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Theorem ovoliun2 18865
Description: The Lebesgue outer measure function is countably sub-additive. (This version is a little easier to read, but does not allow infinite values like ovoliun 18864.) (Contributed by Mario Carneiro, 12-Jun-2014.)
Hypotheses
Ref Expression
ovoliun.t  |-  T  =  seq  1 (  +  ,  G )
ovoliun.g  |-  G  =  ( n  e.  NN  |->  ( vol * `  A
) )
ovoliun.a  |-  ( (
ph  /\  n  e.  NN )  ->  A  C_  RR )
ovoliun.v  |-  ( (
ph  /\  n  e.  NN )  ->  ( vol
* `  A )  e.  RR )
ovoliun2.t  |-  ( ph  ->  T  e.  dom  ~~>  )
Assertion
Ref Expression
ovoliun2  |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A
)  <_  sum_ n  e.  NN  ( vol * `  A ) )
Distinct variable group:    ph, n
Allowed substitution hints:    A( n)    T( n)    G( n)

Proof of Theorem ovoliun2
Dummy variables  k  m  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovoliun.t . . 3  |-  T  =  seq  1 (  +  ,  G )
2 ovoliun.g . . 3  |-  G  =  ( n  e.  NN  |->  ( vol * `  A
) )
3 ovoliun.a . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  A  C_  RR )
4 ovoliun.v . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( vol
* `  A )  e.  RR )
51, 2, 3, 4ovoliun 18864 . 2  |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
6 nnuz 10263 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
7 1z 10053 . . . . . . . . 9  |-  1  e.  ZZ
87a1i 10 . . . . . . . 8  |-  ( ph  ->  1  e.  ZZ )
9 fvex 5539 . . . . . . . . . . 11  |-  ( vol
* `  [_ m  /  n ]_ A )  e. 
_V
10 nfcv 2419 . . . . . . . . . . . . . 14  |-  F/_ m
( vol * `  A )
11 nfcv 2419 . . . . . . . . . . . . . . 15  |-  F/_ n vol *
12 nfcsb1v 3113 . . . . . . . . . . . . . . 15  |-  F/_ n [_ m  /  n ]_ A
1311, 12nffv 5532 . . . . . . . . . . . . . 14  |-  F/_ n
( vol * `  [_ m  /  n ]_ A )
14 csbeq1a 3089 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
1514fveq2d 5529 . . . . . . . . . . . . . 14  |-  ( n  =  m  ->  ( vol * `  A )  =  ( vol * `  [_ m  /  n ]_ A ) )
1610, 13, 15cbvmpt 4110 . . . . . . . . . . . . 13  |-  ( n  e.  NN  |->  ( vol
* `  A )
)  =  ( m  e.  NN  |->  ( vol
* `  [_ m  /  n ]_ A ) )
172, 16eqtri 2303 . . . . . . . . . . . 12  |-  G  =  ( m  e.  NN  |->  ( vol * `  [_ m  /  n ]_ A ) )
1817fvmpt2 5608 . . . . . . . . . . 11  |-  ( ( m  e.  NN  /\  ( vol * `  [_ m  /  n ]_ A )  e.  _V )  -> 
( G `  m
)  =  ( vol
* `  [_ m  /  n ]_ A ) )
199, 18mpan2 652 . . . . . . . . . 10  |-  ( m  e.  NN  ->  ( G `  m )  =  ( vol * `  [_ m  /  n ]_ A ) )
2019adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  =  ( vol * `  [_ m  /  n ]_ A ) )
214ralrimiva 2626 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( vol * `  A
)  e.  RR )
2210nfel1 2429 . . . . . . . . . . . 12  |-  F/ m
( vol * `  A )  e.  RR
2313nfel1 2429 . . . . . . . . . . . 12  |-  F/ n
( vol * `  [_ m  /  n ]_ A )  e.  RR
2415eleq1d 2349 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( vol * `  A )  e.  RR  <->  ( vol * `  [_ m  /  n ]_ A )  e.  RR ) )
2522, 23, 24cbvral 2760 . . . . . . . . . . 11  |-  ( A. n  e.  NN  ( vol * `  A )  e.  RR  <->  A. m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
2621, 25sylib 188 . . . . . . . . . 10  |-  ( ph  ->  A. m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
2726r19.21bi 2641 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol
* `  [_ m  /  n ]_ A )  e.  RR )
2820, 27eqeltrd 2357 . . . . . . . 8  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
296, 8, 28serfre 11075 . . . . . . 7  |-  ( ph  ->  seq  1 (  +  ,  G ) : NN --> RR )
301feq1i 5383 . . . . . . 7  |-  ( T : NN --> RR  <->  seq  1
(  +  ,  G
) : NN --> RR )
