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Theorem uniioombllem3a 22162
Description: Lemma for uniioombl 22167. (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 11625 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
3 uniioombl.g . . . . . . 7  |-  ( ph  ->  G : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
4 inss2 3705 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 rexpssxrxp 9627 . . . . . . . 8  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
64, 5sstri 3498 . . . . . . 7  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
7 fss 5721 . . . . . . 7  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* ) )  ->  G : NN --> ( RR*  X. 
RR* ) )
83, 6, 7sylancl 660 . . . . . 6  |-  ( ph  ->  G : NN --> ( RR*  X. 
RR* ) )
9 fco 5723 . . . . . 6  |-  ( ( (,) : ( RR*  X. 
RR* ) --> ~P RR  /\  G : NN --> ( RR*  X. 
RR* ) )  -> 
( (,)  o.  G
) : NN --> ~P RR )
102, 8, 9sylancr 661 . . . . 5  |-  ( ph  ->  ( (,)  o.  G
) : NN --> ~P RR )
11 ffun 5715 . . . . 5  |-  ( ( (,)  o.  G ) : NN --> ~P RR  ->  Fun  ( (,)  o.  G ) )
12 funiunfv 6135 . . . . 5  |-  ( Fun  ( (,)  o.  G
)  ->  U_ j  e.  ( 1 ... M
) ( ( (,) 
o.  G ) `  j )  =  U. ( ( (,)  o.  G ) " (
1 ... M ) ) )
1310, 11, 123syl 20 . . . 4  |-  ( ph  ->  U_ j  e.  ( 1 ... M ) ( ( (,)  o.  G ) `  j
)  =  U. (
( (,)  o.  G
) " ( 1 ... M ) ) )
14 elfznn 11717 . . . . . 6  |-  ( j  e.  ( 1 ... M )  ->  j  e.  NN )
15 fvco3 5925 . . . . . 6  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  j  e.  NN )  ->  (
( (,)  o.  G
) `  j )  =  ( (,) `  ( G `  j )
) )
163, 14, 15syl2an 475 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  (
( (,)  o.  G
) `  j )  =  ( (,) `  ( G `  j )
) )
1716iuneq2dv 4337 . . . 4  |-  ( ph  ->  U_ j  e.  ( 1 ... M ) ( ( (,)  o.  G ) `  j
)  =  U_ j  e.  ( 1 ... M
) ( (,) `  ( G `  j )
) )
1813, 17eqtr3d 2497 . . 3  |-  ( ph  ->  U. ( ( (,) 
o.  G ) "
( 1 ... M
) )  =  U_ j  e.  ( 1 ... M ) ( (,) `  ( G `
 j ) ) )
191, 18syl5eq 2507 . 2  |-  ( ph  ->  K  =  U_ j  e.  ( 1 ... M
) ( (,) `  ( G `  j )
) )
20 ffvelrn 6005 . . . . . . . . . . . 12  |-  ( ( G : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  j  e.  NN )  ->  ( G `  j )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
213, 14, 20syl2an 475 . . . . . . . . . . 11  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( G `  j )  e.  (  <_  i^i  ( RR  X.  RR ) ) )
224, 21sseldi 3487 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( G `  j )  e.  ( RR  X.  RR ) )
23 1st2nd2 6810 . . . . . . . . . 10  |-  ( ( G `  j )  e.  ( RR  X.  RR )  ->  ( G `
 j )  = 
<. ( 1st `  ( G `  j )
) ,  ( 2nd `  ( G `  j
) ) >. )
2422, 23syl 16 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( G `  j )  =  <. ( 1st `  ( G `  j )
) ,  ( 2nd `  ( G `  j
) ) >. )
2524fveq2d 5852 . . . . . . . 8  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( (,) `  ( G `  j ) )  =  ( (,) `  <. ( 1st `  ( G `
 j ) ) ,  ( 2nd `  ( G `  j )
) >. ) )
26 df-ov 6273 . . . . . . . 8  |-  ( ( 1st `  ( G `
 j ) ) (,) ( 2nd `  ( G `  j )
) )  =  ( (,) `  <. ( 1st `  ( G `  j ) ) ,  ( 2nd `  ( G `  j )
) >. )
2725, 26syl6eqr 2513 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( (,) `  ( G `  j ) )  =  ( ( 1st `  ( G `  j )
) (,) ( 2nd `  ( G `  j
) ) ) )
28 ioossre 11589 . . . . . . 7  |-  ( ( 1st `  ( G `
 j ) ) (,) ( 2nd `  ( G `  j )
) )  C_  RR
2927, 28syl6eqss 3539 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( (,) `  ( G `  j ) )  C_  RR )
3029ralrimiva 2868 . . . . 5  |-  ( ph  ->  A. j  e.  ( 1 ... M ) ( (,) `  ( G `  j )
)  C_  RR )
31 iunss 4356 . . . . 5  |-  ( U_ j  e.  ( 1 ... M ) ( (,) `  ( G `
 j ) ) 
C_  RR  <->  A. j  e.  ( 1 ... M ) ( (,) `  ( G `  j )
)  C_  RR )
3230, 31sylibr 212 . . . 4  |-  ( ph  ->  U_ j  e.  ( 1 ... M ) ( (,) `  ( G `  j )
)  C_  RR )
3319, 32eqsstrd 3523 . . 3  |-  ( ph  ->  K  C_  RR )
34 fzfid 12068 . . . 4  |-  ( ph  ->  ( 1 ... M
)  e.  Fin )
3527fveq2d 5852 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( vol* `  ( (,) `  ( G `  j
