo )  ->  ran  T  C_  ( 0 [,) +oo ) )
75, 6syl 16 . . 3  |-  ( ph  ->  ran  T  C_  (
0 [,) +oo )
)
8 0re 9385 . . . 4  |-  0  e.  RR
9 pnfxr 11091 . . . 4  |- +oo  e.  RR*
10 icossre 11375 . . . 4  |-  ( ( 0  e.  RR  /\ +oo  e.  RR* )  ->  (
0 [,) +oo )  C_  RR )
118, 9, 10mp2an 672 . . 3  |-  ( 0 [,) +oo )  C_  RR
127, 11syl6ss 3367 . 2  |-  ( ph  ->  ran  T  C_  RR )
13 1nn 10332 . . . . 5  |-  1  e.  NN
14 fdm 5562 . . . . . 6  |-  ( T : NN --> ( 0 [,) +oo )  ->  dom  T  =  NN )
155, 14syl 16 . . . . 5  |-  ( ph  ->  dom  T  =  NN )
1613, 15syl5eleqr 2529 . . . 4  |-  ( ph  ->  1  e.  dom  T
)
17 ne0i 3642 . . . 4  |-  ( 1  e.  dom  T  ->  dom  T  =/=  (/) )
1816, 17syl 16 . . 3  |-  ( ph  ->  dom  T  =/=  (/) )
19 dm0rn0 5055 . . . 4  |-  ( dom 
T  =  (/)  <->  ran  T  =  (/) )
2019necon3bii 2639 . . 3  |-  ( dom 
T  =/=  (/)  <->  ran  T  =/=  (/) )
2118, 20sylib 196 . 2  |-  ( ph  ->  ran  T  =/=  (/) )
22 icossxr 11379 . . . . 5  |-  ( 0 [,) +oo )  C_  RR*
237, 22syl6ss 3367 . . . 4  |-  ( ph  ->  ran  T  C_  RR* )
24 supxrcl 11276 . . . 4  |-  ( ran 
T  C_  RR*  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
2523, 24syl 16 . . 3  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
26 uniioombl.e . . . . 5  |-  ( ph  ->  ( vol* `  E )  e.  RR )
27 uniioombl.c . . . . . 6  |-  ( ph  ->  C  e.  RR+ )
2827rpred 11026 . . . . 5  |-  ( ph  ->  C  e.  RR )
2926, 28readdcld 9412 . . . 4  |-  ( ph  ->  ( ( vol* `  E )  +  C
)  e.  RR )
3029rexrd 9432 . . 3  |-  ( ph  ->  ( ( vol* `  E )  +  C
)  e.  RR* )
319a1i 11 . . 3  |-  ( ph  -> +oo  e.  RR* )
32 uniioombl.v . . 3  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol* `  E )  +  C
) )
33 ltpnf 11101 . . . 4  |-  ( ( ( vol* `  E )  +  C
)  e.  RR  ->  ( ( vol* `  E )  +  C
)  < +oo )
3429, 33syl 16 . . 3  |-  ( ph  ->  ( ( vol* `  E )  +  C
)  < +oo )
3525, 30, 31, 32, 34xrlelttrd 11133 . 2  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  < +oo )
36 supxrbnd 11290 . 2  |-  ( ( ran  T  C_  RR  /\ 
ran  T  =/=  (/)  /\  sup ( ran  T ,  RR* ,  <  )  < +oo )  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
3712, 21, 35, 36syl3anc 1218 1  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    =/= wne 2605    i^i cin 3326    C_ wss 3327   (/)c0 3636   U.cuni 4090  Disj wdisj 4261   class class class wbr 4291    X. cxp 4837   dom cdm 4839   ran crn 4840    o. ccom 4843   -->wf 5413   ` cfv 5417  (class class class)co 6090   supcsup 7689   RRcr 9280   0cc0 9281   1c1 9282    + caddc 9284   +oocpnf 9414   RR*cxr 9416    < clt 9417    <_ cle 9418    - cmin 9594   NNcn 10321   RR+crp 10990   (,)cioo 11299   [,)cico 11301    seqcseq 11805   abscabs 12722   vol*covol 20945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-n0 10579  df-z 10646  df-uz 10861  df-rp 10991  df-ico 11305  df-fz 11437  df-seq 11806  df-exp 11865  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724
This theorem is referenced by:  uniioombllem3  21064  uniioombllem4  21065  uniioombllem5  21066  uniioombllem6  21067
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