3129, 30sylibr 203 . . . . . 6  |-  ( ph  ->  T : NN --> RR )
32 frn 5395 . . . . . 6  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
3331, 32syl 15 . . . . 5  |-  ( ph  ->  ran  T  C_  RR )
34 1nn 9757 . . . . . . . 8  |-  1  e.  NN
35 fdm 5393 . . . . . . . . 9  |-  ( T : NN --> RR  ->  dom 
T  =  NN )
3631, 35syl 15 . . . . . . . 8  |-  ( ph  ->  dom  T  =  NN )
3734, 36syl5eleqr 2370 . . . . . . 7  |-  ( ph  ->  1  e.  dom  T
)
38 ne0i 3461 . . . . . . 7  |-  ( 1  e.  dom  T  ->  dom  T  =/=  (/) )
3937, 38syl 15 . . . . . 6  |-  ( ph  ->  dom  T  =/=  (/) )
40 dm0rn0 4895 . . . . . . 7  |-  ( dom 
T  =  (/)  <->  ran  T  =  (/) )
4140necon3bii 2478 . . . . . 6  |-  ( dom 
T  =/=  (/)  <->  ran  T  =/=  (/) )
4239, 41sylib 188 . . . . 5  |-  ( ph  ->  ran  T  =/=  (/) )
43 ovoliun2.t . . . . . . . . 9  |-  ( ph  ->  T  e.  dom  ~~>  )
441, 43syl5eqelr 2368 . . . . . . . 8  |-  ( ph  ->  seq  1 (  +  ,  G )  e. 
dom 
~~>  )
456, 8, 20, 27, 44isumrecl 12228 . . . . . . 7  |-  ( ph  -> 
sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
46 elfznn 10819 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... k )  ->  m  e.  NN )
4746adantl 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  m  e.  NN )
4847, 19syl 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( G `  m )  =  ( vol * `  [_ m  /  n ]_ A ) )
49 simpr 447 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  NN )
5049, 6syl6eleq 2373 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  ( ZZ>= `  1 )
)
51 simpl 443 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  NN )  ->  ph )
5251, 46, 27syl2an 463 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
5352recnd 8861 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( vol * `  [_ m  /  n ]_ A )  e.  CC )
5448, 50, 53fsumser 12203 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol * `  [_ m  /  n ]_ A )  =  (  seq  1 (  +  ,  G ) `  k ) )
551fveq1i 5526 . . . . . . . . . 10  |-  ( T `
 k )  =  (  seq  1 (  +  ,  G ) `
 k )
5654, 55syl6eqr 2333 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol * `  [_ m  /  n ]_ A )  =  ( T `  k ) )
57 fzfid 11035 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... k
)  e.  Fin )
58 elfznn 10819 . . . . . . . . . . . . 13  |-  ( n  e.  ( 1 ... k )  ->  n  e.  NN )
5958ssriv 3184 . . . . . . . . . . . 12  |-  ( 1 ... k )  C_  NN
6059a1i 10 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... k
)  C_  NN )
613ralrimiva 2626 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. n  e.  NN  A  C_  RR )
62 nfv 1605 . . . . . . . . . . . . . . 15  |-  F/ m  A  C_  RR
63 nfcv 2419 . . . . . . . . . . . . . . . 16  |-  F/_ n RR
6412, 63nfss 3173 . . . . . . . . . . . . . . 15  |-  F/ n [_ m  /  n ]_ A  C_  RR
6514sseq1d 3205 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  ( A  C_  RR  <->  [_ m  /  n ]_ A  C_  RR ) )
6662, 64, 65cbvral 2760 . . . . . . . . . . . . . 14  |-  ( A. n  e.  NN  A  C_  RR  <->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
6761, 66sylib 188 . . . . . . . . . . . . 13  |-  ( ph  ->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
6867r19.21bi 2641 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  [_ m  /  n ]_ A  C_  RR )
69 ovolge0 18840 . . . . . . . . . . . 12  |-  ( [_ m  /  n ]_ A  C_  RR  ->  0  <_  ( vol * `  [_ m  /  n ]_ A ) )
7068, 69syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  0  <_ 
( vol * `  [_ m  /  n ]_ A ) )
716, 8, 57, 60, 20, 27, 70, 44isumless 12304 . . . . . . . . . 10  |-  ( ph  -> 
sum_ m  e.  (
1 ... k ) ( vol * `  [_ m  /  n ]_ A )  <_  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) )
