MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovoliun2 Structured version   Unicode version

Theorem ovoliun2 20992
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 20991.) (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 20991 . 2  |-  ( ph  ->  ( vol* `  U_ n  e.  NN  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
6 nnuz 10899 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
7 1zzd 10680 . . . . . . . 8  |-  ( ph  ->  1  e.  ZZ )
8 fvex 5704 . . . . . . . . . . 11  |-  ( vol* `  [_ m  /  n ]_ A )  e. 
_V
9 nfcv 2582 . . . . . . . . . . . . . 14  |-  F/_ m
( vol* `  A )
10 nfcv 2582 . . . . . . . . . . . . . . 15  |-  F/_ n vol*
11 nfcsb1v 3307 . . . . . . . . . . . . . . 15  |-  F/_ n [_ m  /  n ]_ A
1210, 11nffv 5701 . . . . . . . . . . . . . 14  |-  F/_ n
( vol* `  [_ m  /  n ]_ A )
13 csbeq1a 3300 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
1413fveq2d 5698 . . . . . . . . . . . . . 14  |-  ( n  =  m  ->  ( vol* `  A )  =  ( vol* `  [_ m  /  n ]_ A ) )
159, 12, 14cbvmpt 4385 . . . . . . . . . . . . 13  |-  ( n  e.  NN  |->  ( vol* `  A )
)  =  ( m  e.  NN  |->  ( vol* `  [_ m  /  n ]_ A ) )
162, 15eqtri 2463 . . . . . . . . . . . 12  |-  G  =  ( m  e.  NN  |->  ( vol* `  [_ m  /  n ]_ A ) )
1716fvmpt2 5784 . . . . . . . . . . 11  |-  ( ( m  e.  NN  /\  ( vol* `  [_ m  /  n ]_ A )  e.  _V )  -> 
( G `  m
)  =  ( vol* `  [_ m  /  n ]_ A ) )
188, 17mpan2 671 . . . . . . . . . 10  |-  ( m  e.  NN  ->  ( G `  m )  =  ( vol* `  [_ m  /  n ]_ A ) )
1918adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  =  ( vol* `  [_ m  /  n ]_ A ) )
204ralrimiva 2802 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( vol* `  A
)  e.  RR )
219nfel1 2592 . . . . . . . . . . . 12  |-  F/ m
( vol* `  A )  e.  RR
2212nfel1 2592 . . . . . . . . . . . 12  |-  F/ n
( vol* `  [_ m  /  n ]_ A )  e.  RR
2314eleq1d 2509 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( vol* `  A )  e.  RR  <->  ( vol* `  [_ m  /  n ]_ A )  e.  RR ) )
2421, 22, 23cbvral 2946 . . . . . . . . . . 11  |-  ( A. n  e.  NN  ( vol* `  A )  e.  RR  <->  A. m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
2520, 24sylib 196 . . . . . . . . . 10  |-  ( ph  ->  A. m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
2625r19.21bi 2817 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
2719, 26eqeltrd 2517 . . . . . . . 8  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
286, 7, 27serfre 11838 . . . . . . 7  |-  ( ph  ->  seq 1 (  +  ,  G ) : NN --> RR )
291feq1i 5554 . . . . . . 7  |-  ( T : NN --> RR  <->  seq 1
(  +  ,  G
) : NN --> RR )
3028, 29sylibr 212 . . . . . 6  |-  ( ph  ->  T : NN --> RR )
31 frn 5568 . . . . . 6  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
3230, 31syl 16 . . . . 5  |-  ( ph  ->  ran  T  C_  RR )
33 1nn 10336 . . . . . . . 8  |-  1  e.  NN
34 fdm 5566 . . . . . . . . 9  |-  ( T : NN --> RR  ->  dom 
T  =  NN )
3530, 34syl 16 . . . . . . . 8  |-  ( ph  ->  dom  T  =  NN )
3633, 35syl5eleqr 2530 . . . . . . 7  |-  ( ph  ->  1  e.  dom  T
)
37 ne0i 3646 . . . . . . 7  |-  ( 1  e.  dom  T  ->  dom  T  =/=  (/) )
3836, 37syl 16 . . . . . 6  |-  ( ph  ->  dom  T  =/=  (/) )
39 dm0rn0 5059 . . . . . . 7  |-  ( dom 
T  =  (/)  <->  ran  T  =  (/) )
4039necon3bii 2643 . . . . . 6  |-  ( dom 
T  =/=  (/)  <->  ran  T  =/=  (/) )
4138, 40sylib 196 . . . . 5  |-  ( ph  ->  ran  T  =/=  (/) )
42 ovoliun2.t . . . . . . . . 9  |-  ( ph  ->  T  e.  dom  ~~>  )
431, 42syl5eqelr 2528 . . . . . . . 8  |-  ( ph  ->  seq 1 (  +  ,  G )  e. 
