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Theorem uniioombllem3a 19429
Description: Lemma for uniioombl 19434. (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 10958 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
3 uniioombl.g . . . . . . 7  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
4 inss2 3522 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 ressxr 9085 . . . . . . . . 9  |-  RR  C_  RR*
6 xpss12 4940 . . . . . . . . 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 3317 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
9 fss 5558 . . . . . . 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 5559 . . . . . 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 5552 . . . . 5  |-  ( ( (,)  o.  G ) : NN --> ~P RR  ->  Fun  ( (,)  o.  G ) )
14 funiunfv 5954 . . . . 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 11036 . . . . . 6  |-  ( j  e.  ( 1 ... M )  ->  j  e.  NN )
17 fvco3 5759 . . . . . 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 4074 . . . 4  |-  ( ph  ->  U_ j  e.  ( 1 ... M ) ( ( (,)  o.  G ) `  j
)  =  U_ j  e.  ( 1 ... M
) ( (,) `  ( G `  j )
) )
2015, 19eqtr3d 2438 . . 3  |-  ( ph  ->  U. ( ( (,) 
o.  G ) "
( 1 ... M
) )  =  U_ j  e.  ( 1 ... M ) ( (,) `  ( G `
 j ) ) )
211, 20syl5eq 2448 . 2  |-  ( ph  ->  K  =  U_ j  e.  ( 1 ... M
) ( (,) `  ( G `  j )
) )
22 ffvelrn 5827 . . . . . . . . . . . 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 3306 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( G `  j )  e.  ( RR  X.  RR ) )
25 1st2nd2 6345 . . . . . . . . . 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 5691 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( (,) `  ( G `  j ) )  =  ( (,) `  <. ( 1st `  ( G `
 j ) ) ,  ( 2nd `  ( G `  j )
) >. ) )
28 df-ov 6043 . . . . . . . 8  |-  ( ( 1st `  ( G `
 j ) ) (,) ( 2nd `  ( G `  j )
) )  =  ( (,) `  <. ( 1st `  ( G `  j ) ) ,  ( 2nd `  ( G `  j )
) >. )
2927, 28syl6eqr 2454 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( (,) `  ( G `  j ) )  =  ( ( 1st `  ( G `  j )
) (,) ( 2nd `  ( G `  j
) ) ) )
30 ioossre 10928 . . . . . . 7  |-  ( ( 1st `  ( G `
 j ) ) (,) ( 2nd `  ( G `  j )
) )  C_  RR
3129, 30syl6eqss 3358 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( (,) `  ( G `  j ) )  C_  RR )
3231ralrimiva 2749 . . . . 5  |-  ( ph  ->  A. j  e.  ( 1 ... M ) ( (,) `  ( G `  j )
)  C_  RR )
33 iunss 4092 . . . . 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 3342 . . 3  |-  ( ph  ->  K  C_  RR )
36 fzfid 11267 . . . 4  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
3729fveq2d 5691 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( vol * `  ( (,) `  ( G `  j
) ) )  =  ( vol * `  ( ( 1st `  ( G `  j )
) (,) ( 2nd `  ( G `  j
) ) ) ) )
38 ovolfcl 19316 . . . . . . . 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 19415 . . . . . . 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 2436 . . . . 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 9421 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  (
( 2nd `  ( G `  j )
)  -  ( 1st `  ( G `  j
) ) )  e.  RR )
4642, 45eqeltrd 2478 . . . 4  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( vol * `  ( (,) `  ( G `  j
) ) )  e.  RR )
4736, 46fsumrecl 12483 . . 3  |-  ( ph  -> 
sum_ j  e.  ( 1 ... M ) ( vol * `  ( (,) `  ( G `
 j ) ) )  e.  RR )
4821fveq2d 5691 . . . 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 2749 . . . . 5  |-  ( ph  ->  A. j  e.  ( 1 ... M ) ( ( (,) `  ( G `  j )
)  C_  RR  /\  ( vol * `  ( (,) `  ( G `  j
) ) )  e.  RR ) )
51 ovolfiniun 19350 . . . . 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 4192 . . 3  |-  ( ph  ->  ( vol * `  K )  <_  sum_ j  e.  ( 1 ... M
) ( vol * `  ( (,) `  ( G `  j )
) ) )
54 ovollecl 19332 . . 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 2666    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   <.cop 3777   U.cuni 3975   U_ciun 4053  Disj wdisj 4142   class class class wbr 4172    X. cxp 4835   ran crn 4838   "cima 4840    o. ccom 4841   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   Fincfn 7068   supcsup 7403   RRcr 8945   1c1 8947    + caddc 8949   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247   NNcn 9956   RR+crp 10568   (,)cioo 10872   ...cfz 10999    seq cseq 11278   abscabs 11994   sum_csu 12434   vol
*covol 19312
This theorem is referenced by:  uniioombllem3  19430
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-rest 13605  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-top 16918  df-bases 16920  df-topon 16921  df-cmp 17404  df-ovol 19314  df-vol 19315
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