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Theorem uniioombllem1 22159
Description: Lemma for uniioombl 22167. (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 2454 . . . . . 6  |-  ( ( abs  o.  -  )  o.  G )  =  ( ( abs  o.  -  )  o.  G )
3 uniioombl.t . . . . . 6  |-  T  =  seq 1 (  +  ,  ( ( abs 
o.  -  )  o.  G ) )
42, 3ovolsf 22053 . . . . 5  |-  ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  ->  T : NN --> ( 0 [,) +oo ) )
51, 4syl 16 . . . 4  |-  ( ph  ->  T : NN --> ( 0 [,) +oo ) )
6 frn 5719 . . . 4  |-  ( T : NN --> ( 0 [,) +oo )  ->  ran  T  C_  ( 0 [,) +oo ) )
75, 6syl 16 . . 3  |-  ( ph  ->  ran  T  C_  (
0 [,) +oo )
)
8 rge0ssre 11631 . . 3  |-  ( 0 [,) +oo )  C_  RR
97, 8syl6ss 3501 . 2  |-  ( ph  ->  ran  T  C_  RR )
10 1nn 10542 . . . . 5  |-  1  e.  NN
11 fdm 5717 . . . . . 6  |-  ( T : NN --> ( 0 [,) +oo )  ->  dom  T  =  NN )
125, 11syl 16 . . . . 5  |-  ( ph  ->  dom  T  =  NN )
1310, 12syl5eleqr 2549 . . . 4  |-  ( ph  ->  1  e.  dom  T
)
14 ne0i 3789 . . . 4  |-  ( 1  e.  dom  T  ->  dom  T  =/=  (/) )
1513, 14syl 16 . . 3  |-  ( ph  ->  dom  T  =/=  (/) )
16 dm0rn0 5208 . . . 4  |-  ( dom 
T  =  (/)  <->  ran  T  =  (/) )
1716necon3bii 2722 . . 3  |-  ( dom 
T  =/=  (/)  <->  ran  T  =/=  (/) )
1815, 17sylib 196 . 2  |-  ( ph  ->  ran  T  =/=  (/) )
19 icossxr 11612 . . . . 5  |-  ( 0 [,) +oo )  C_  RR*
207, 19syl6ss 3501 . . . 4  |-  ( ph  ->  ran  T  C_  RR* )
21 supxrcl 11509 . . . 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 11259 . . . . 5  |-  ( ph  ->  C  e.  RR )
2623, 25readdcld 9612 . . . 4  |-  ( ph  ->  ( ( vol* `  E )  +  C
)  e.  RR )
2726rexrd 9632 . . 3  |-  ( ph  ->  ( ( vol* `  E )  +  C
)  e.  RR* )
28 pnfxr 11324 . . . 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 11334 . . . 4  |-  ( ( ( vol* `  E )  +  C
)  e.  RR  ->  ( ( vol* `  E )  +  C
)  < +oo )
3226, 31syl 16 . . 3  |-  ( ph  ->  ( ( vol* `  E )  +  C
)  < +oo )
3322, 27, 29, 30, 32xrlelttrd 11366 . 2  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  < +oo )
34 supxrbnd 11523 . 2  |-  ( ( ran  T  C_  RR  /\ 
ran  T  =/=  (/)  /\  sup ( ran  T ,  RR* ,  <  )  < +oo )  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
359, 18, 33, 34syl3anc 1226 1  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823    =/= wne 2649    i^i cin 3460    C_ wss 3461   (/)c0 3783   U.cuni 4235  Disj wdisj 4410   class class class wbr 4439    X. cxp 4986   dom cdm 4988   ran crn 4989    o. ccom 4992   -->wf 5566   ` cfv 5570  (class class class)co 6270   supcsup 7892   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484   +oocpnf 9614   RR*cxr 9616    < clt 9617    <_ cle 9618    - cmin 9796   NNcn 10531   RR+crp 11221   (,)cioo 11532   [,)cico 11534    seqcseq 12092   abscabs 13152   vol*covol 22043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-ico 11538  df-fz 11676  df-seq 12093  df-exp 12152  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154
This theorem is referenced by:  uniioombllem3  22163  uniioombllem4  22164  uniioombllem5  22165  uniioombllem6  22166
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