dom 
~~>  )
446, 7, 19, 26, 43isumrecl 13235 . . . . . . 7  |-  ( ph  -> 
sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
45 elfznn 11481 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... k )  ->  m  e.  NN )
4645adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  m  e.  NN )
4746, 18syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( G `  m )  =  ( vol* `  [_ m  /  n ]_ A ) )
48 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  NN )
4948, 6syl6eleq 2533 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  ( ZZ>= `  1 )
)
50 simpl 457 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  NN )  ->  ph )
5150, 45, 26syl2an 477 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
5251recnd 9415 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( vol* `  [_ m  /  n ]_ A )  e.  CC )
5347, 49, 52fsumser 13210 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol* `  [_ m  /  n ]_ A )  =  (  seq 1 (  +  ,  G ) `  k ) )
541fveq1i 5695 . . . . . . . . . 10  |-  ( T `
 k )  =  (  seq 1 (  +  ,  G ) `
 k )
5553, 54syl6eqr 2493 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol* `  [_ m  /  n ]_ A )  =  ( T `  k ) )
56 fzfid 11798 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... k
)  e.  Fin )
57 elfznn 11481 . . . . . . . . . . . . 13  |-  ( n  e.  ( 1 ... k )  ->  n  e.  NN )
5857ssriv 3363 . . . . . . . . . . . 12  |-  ( 1 ... k )  C_  NN
5958a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... k
)  C_  NN )
603ralrimiva 2802 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. n  e.  NN  A  C_  RR )
61 nfv 1673 . . . . . . . . . . . . . . 15  |-  F/ m  A  C_  RR
62 nfcv 2582 . . . . . . . . . . . . . . . 16  |-  F/_ n RR
6311, 62nfss 3352 . . . . . . . . . . . . . . 15  |-  F/ n [_ m  /  n ]_ A  C_  RR
6413sseq1d 3386 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  ( A  C_  RR  <->  [_ m  /  n ]_ A  C_  RR ) )
6561, 63, 64cbvral 2946 . . . . . . . . . . . . . 14  |-  ( A. n  e.  NN  A  C_  RR  <->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
6660, 65sylib 196 . . . . . . . . . . . . 13  |-  ( ph  ->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
6766r19.21bi 2817 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  [_ m  /  n ]_ A  C_  RR )
68 ovolge0 20967 . . . . . . . . . . . 12  |-  ( [_ m  /  n ]_ A  C_  RR  ->  0  <_  ( vol* `  [_ m  /  n ]_ A ) )
6967, 68syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  0  <_ 
( vol* `  [_ m  /  n ]_ A ) )
706, 7, 56, 59, 19, 26, 69, 43isumless 13311 . . . . . . . . . 10  |-  ( ph  -> 
sum_ m  e.  (
1 ... k ) ( vol* `  [_ m  /  n ]_ A )  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
7170adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol* `  [_ m  /  n ]_ A )  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
7255, 71eqbrtrrd 4317 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( T `
 k )  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
7372ralrimiva 2802 . . . . . . 7  |-  ( ph  ->  A. k  e.  NN  ( T `  k )  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
74 breq2 4299 . . . . . . . . 9  |-  ( x  =  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  ->  ( ( T `
