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Theorem uniioombllem3a 18939
Description: Lemma for uniioombl 18944. (Contributed by Mario Carneiro, 8-May-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
) )
uniioombl.m  |-  ( ph  ->  M  e.  NN )
uniioombl.m2  |-  ( ph  ->  ( abs `  (
( T `  M
)  -  sup ( ran  T ,  RR* ,  <  ) ) )  <  C
)
uniioombl.k  |-  K  = 
U. ( ( (,) 
o.  G ) "
( 1 ... M
) )
Assertion
Ref Expression
uniioombllem3a  |-  ( ph  ->  ( K  =  U_ j  e.  ( 1 ... M ) ( (,) `  ( G `
 j ) )  /\  ( vol * `  K )  e.  RR ) )
Distinct variable groups:    x, j, F    j, G, x    j, K, x    A, j, x    C, j, x    j, M, x    ph, j, x    T, j, x
Allowed substitution hints:    S( x, j)    E( x, j)

Proof of Theorem uniioombllem3a
StepHypRef Expression
1 uniioombl.k . . 3  |-  K  = 
U. ( ( (,) 
o.  G ) "
( 1 ... M
) )
2 ioof 10741 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
3 uniioombl.g . . . . . . 7  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
4 inss2 3390 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 ressxr 8876 . . . . . . . . 9  |-  RR  C_  RR*
6 xpss12 4792 . . . . . . . . 9  |-  ( ( RR  C_  RR*  /\  RR  C_ 
RR* )  ->  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
)
75, 5, 6mp2an 653 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
84, 7sstri 3188 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
9 fss 5397 . . . . . . 7  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* ) )  ->  G : NN --> ( RR*  X. 
RR* ) )
103, 8, 9sylancl 643 . . . . . 6  |-  ( ph  ->  G : NN --> ( RR*  X. 
RR* ) )
11 fco 5398 . . . . . 6  |-  ( ( (,) : ( RR*  X. 
RR* ) --> ~P RR  /\  G : NN --> ( RR*  X. 
RR* ) )  -> 
( (,)  o.  G
) : NN --> ~P RR )
122, 10, 11sylancr 644 . . . . 5  |-  ( ph  ->  ( (,)  o.  G
) : NN --> ~P RR )
13 ffun 5391 . . . . 5  |-  ( ( (,)  o.  G ) : NN --> ~P RR  ->  Fun  ( (,)  o.  G ) )
14 funiunfv 5774 . . . . 5  |-  ( Fun  ( (,)  o.  G
)  ->  U_ j  e.  ( 1 ... M
) ( ( (,) 
o.  G ) `  j )  =  U. ( ( (,)  o.  G ) " (
1 ... M ) ) )
1512, 13, 143syl 18 . . . 4  |-  ( ph  ->  U_ j  e.  ( 1 ... M ) ( ( (,)  o.  G ) `  j
)  =  U. (
( (,)  o.  G
) " ( 1 ... M ) ) )
16 elfznn 10819 . . . . . 6  |-  ( j  e.  ( 1 ... M )  ->  j  e.  NN )
17 fvco3 5596 . . . . . 6  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  j  e.  NN )  ->  (
( (,)  o.  G
) `  j )  =  ( (,) `  ( G `  j )
) )
183, 16, 17syl2an 463 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  (
( (,)  o.  G
) `  j )  =  ( (,) `  ( G `  j )
) )
1918iuneq2dv 3926 . . . 4  |-  ( ph  ->  U_ j  e.  ( 1 ... M ) ( ( (,)  o.  G ) `  j
)  =  U_ j  e.  ( 1 ... M
) ( (,) `  ( G `  j )
) )
2015, 19eqtr3d 2317 . . 3  |-  ( ph  ->  U. ( ( (,) 
o.  G ) "
( 1 ... M
) )  =  U_ j  e.  ( 1 ... M ) ( (,) `  ( G `
 j ) ) )
211, 20syl5eq 2327 . 2  |-  ( ph  ->  K  =  U_ j  e.  ( 1 ... M
) ( (,) `  ( G `  j )
) )
22 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  j  e.  NN )  ->  ( G `  j )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
