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Theorem uniioombllem3a 19433
Description: Lemma for uniioombl 19438. (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 10962 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
3 uniioombl.g . . . . . . 7  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
4 inss2 3526 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 ressxr 9089 . . . . . . . . 9  |-  RR  C_  RR*
6 xpss12 4944 . . . . . . . . 9  |-  ( ( RR  C_  RR*  /\  RR  C_ 
RR* )  ->  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
)
75, 5, 6mp2an 654 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
84, 7sstri 3321 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
9 fss 5562 . . . . . . 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 644 . . . . . 6  |-  ( ph  ->  G : NN --> ( RR*  X. 
RR* ) )
11 fco 5563 . . . . . 6  |-  ( ( (,) : ( RR*  X. 
RR* ) --> ~P RR  /\  G : NN --> ( RR*  X. 
RR* ) )  -> 
( (,)  o.  G
) : NN --> ~P RR )
122, 10, 11sylancr 645 . . . . 5  |-  ( ph  ->  ( (,)  o.  G
) : NN --> ~P RR )
13 ffun 5556 . . . . 5  |-  ( ( (,)  o.  G ) : NN --> ~P RR  ->  Fun  ( (,)  o.  G ) )
14 funiunfv 5958 . . . . 5  |-  ( Fun  ( (,)  o.  G
)  ->  U_ j  e.  ( 1 ... M
) ( ( (,) 
o.  G ) `  j )  =  U. ( ( (,)  o.  G ) " (
1 ... M ) ) )
1512, 13, 143syl 19 . . . 4  |-  ( ph  ->  U_ j  e.  ( 1 ... M ) ( ( (,)  o.  G ) `  j
)  =  U. (
( (,)  o.  G
) " ( 1 ... M ) ) )
16 elfznn 11040 . . . . . 6  |-  ( j  e.  ( 1 ... M )  ->  j  e.  NN )
17 fvco3 5763 . . . . . 6  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  j  e.  NN )  ->  (
( (,)  o.  G
) `  j )  =  ( (,) `  ( G `  j )
) )
183, 16, 17syl2an 464 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  (
( (,)  o.  G
) `  j )  =  ( (,) `  ( G `  j )
) )
1918iuneq2dv 4078 . . . 4  |-  ( ph  ->  U_ j  e.  ( 1 ... M ) ( ( (,)  o.  G ) `  j
)  =  U_ j  e.  ( 1 ... M
) ( (,) `  ( G `  j )
) )
2015, 19eqtr3d 2442 . . 3  |-  ( ph  ->  U. ( ( (,) 
o.  G ) "
( 1 ... M
) )  =  U_ j  e.  ( 1 ... M ) ( (,) `  ( G `
 j ) ) )
211, 20syl5eq 2452 . 2  |-  ( ph  ->  K  =  U_ j  e.  ( 1 ... M
) ( (,) `  ( G `  j )
) )
22 ffvelrn 5831 . . . . . . . . . . . 12  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  j  e.  NN )  ->  ( G `  j )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
233, 16, 22syl2an 464 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( G `  j )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
244, 23sseldi 3310 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( G `  j )  e.  ( RR  X.  RR ) )
25 1st2nd2 6349 . . . . . . . . . 10  |-  ( ( G `  j )  e.  ( RR  X.  RR )  ->  ( G `
 j )  = 
<. ( 1st `  ( G `  j )
) ,  ( 2nd `  ( G `  j
) ) >. )
2624, 25syl 16 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( G `  j )  =  <. ( 1st `  ( G `  j )
) ,  ( 2nd `  ( G `  j
) ) >. )
2726fveq2d 5695 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( (,) `  ( G `  j ) )  =  ( (,) `  <. ( 1st `  ( G `
 j ) ) ,  ( 2nd `  ( G `  j )
) >. ) )
28 df-ov 6047 . . . . . . . 8  |-  ( ( 1st `  ( G `
 j ) ) (,) ( 2nd `  ( G `  j )
) )  =  ( (,) `  <. ( 1st `  ( G `  j ) ) ,  ( 2nd `  ( G `  j )
) >. )
2927, 28syl6eqr 2458 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( (,) `  ( G `  j ) )  =  ( ( 1st `  ( G `  j )
) (,) ( 2nd `  ( G `  j
) ) ) )
30 ioossre 10932 . . . . . . 7  |-  ( ( 1st `  ( G `
 j ) ) (,) ( 2nd `  ( G `  j )
) )  C_  RR
3129, 30syl6eqss 3362 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( (,) `  ( G `  j ) )  C_  RR )
3231ralrimiva 2753 . . . . 5  |-  ( ph  ->  A. j  e.  ( 1 ... M ) ( (,) `  ( G `  j )
)  C_  RR )
33 iunss 4096 . . . . 5  |-  ( U_ j  e.  ( 1 ... M ) ( (,) `  ( G `
 j ) ) 
C_  RR  <->  A. j  e.  ( 1 ... M ) ( (,) `  ( G `  j )
)  C_  RR )
3432, 33sylibr 204 . . . 4  |-  ( ph  ->  U_ j  e.  ( 1 ... M ) ( (,) `  ( G `  j )
)  C_  RR )
3521, 34eqsstrd 3346 . . 3  |-  ( ph  ->  K  C_  RR )
