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Theorem uniioombllem1 19034
Description: Lemma for uniioombl 19042. (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 2358 . . . . . 6  |-  ( ( abs  o.  -  )  o.  G )  =  ( ( abs  o.  -  )  o.  G )
3 uniioombl.t . . . . . 6  |-  T  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )
42, 3ovolsf 18930 . . . . 5  |-  ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  T : NN --> ( 0 [,) 
+oo ) )
51, 4syl 15 . . . 4  |-  ( ph  ->  T : NN --> ( 0 [,)  +oo ) )
6 frn 5475 . . . 4  |-  ( T : NN --> ( 0 [,)  +oo )  ->  ran  T 
C_  ( 0 [,) 
+oo ) )
75, 6syl 15 . . 3  |-  ( ph  ->  ran  T  C_  (
0 [,)  +oo ) )
8 0re 8925 . . . 4  |-  0  e.  RR
9 pnfxr 10544 . . . 4  |-  +oo  e.  RR*
10 icossre 10819 . . . 4  |-  ( ( 0  e.  RR  /\  +oo 
e.  RR* )  ->  (
0 [,)  +oo )  C_  RR )
118, 9, 10mp2an 653 . . 3  |-  ( 0 [,)  +oo )  C_  RR
127, 11syl6ss 3267 . 2  |-  ( ph  ->  ran  T  C_  RR )
13 1nn 9844 . . . . 5  |-  1  e.  NN
14 fdm 5473 . . . . . 6  |-  ( T : NN --> ( 0 [,)  +oo )  ->  dom  T  =  NN )
155, 14syl 15 . . . . 5  |-  ( ph  ->  dom  T  =  NN )
1613, 15syl5eleqr 2445 . . . 4  |-  ( ph  ->  1  e.  dom  T
)
17 ne0i 3537 . . . 4  |-  ( 1  e.  dom  T  ->  dom  T  =/=  (/) )
1816, 17syl 15 . . 3  |-  ( ph  ->  dom  T  =/=  (/) )
19 dm0rn0 4974 . . . 4  |-  ( dom 
T  =  (/)  <->  ran  T  =  (/) )
2019necon3bii 2553 . . 3  |-  ( dom 
T  =/=  (/)  <->  ran  T  =/=  (/) )
2118, 20sylib 188 . 2  |-  ( ph  ->  ran  T  =/=  (/) )
22 icossxr 10823 . . . . 5  |-  ( 0 [,)  +oo )  C_  RR*
237, 22syl6ss 3267 . . . 4  |-  ( ph  ->  ran  T  C_  RR* )
24 supxrcl 10722 . . . 4  |-  ( ran 
T  C_  RR*  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
2523, 24syl 15 . . 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 10479 . . . . 5  |-  ( ph  ->  C  e.  RR )
2926, 28readdcld 8949 . . . 4  |-  ( ph  ->  ( ( vol * `  E )  +  C
)  e.  RR )
3029rexrd 8968 . . 3  |-  ( ph  ->  ( ( vol * `  E )  +  C
)  e.  RR* )
319a1i 10 . . 3  |-  ( ph  ->  +oo  e.  RR* )
32 uniioombl.v . . 3  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <_  ( ( vol * `  E )  +  C
) )
33 ltpnf 10552 . . . 4  |-  ( ( ( vol * `  E )  +  C
)  e.  RR  ->  ( ( vol * `  E )  +  C
)  <  +oo )
3429, 33syl 15 . . 3  |-  ( ph  ->  ( ( vol * `  E )  +  C
)  <  +oo )
3525, 30, 31, 32, 34xrlelttrd 10580 . 2  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  <  +oo )
36 supxrbnd 10736 . 2  |-  ( ( ran  T  C_  RR  /\ 
ran  T  =/=  (/)  /\  sup ( ran  T ,  RR* ,  <  )  <  +oo )  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
3712, 21, 35, 36syl3anc 1182 1  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710    =/= wne 2521    i^i cin 3227    C_ wss 3228   (/)c0 3531   U.cuni 3906  Disj wdisj 4072   class class class wbr 4102    X. cxp 4766   dom cdm 4768   ran crn 4769    o. ccom 4772   -->wf 5330   ` cfv 5334  (class class class)co 5942   supcsup 7280   RRcr 8823   0cc0 8824   1c1 8825    + caddc 8827    +oocpnf 8951   RR*cxr 8953    < clt 8954    <_ cle 8955    - cmin 9124   NNcn 9833   RR+crp 10443   (,)cioo 10745   [,)cico 10747    seq cseq 11135   abscabs 11809   vol
*covol 18920
This theorem is referenced by:  uniioombllem3  19038  uniioombllem4  19039  uniioombllem5  19040  uniioombllem6  19041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-sup 7281  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-rp 10444  df-ico 10751  df-fz 10872  df-seq 11136  df-exp 11195  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811
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