) ) )  =  ( vol* `  ( ( 1st `  ( G `  j )
) (,) ( 2nd `  ( G `  j
) ) ) ) )
36 ovolfcl 22047 . . . . . . . 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 )
) ) )
373, 14, 36syl2an 475 . . . . . . 7  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  (
( 1st `  ( G `  j )
)  e.  RR  /\  ( 2nd `  ( G `
 j ) )  e.  RR  /\  ( 1st `  ( G `  j ) )  <_ 
( 2nd `  ( G `  j )
) ) )
38 ovolioo 22147 . . . . . . 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
) ) ) )
3937, 38syl 16 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( vol* `  ( ( 1st `  ( G `
 j ) ) (,) ( 2nd `  ( G `  j )
) ) )  =  ( ( 2nd `  ( G `  j )
)  -  ( 1st `  ( G `  j
) ) ) )
4035, 39eqtrd 2495 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( vol* `  ( (,) `  ( G `  j
) ) )  =  ( ( 2nd `  ( G `  j )
)  -  ( 1st `  ( G `  j
) ) ) )
4137simp2d 1007 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( 2nd `  ( G `  j ) )  e.  RR )
4237simp1d 1006 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( 1st `  ( G `  j ) )  e.  RR )
4341, 42resubcld 9983 . . . . 5  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  (
( 2nd `  ( G `  j )
)  -  ( 1st `  ( G `  j
) ) )  e.  RR )
4440, 43eqeltrd 2542 . . . 4  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  ( vol* `  ( (,) `  ( G `  j
) ) )  e.  RR )
4534, 44fsumrecl 13641 . . 3  |-  ( ph  -> 
sum_ j  e.  ( 1 ... M ) ( vol* `  ( (,) `  ( G `
 j ) ) )  e.  RR )
4619fveq2d 5852 . . . 4  |-  ( ph  ->  ( vol* `  K )  =  ( vol* `  U_ j  e.  ( 1 ... M
) ( (,) `  ( G `  j )
) ) )
4729, 44jca 530 . . . . . 6  |-  ( (
ph  /\  j  e.  ( 1 ... M
) )  ->  (
( (,) `  ( G `  j )
)  C_  RR  /\  ( vol* `  ( (,) `  ( G `  j
) ) )  e.  RR ) )
4847ralrimiva 2868 . . . . 5  |-  ( ph  ->  A. j  e.  ( 1 ... M ) ( ( (,) `  ( G `  j )
)  C_  RR  /\  ( vol* `  ( (,) `  ( G `  j
) ) )  e.  RR ) )
49 ovolfiniun 22081 . . . . 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 )
) ) )
5034, 48, 49syl2anc 659 . . . 4  |-  ( ph  ->  ( vol* `  U_ j  e.  ( 1 ... M ) ( (,) `  ( G `
 j ) ) )  <_  sum_ j  e.  ( 1 ... M
) ( vol* `  ( (,) `  ( G `  j )
) ) )
5146, 50eqbrtrd 4459 . . 3  |-  ( ph  ->  ( vol* `  K )  <_  sum_ j  e.  ( 1 ... M
) ( vol* `  ( (,) `  ( G `  j )
) ) )
52 ovollecl 22063 . . 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 )
5333, 45, 51, 52syl3anc 1226 . 2  |-  ( ph  ->  ( vol* `  K )  e.  RR )
5419, 53jca 530 1  |-  ( ph  ->  ( K  =  U_ j  e.  ( 1 ... M ) ( (,) `  ( G `
 j ) )  /\  ( vol* `  K )  e.  RR ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804    i^i cin 3460    C_ wss 3461   ~Pcpw 3999   <.cop 4022   U.cuni 4235   U_ciun 4315  Disj wdisj 4410   class class class wbr 4439    X. cxp 4986   ran crn 4989   "cima 4991    o. ccom 4992   Fun wfun 5564   -->wf 5566   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   Fincfn 7509   supcsup 7892   RRcr 9480   1c1 9482    + caddc 9484   RR*cxr 9616    < clt 9617    <_ cle 9618    - cmin 9796   NNcn 10531   RR+crp 11221   (,)cioo 11532   ...cfz 11675    seqcseq 12092   abscabs 13152   sum_csu 13593   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-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  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-fal 1404  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-int 4272  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-se 4828  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-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  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-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12093  df-exp 12152  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-clim 13396  df-rlim 13397  df-sum 13594  df-rest 14915  df-topgen 14936  df-psmet 18609  df-xmet 18610  df-met 18611  df-bl 18612  df-mopn 18613  df-top 19569  df-bases 19571  df-topon 19572  df-cmp 20057  df-ovol 22045  df-vol 22046
This theorem is referenced by:  uniioombllem3  22163
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