36 fzfid 11271 . . . 4  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
3729fveq2d 5695 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( vol * `  ( (,) `  ( G `  j
) ) )  =  ( vol * `  ( ( 1st `  ( G `  j )
) (,) ( 2nd `  ( G `  j
) ) ) ) )
38 ovolfcl 19320 . . . . . . . 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 )
) ) )
393, 16, 38syl2an 464 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  (
( 1st `  ( G `  j )
)  e.  RR  /\  ( 2nd `  ( G `
 j ) )  e.  RR  /\  ( 1st `  ( G `  j ) )  <_ 
( 2nd `  ( G `  j )
) ) )
40 ovolioo 19419 . . . . . . 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
) ) ) )
4139, 40syl 16 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( vol * `  ( ( 1st `  ( G `
 j ) ) (,) ( 2nd `  ( G `  j )
) ) )  =  ( ( 2nd `  ( G `  j )
)  -  ( 1st `  ( G `  j
) ) ) )
4237, 41eqtrd 2440 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( vol * `  ( (,) `  ( G `  j
) ) )  =  ( ( 2nd `  ( G `  j )
)  -  ( 1st `  ( G `  j
) ) ) )
4339simp2d 970 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( 2nd `  ( G `  j ) )  e.  RR )
4439simp1d 969 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( 1st `  ( G `  j ) )  e.  RR )
4543, 44resubcld 9425 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  (
( 2nd `  ( G `  j )
)  -  ( 1st `  ( G `  j
) ) )  e.  RR )
4642, 45eqeltrd 2482 . . . 4  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( vol * `  ( (,) `  ( G `  j
) ) )  e.  RR )
4736, 46fsumrecl 12487 . . 3  |-  ( ph  -> 
sum_ j  e.  ( 1 ... M ) ( vol * `  ( (,) `  ( G `
 j ) ) )  e.  RR )
4821fveq2d 5695 . . . 4  |-  ( ph  ->  ( vol * `  K )  =  ( vol * `  U_ j  e.  ( 1 ... M
) ( (,) `  ( G `  j )
) ) )
4931, 46jca 519 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  (
( (,) `  ( G `  j )
)  C_  RR  /\  ( vol * `  ( (,) `  ( G `  j
) ) )  e.  RR ) )
5049ralrimiva 2753 . . . . 5  |-  ( ph  ->  A. j  e.  ( 1 ... M ) ( ( (,) `  ( G `  j )
)  C_  RR  /\  ( vol * `  ( (,) `  ( G `  j
) ) )  e.  RR ) )
51 ovolfiniun 19354 . . . . 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 )
) ) )
5236, 50, 51syl2anc 643 . . . 4  |-  ( ph  ->  ( vol * `  U_ j  e.  ( 1 ... M ) ( (,) `  ( G `
 j ) ) )  <_  sum_ j  e.  ( 1 ... M
) ( vol * `  ( (,) `  ( G `  j )
) ) )
5348, 52eqbrtrd 4196 . . 3  |-  ( ph  ->  ( vol * `  K )  <_  sum_ j  e.  ( 1 ... M
) ( vol * `  ( (,) `  ( G `  j )
) ) )
54 ovollecl 19336 . . 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 )
5535, 47, 53, 54syl3anc 1184 . 2  |-  ( ph  ->  ( vol * `  K )  e.  RR )
5621, 55jca 519 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670    i^i cin 3283    C_ wss 3284   ~Pcpw 3763   <.cop 3781   U.cuni 3979   U_ciun 4057  Disj wdisj 4146   class class class wbr 4176    X. cxp 4839   ran crn 4842   "cima 4844    o. ccom 4845   Fun wfun 5411   -->wf 5413   ` cfv 5417  (class class class)co 6044   1stc1st 6310   2ndc2nd 6311   Fincfn 7072   supcsup 7407   RRcr 8949   1c1 8951    + caddc 8953   RR*cxr 9079    < clt 9080    <_ cle 9081    - cmin 9251   NNcn 9960   RR+crp 10572   (,)cioo 10876   ...cfz 11003    seq cseq 11282   abscabs 11998   sum_csu 12438   vol
*covol 19316
This theorem is referenced by:  uniioombllem3  19434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6868  df-map 6983  df-pm 6984  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-fi 7378  df-sup 7408  df-oi 7439  df-card 7786  df-cda 8008  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-q 10535  df-rp 10573  df-xneg 10670  df-xadd 10671  df-xmul 10672  df-ioo 10880  df-ico 10882  df-icc 10883  df-fz 11004  df-fzo 11095  df-fl 11161  df-seq 11283  df-exp 11342  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-rlim 12242  df-sum 12439  df-rest 13609  df-topgen 13626  df-psmet 16653  df-xmet 16654  df-met 16655  df-bl 16656  df-mopn 16657  df-top 16922  df-bases 16924  df-topon 16925  df-cmp 17408  df-ovol 19318  df-vol 19319
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