7271adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol * `  [_ m  /  n ]_ A )  <_  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) )
7356, 72eqbrtrrd 4045 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( T `
 k )  <_  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) )
7473ralrimiva 2626 . . . . . . 7  |-  ( ph  ->  A. k  e.  NN  ( T `  k )  <_  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) )
75 breq2 4027 . . . . . . . . 9  |-  ( x  =  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  ->  ( ( T `
 k )  <_  x 
<->  ( T `  k
)  <_  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) ) )
7675ralbidv 2563 . . . . . . . 8  |-  ( x  =  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  ->  ( A. k  e.  NN  ( T `  k )  <_  x  <->  A. k  e.  NN  ( T `  k )  <_ 
sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) ) )
7776rspcev 2884 . . . . . . 7  |-  ( (
sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  e.  RR  /\  A. k  e.  NN  ( T `  k )  <_ 
sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) )  ->  E. x  e.  RR  A. k  e.  NN  ( T `  k )  <_  x
)
7845, 74, 77syl2anc 642 . . . . . 6  |-  ( ph  ->  E. x  e.  RR  A. k  e.  NN  ( T `  k )  <_  x )
79 ffn 5389 . . . . . . . . 9  |-  ( T : NN --> RR  ->  T  Fn  NN )
8031, 79syl 15 . . . . . . . 8  |-  ( ph  ->  T  Fn  NN )
81 breq1 4026 . . . . . . . . 9  |-  ( z  =  ( T `  k )  ->  (
z  <_  x  <->  ( T `  k )  <_  x
) )
8281ralrn 5668 . . . . . . . 8  |-  ( T  Fn  NN  ->  ( A. z  e.  ran  T  z  <_  x  <->  A. k  e.  NN  ( T `  k )  <_  x
) )
8380, 82syl 15 . . . . . . 7  |-  ( ph  ->  ( A. z  e. 
ran  T  z  <_  x  <->  A. k  e.  NN  ( T `  k )  <_  x ) )
8483rexbidv 2564 . . . . . 6  |-  ( ph  ->  ( E. x  e.  RR  A. z  e. 
ran  T  z  <_  x  <->  E. x  e.  RR  A. k  e.  NN  ( T `  k )  <_  x ) )
8578, 84mpbird 223 . . . . 5  |-  ( ph  ->  E. x  e.  RR  A. z  e.  ran  T  z  <_  x )
86 supxrre 10646 . . . . 5  |-  ( ( ran  T  C_  RR  /\ 
ran  T  =/=  (/)  /\  E. x  e.  RR  A. z  e.  ran  T  z  <_  x )  ->  sup ( ran  T ,  RR* ,  <  )  =  sup ( ran  T ,  RR ,  <  ) )
8733, 42, 85, 86syl3anc 1182 . . . 4  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sup ( ran  T ,  RR ,  <  )
)
886, 1, 8, 20, 27, 70, 78isumsup 12306 . . . 4  |-  ( ph  -> 
sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  =  sup ( ran 
T ,  RR ,  <  ) )
8987, 88eqtr4d 2318 . . 3  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) )
9010, 13, 15cbvsumi 12170 . . 3  |-  sum_ n  e.  NN  ( vol * `  A )  =  sum_ m  e.  NN  ( vol
* `  [_ m  /  n ]_ A )
9189, 90syl6eqr 2333 . 2  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sum_ n  e.  NN  ( vol * `  A
) )
925, 91breqtrd 4047 1  |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A
)  <_  sum_ n  e.  NN  ( vol * `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788   [_csb 3081    C_ wss 3152   (/)c0 3455   U_ciun 3905   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740   RR*cxr 8866    < clt 8867    <_ cle 8868   NNcn 9746   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046    ~~> cli 11958   sum_csu 12158   vol *covol 18822
This theorem is referenced by:  ovoliunnul  18866  vitalilem5  18967
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-cc 8061  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-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  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-sup 7194  df-oi 7225  df-card 7572  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-ioo 10660  df-ico 10662  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-ovol 18824
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