233, 16, 22syl2an 463 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( G `  j )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
244, 23sseldi 3178 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( G `  j )  e.  ( RR  X.  RR ) )
25 1st2nd2 6159 . . . . . . . . . 10  |-  ( ( G `  j )  e.  ( RR  X.  RR )  ->  ( G `
 j )  = 
<. ( 1st `  ( G `  j )
) ,  ( 2nd `  ( G `  j
) ) >. )
2624, 25syl 15 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( G `  j )  =  <. ( 1st `  ( G `  j )
) ,  ( 2nd `  ( G `  j
) ) >. )
2726fveq2d 5529 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( (,) `  ( G `  j ) )  =  ( (,) `  <. ( 1st `  ( G `
 j ) ) ,  ( 2nd `  ( G `  j )
) >. ) )
28 df-ov 5861 . . . . . . . 8  |-  ( ( 1st `  ( G `
 j ) ) (,) ( 2nd `  ( G `  j )
) )  =  ( (,) `  <. ( 1st `  ( G `  j ) ) ,  ( 2nd `  ( G `  j )
) >. )
2927, 28syl6eqr 2333 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( (,) `  ( G `  j ) )  =  ( ( 1st `  ( G `  j )
) (,) ( 2nd `  ( G `  j
) ) ) )
30 ioossre 10712 . . . . . . . 8  |-  ( ( 1st `  ( G `
 j ) ) (,) ( 2nd `  ( G `  j )
) )  C_  RR
3130a1i 10 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  (
( 1st `  ( G `  j )
) (,) ( 2nd `  ( G `  j
) ) )  C_  RR )
3229, 31eqsstrd 3212 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( (,) `  ( G `  j ) )  C_  RR )
3332ralrimiva 2626 . . . . 5  |-  ( ph  ->  A. j  e.  ( 1 ... M ) ( (,) `  ( G `  j )
)  C_  RR )
34 iunss 3943 . . . . 5  |-  ( U_ j  e.  ( 1 ... M ) ( (,) `  ( G `
 j ) ) 
C_  RR  <->  A. j  e.  ( 1 ... M ) ( (,) `  ( G `  j )
)  C_  RR )
3533, 34sylibr 203 . . . 4  |-  ( ph  ->  U_ j  e.  ( 1 ... M ) ( (,) `  ( G `  j )
)  C_  RR )
3621, 35eqsstrd 3212 . . 3  |-  ( ph  ->  K  C_  RR )
37 fzfid 11035 . . . 4  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
3829fveq2d 5529 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( vol * `  ( (,) `  ( G `  j
) ) )  =  ( vol * `  ( ( 1st `  ( G `  j )
) (,) ( 2nd `  ( G `  j
) ) ) ) )
39 ovolfcl 18826 . . . . . . . 8  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  j  e.  NN )  ->  (
( 1st `  ( G `  j )
)  e.  RR  /\  ( 2nd `  ( G `
 j ) )  e.  RR  /\  ( 1st `  ( G `  j ) )  <_ 
( 2nd `  ( G `  j )
) ) )
403, 16, 39syl2an 463 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  (
( 1st `  ( G `  j )
)  e.  RR  /\  ( 2nd `  ( G `
 j ) )  e.  RR  /\  ( 1st `  ( G `  j ) )  <_ 
( 2nd `  ( G `  j )
) ) )
41 ovolioo 18925 . . . . . . 7  |-  ( ( ( 1st `  ( G `  j )
)  e.  RR  /\  ( 2nd `  ( G `
 j ) )  e.  RR  /\  ( 1st `  ( G `  j ) )  <_ 
( 2nd `  ( G `  j )
) )  ->  ( vol * `  ( ( 1st `  ( G `
 j ) ) (,) ( 2nd `  ( G `  j )
) ) )  =  ( ( 2nd `  ( G `  j )
)  -  ( 1st `  ( G `  j
) ) ) )
4240, 41syl 15 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( vol * `  ( ( 1st `  ( G `
 j ) ) (,) ( 2nd `  ( G `  j )
) ) )  =  ( ( 2nd `  ( G `  j )
)  -  ( 1st `  ( G `  j
