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Theorem uniioombllem1 19478
Description: Lemma for uniioombl 19486. (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 2438 . . . . . 6  |-  ( ( abs  o.  -  )  o.  G )  =  ( ( abs  o.  -  )  o.  G )
3 uniioombl.t . . . . . 6  |-  T  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )
42, 3ovolsf 19374 . . . . 5  |-  ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  T : NN --> ( 0 [,) 
+oo ) )
51, 4syl 16 . . . 4  |-  ( ph  ->  T : NN --> ( 0 [,)  +oo ) )
6 frn 5600 . . . 4  |-  ( T : NN --> ( 0 [,)  +oo )  ->  ran  T 
C_  ( 0 [,) 
+oo ) )
75, 6syl 16 . . 3  |-  ( ph  ->  ran  T  C_  (
0 [,)  +oo ) )
8 0re 9096 . . . 4  |-  0  e.  RR
9 pnfxr 10718 . . . 4  |-  +oo  e.  RR*
10 icossre 10996 . . . 4  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
118, 9, 10mp2an 655 . . 3  |-  ( 0 [,)  +oo )  C_  RR
127, 11syl6ss 3362 . 2  |-  ( ph  ->  ran  T  C_  RR )
13 1nn 10016 . . . . 5  |-  1  e.  NN
14 fdm 5598 . . . . . 6  |-  ( T : NN --> ( 0 [,)  +oo )  ->  dom  T  =  NN )
155, 14syl 16 . . . . 5  |-  ( ph  ->  dom  T  =  NN )
1613, 15syl5eleqr 2525 . . . 4  |-  ( ph  ->  1  e.  dom  T
)
17 ne0i 3636 . . . 4  |-  ( 1  e.  dom  T  ->  dom  T  =/=  (/) )
1816, 17syl 16 . . 3  |-  ( ph  ->  dom  T  =/=  (/) )
19 dm0rn0 5089 . . . 4  |-  ( dom 
T  =  (/)  <->  ran  T  =  (/) )
2019necon3bii 2635 . . 3  |-  ( dom 
T  =/=  (/)  <->  ran  T  =/=  (/) )
2118, 20sylib 190 . 2  |-  ( ph  ->  ran  T  =/=  (/) )
22 icossxr 11000 . . . . 5  |-  ( 0 [,)  +oo )  C_  RR*
237, 22syl6ss 3362 . . . 4  |-  ( ph  ->  ran  T  C_  RR* )
24 supxrcl 10898 . . . 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 10653 . . . . 5  |-  ( ph  ->  C  e.  RR )
2926, 28readdcld 9120 . . . 4  |-  ( ph  ->  ( ( vol * `  E )  +  C
)  e.  RR )
3029rexrd 9139 . . 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 10726 . . . 4  |-  ( ( ( vol * `  E )  +  C
)  e.  RR  ->  ( ( vol * `  E )  +  C
)  <  +oo )
3429, 33syl 16 . . 3  |-  ( ph  ->  ( ( vol * `  E )  +  C
)  <  +oo )
3525, 30, 31, 32, 34xrlelttrd 10755 . 2  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <  +oo )
36 supxrbnd 10912 . 2  |-  ( ( ran  T  C_  RR  /\ 
ran  T  =/=  (/)  /\  sup ( ran  T ,  RR* ,  <  )  <  +oo )  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
3712, 21, 35, 36syl3anc 1185 1  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    =/= wne 2601    i^i cin 3321    C_ wss 3322   (/)c0 3630   U.cuni 4017  Disj wdisj 4185   class class class wbr 4215    X. cxp 4879   dom cdm 4881   ran crn 4882    o. ccom 4885   -->wf 5453   ` cfv 5457  (class class class)co 6084   supcsup 7448   RRcr 8994   0cc0 8995   1c1 8996    + caddc 8998    +oocpnf 9122   RR*cxr 9124    < clt 9125    <_ cle 9126    - cmin 9296   NNcn 10005   RR+crp 10617   (,)cioo 10921   [,)cico 10923    seq cseq 11328   abscabs 12044   vol
*covol 19364
This theorem is referenced by:  uniioombllem3  19482  uniioombllem4  19483  uniioombllem5  19484  uniioombllem6  19485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-ico 10927  df-fz 11049  df-seq 11329  df-exp 11388  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046
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