 k )  <_  x 
<->  ( T `  k
)  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) ) )
7574ralbidv 2738 . . . . . . . 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 ) ) )
7675rspcev 3076 . . . . . . 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
)
7744, 73, 76syl2anc 661 . . . . . 6  |-  ( ph  ->  E. x  e.  RR  A. k  e.  NN  ( T `  k )  <_  x )
78 ffn 5562 . . . . . . . . 9  |-  ( T : NN --> RR  ->  T  Fn  NN )
7930, 78syl 16 . . . . . . . 8  |-  ( ph  ->  T  Fn  NN )
80 breq1 4298 . . . . . . . . 9  |-  ( z  =  ( T `  k )  ->  (
z  <_  x  <->  ( T `  k )  <_  x
) )
8180ralrn 5849 . . . . . . . 8  |-  ( T  Fn  NN  ->  ( A. z  e.  ran  T  z  <_  x  <->  A. k  e.  NN  ( T `  k )  <_  x
) )
8279, 81syl 16 . . . . . . 7  |-  ( ph  ->  ( A. z  e. 
ran  T  z  <_  x  <->  A. k  e.  NN  ( T `  k )  <_  x ) )
8382rexbidv 2739 . . . . . 6  |-  ( ph  ->  ( E. x  e.  RR  A. z  e. 
ran  T  z  <_  x  <->  E. x  e.  RR  A. k  e.  NN  ( T `  k )  <_  x ) )
8477, 83mpbird 232 . . . . 5  |-  ( ph  ->  E. x  e.  RR  A. z  e.  ran  T  z  <_  x )
85 supxrre 11293 . . . . 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 ,  <  ) )
8632, 41, 84, 85syl3anc 1218 . . . 4  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sup ( ran  T ,  RR ,  <  )
)
876, 1, 7, 19, 26, 69, 77isumsup 13313 . . . 4  |-  ( ph  -> 
sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  =  sup ( ran 
T ,  RR ,  <  ) )
8886, 87eqtr4d 2478 . . 3  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
899, 12, 14cbvsumi 13177 . . 3  |-  sum_ n  e.  NN  ( vol* `  A )  =  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )
9088, 89syl6eqr 2493 . 2  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sum_ n  e.  NN  ( vol* `  A
) )
915, 90breqtrd 4319 1  |-  ( ph  ->  ( vol* `  U_ n  e.  NN  A
)  <_  sum_ n  e.  NN  ( vol* `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2609   A.wral 2718   E.wrex 2719   _Vcvv 2975   [_csb 3291    C_ wss 3331   (/)c0 3640   U_ciun 4174   class class class wbr 4295    e. cmpt 4353   dom cdm 4843   ran crn 4844    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6094   supcsup 7693   RRcr 9284   0cc0 9285   1c1 9286    + caddc 9288   RR*cxr 9420    < clt 9421    <_ cle 9422   NNcn 10325   ZZ>=cuz 10864   ...cfz 11440    seqcseq 11809    ~~> cli 12965   sum_csu 13166   vol*covol 20949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cc 8607  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-pre-sup 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-oadd 6927  df-er 7104  df-map 7219  df-pm 7220  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-sup 7694  df-oi 7727  df-card 8112  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-q 10957  df-rp 10995  df-ioo 11307  df-ico 11309  df-fz 11441  df-fzo 11552  df-fl 11645  df-seq 11810  df-exp 11869  df-hash 12107  df-cj 12591  df-re 12592  df-im 12593  df-sqr 12727  df-abs 12728  df-clim 12969  df-rlim 12970  df-sum 13167  df-ovol 20951
This theorem is referenced by:  ovoliunnul  20993  vitalilem5  21095
  Copyright terms: Public domain W3C validator