) ) ) )
4338, 42eqtrd 2315 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( vol * `  ( (,) `  ( G `  j
) ) )  =  ( ( 2nd `  ( G `  j )
)  -  ( 1st `  ( G `  j
) ) ) )
4440simp2d 968 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( 2nd `  ( G `  j ) )  e.  RR )
4540simp1d 967 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( 1st `  ( G `  j ) )  e.  RR )
4644, 45resubcld 9211 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  (
( 2nd `  ( G `  j )
)  -  ( 1st `  ( G `  j
) ) )  e.  RR )
4743, 46eqeltrd 2357 . . . 4  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( vol * `  ( (,) `  ( G `  j
) ) )  e.  RR )
4837, 47fsumrecl 12207 . . 3  |-  ( ph  -> 
sum_ j  e.  ( 1 ... M ) ( vol * `  ( (,) `  ( G `
 j ) ) )  e.  RR )
4921fveq2d 5529 . . . 4  |-  ( ph  ->  ( vol * `  K )  =  ( vol * `  U_ j  e.  ( 1 ... M
) ( (,) `  ( G `  j )
) ) )
5032, 47jca 518 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  (
( (,) `  ( G `  j )
)  C_  RR  /\  ( vol * `  ( (,) `  ( G `  j
) ) )  e.  RR ) )
5150ralrimiva 2626 . . . . 5  |-  ( ph  ->  A. j  e.  ( 1 ... M ) ( ( (,) `  ( G `  j )
)  C_  RR  /\  ( vol * `  ( (,) `  ( G `  j
) ) )  e.  RR ) )
52 ovolfiniun 18860 . . . . 5  |-  ( ( ( 1 ... M
)  e.  Fin  /\  A. j  e.  ( 1 ... M ) ( ( (,) `  ( G `  j )
)  C_  RR  /\  ( vol * `  ( (,) `  ( G `  j
) ) )  e.  RR ) )  -> 
( vol * `  U_ j  e.  ( 1 ... M ) ( (,) `  ( G `
 j ) ) )  <_  sum_ j  e.  ( 1 ... M
) ( vol * `  ( (,) `  ( G `  j )
) ) )
5337, 51, 52syl2anc 642 . . . 4  |-  ( ph  ->  ( vol * `  U_ j  e.  ( 1 ... M ) ( (,) `  ( G `
 j ) ) )  <_  sum_ j  e.  ( 1 ... M
) ( vol * `  ( (,) `  ( G `  j )
) ) )
5449, 53eqbrtrd 4043 . . 3  |-  ( ph  ->  ( vol * `  K )  <_  sum_ j  e.  ( 1 ... M
) ( vol * `  ( (,) `  ( G `  j )
) ) )
55 ovollecl 18842 . . 3  |-  ( ( K  C_  RR  /\  sum_ j  e.  ( 1 ... M ) ( vol * `  ( (,) `  ( G `  j ) ) )  e.  RR  /\  ( vol * `  K )  <_  sum_ j  e.  ( 1 ... M ) ( vol * `  ( (,) `  ( G `
 j ) ) ) )  ->  ( vol * `  K )  e.  RR )
5636, 48, 54, 55syl3anc 1182 . 2  |-  ( ph  ->  ( vol * `  K )  e.  RR )
5721, 56jca 518 1  |-  ( ph  ->  ( K  =  U_ j  e.  ( 1 ... M ) ( (,) `  ( G `
 j ) )  /\  ( vol * `  K )  e.  RR ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   <.cop 3643   U.cuni 3827   U_ciun 3905  Disj wdisj 3993   class class class wbr 4023    X. cxp 4687   ran crn 4690   "cima 4692    o. ccom 4693   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   Fincfn 6863   supcsup 7193   RRcr 8736   1c1 8738    + caddc 8740   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   RR+crp 10354   (,)cioo 10656   ...cfz 10782    seq cseq 11046   abscabs 11719   sum_csu 12158   vol
*covol 18822
This theorem is referenced by:  uniioombllem3  18940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-rest 13327  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cmp 17114  df-ovol 18824  df-vol 18825
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