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Theorem uniioombllem3a 19154
Description: Lemma for uniioombl 19159. (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 10894 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
3 uniioombl.g . . . . . . 7  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
4 inss2 3478 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 ressxr 9023 . . . . . . . . 9  |-  RR  C_  RR*
6 xpss12 4895 . . . . . . . . 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 3274 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
9 fss 5503 . . . . . . 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 5504 . . . . . 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 5497 . . . . 5  |-  ( ( (,)  o.  G ) : NN --> ~P RR  ->  Fun  ( (,)  o.  G ) )
14 funiunfv 5895 . . . . 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 10972 . . . . . 6  |-  ( j  e.  ( 1 ... M )  ->  j  e.  NN )
17 fvco3 5703 . . . . . 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 4028 . . . 4  |-  ( ph  ->  U_ j  e.  ( 1 ... M ) ( ( (,)  o.  G ) `  j
)  =  U_ j  e.  ( 1 ... M
) ( (,) `  ( G `  j )
) )
2015, 19eqtr3d 2400 . . 3  |-  ( ph  ->  U. ( ( (,) 
o.  G ) "
( 1 ... M
) )  =  U_ j  e.  ( 1 ... M ) ( (,) `  ( G `
 j ) ) )
211, 20syl5eq 2410 . 2  |-  ( ph  ->  K  =  U_ j  e.  ( 1 ... M
) ( (,) `  ( G `  j )
) )
22 ffvelrn 5770 . . . . . . . . . . . 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 3264 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( G `  j )  e.  ( RR  X.  RR ) )
25 1st2nd2 6286 . . . . . . . . . 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 5636 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( (,) `  ( G `  j ) )  =  ( (,) `  <. ( 1st `  ( G `
 j ) ) ,  ( 2nd `  ( G `  j )
) >. ) )
28 df-ov 5984 . . . . . . . 8  |-  ( ( 1st `  ( G `
 j ) ) (,) ( 2nd `  ( G `  j )
) )  =  ( (,) `  <. ( 1st `  ( G `  j ) ) ,  ( 2nd `  ( G `  j )
) >. )
2927, 28syl6eqr 2416 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( (,) `  ( G `  j ) )  =  ( ( 1st `  ( G `  j )
) (,) ( 2nd `  ( G `  j
) ) ) )
30 ioossre 10865 . . . . . . . 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 3298 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( (,) `  ( G `  j ) )  C_  RR )
3332ralrimiva 2711 . . . . 5  |-  ( ph  ->  A. j  e.  ( 1 ... M ) ( (,) `  ( G `  j )
)  C_  RR )
34 iunss 4045 . . . . 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 3298 . . 3  |-  ( ph  ->  K  C_  RR )
37 fzfid 11199 . . . 4  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
3829fveq2d 5636 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( vol * `  ( (,) `  ( G `  j
) ) )  =  ( vol * `  ( ( 1st `  ( G `  j )
) (,) ( 2nd `  ( G `  j
) ) ) ) )
39 ovolfcl 19041 . . . . . . . 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 19140 . . . . . . 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 2398 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( vol * `  ( (,) `  ( G `  j
) ) )  =  ( ( 2nd `  ( G `  j )
)  -  ( 1st `  ( G `  j
) ) ) )
4440simp2d 969 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( 2nd `  ( G `  j ) )  e.  RR )
4540simp1d 968 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( 1st `  ( G `  j ) )  e.  RR )
4644, 45resubcld 9358 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  (
( 2nd `  ( G `  j )
)  -  ( 1st `  ( G `  j
) ) )  e.  RR )
4743, 46eqeltrd 2440 . . . 4  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( vol * `  ( (,) `  ( G `  j
) ) )  e.  RR )
4837, 47fsumrecl 12415 . . 3  |-  ( ph  -> 
sum_ j  e.  ( 1 ... M ) ( vol * `  ( (,) `  ( G `
 j ) ) )  e.  RR )
4921fveq2d 5636 . . . 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 2711 . . . . 5  |-  ( ph  ->  A. j  e.  ( 1 ... M ) ( ( (,) `  ( G `  j )
)  C_  RR  /\  ( vol * `  ( (,) `  ( G `  j
) ) )  e.  RR ) )
52 ovolfiniun 19075 . . . . 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 4145 . . 3  |-  ( ph  ->  ( vol * `  K )  <_  sum_ j  e.  ( 1 ... M
) ( vol * `  ( (,) `  ( G `  j )
) ) )
55 ovollecl 19057 . . 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 1183 . 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 935    = wceq 1647    e. wcel 1715   A.wral 2628    i^i cin 3237    C_ wss 3238   ~Pcpw 3714   <.cop 3732   U.cuni 3929   U_ciun 4007  Disj wdisj 4095   class class class wbr 4125    X. cxp 4790   ran crn 4793   "cima 4795    o. ccom 4796   Fun wfun 5352   -->wf 5354   ` cfv 5358  (class class class)co 5981   1stc1st 6247   2ndc2nd 6248   Fincfn 7006   supcsup 7340   RRcr 8883   1c1 8885    + caddc 8887   RR*cxr 9013    < clt 9014    <_ cle 9015    - cmin 9184   NNcn 9893   RR+crp 10505   (,)cioo 10809   ...cfz 10935    seq cseq 11210   abscabs 11926   sum_csu 12366   vol
*covol 19037
This theorem is referenced by:  uniioombllem3  19155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-map 6917  df-pm 6918  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-fi 7312  df-sup 7341  df-oi 7372  df-card 7719  df-cda 7941  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-n0 10115  df-z 10176  df-uz 10382  df-q 10468  df-rp 10506  df-xneg 10603  df-xadd 10604  df-xmul 10605  df-ioo 10813  df-ico 10815  df-icc 10816  df-fz 10936  df-fzo 11026  df-fl 11089  df-seq 11211  df-exp 11270  df-hash 11506  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-clim 12169  df-rlim 12170  df-sum 12367  df-rest 13537  df-topgen 13554  df-xmet 16586  df-met 16587  df-bl 16588  df-mopn 16589  df-top 16853  df-bases 16855  df-topon 16856  df-cmp 17331  df-ovol 19039  df-vol 19040
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