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Theorem ovoliun2 21652
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 21651.) (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 21651 . 2  |-  ( ph  ->  ( vol* `  U_ n  e.  NN  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
6 nnuz 11113 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
7 1zzd 10891 . . . . . . . 8  |-  ( ph  ->  1  e.  ZZ )
8 fvex 5874 . . . . . . . . . . 11  |-  ( vol* `  [_ m  /  n ]_ A )  e. 
_V
9 nfcv 2629 . . . . . . . . . . . . . 14  |-  F/_ m
( vol* `  A )
10 nfcv 2629 . . . . . . . . . . . . . . 15  |-  F/_ n vol*
11 nfcsb1v 3451 . . . . . . . . . . . . . . 15  |-  F/_ n [_ m  /  n ]_ A
1210, 11nffv 5871 . . . . . . . . . . . . . 14  |-  F/_ n
( vol* `  [_ m  /  n ]_ A )
13 csbeq1a 3444 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
1413fveq2d 5868 . . . . . . . . . . . . . 14  |-  ( n  =  m  ->  ( vol* `  A )  =  ( vol* `  [_ m  /  n ]_ A ) )
159, 12, 14cbvmpt 4537 . . . . . . . . . . . . 13  |-  ( n  e.  NN  |->  ( vol* `  A )
)  =  ( m  e.  NN  |->  ( vol* `  [_ m  /  n ]_ A ) )
162, 15eqtri 2496 . . . . . . . . . . . 12  |-  G  =  ( m  e.  NN  |->  ( vol* `  [_ m  /  n ]_ A ) )
1716fvmpt2 5955 . . . . . . . . . . 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 2878 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( vol* `  A
)  e.  RR )
219nfel1 2645 . . . . . . . . . . . 12  |-  F/ m
( vol* `  A )  e.  RR
2212nfel1 2645 . . . . . . . . . . . 12  |-  F/ n
( vol* `  [_ m  /  n ]_ A )  e.  RR
2314eleq1d 2536 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( vol* `  A )  e.  RR  <->  ( vol* `  [_ m  /  n ]_ A )  e.  RR ) )
2421, 22, 23cbvral 3084 . . . . . . . . . . 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 2833 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
2719, 26eqeltrd 2555 . . . . . . . 8  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
286, 7, 27serfre 12100 . . . . . . 7  |-  ( ph  ->  seq 1 (  +  ,  G ) : NN --> RR )
291feq1i 5721 . . . . . . 7  |-  ( T : NN --> RR  <->  seq 1
(  +  ,  G
) : NN --> RR )
3028, 29sylibr 212 . . . . . 6  |-  ( ph  ->  T : NN --> RR )
31 frn 5735 . . . . . 6  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
3230, 31syl 16 . . . . 5  |-  ( ph  ->  ran  T  C_  RR )
33 1nn 10543 . . . . . . . 8  |-  1  e.  NN
34 fdm 5733 . . . . . . . . 9  |-  ( T : NN --> RR  ->  dom 
T  =  NN )
3530, 34syl 16 . . . . . . . 8  |-  ( ph  ->  dom  T  =  NN )
3633, 35syl5eleqr 2562 . . . . . . 7  |-  ( ph  ->  1  e.  dom  T
)
37 ne0i 3791 . . . . . . 7  |-  ( 1  e.  dom  T  ->  dom  T  =/=  (/) )
3836, 37syl 16 . . . . . 6  |-  ( ph  ->  dom  T  =/=  (/) )
39 dm0rn0 5217 . . . . . . 7  |-  ( dom 
T  =  (/)  <->  ran  T  =  (/) )
4039necon3bii 2735 . . . . . 6  |-  ( dom 
T  =/=  (/)  <->  ran  T  =/=  (/) )
4138, 40sylib 196 . . . . 5  |-  ( ph  ->  ran  T  =/=  (/) )
42 ovoliun2.t . . . . . . . . 9  |-  ( ph  ->  T  e.  dom  ~~>  )
431, 42syl5eqelr 2560 . . . . . . . 8  |-  ( ph  ->  seq 1 (  +  ,  G )  e. 
dom 
~~>  )
446, 7, 19, 26, 43isumrecl 13539 . . . . . . 7  |-  ( ph  -> 
sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
45 elfznn 11710 . . . . . . . . . . . . 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 2565 . . . . . . . . . . 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 9618 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( vol* `  [_ m  /  n ]_ A )  e.  CC )
5347, 49, 52fsumser 13511 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol* `  [_ m  /  n ]_ A )  =  (  seq 1 (  +  ,  G ) `  k ) )
541fveq1i 5865 . . . . . . . . . 10  |-  ( T `
 k )  =  (  seq 1 (  +  ,  G ) `
 k )
5553, 54syl6eqr 2526 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol* `  [_ m  /  n ]_ A )  =  ( T `  k ) )
56 fzfid 12047 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... k
)  e.  Fin )
57 elfznn 11710 . . . . . . . . . . . . 13  |-  ( n  e.  ( 1 ... k )  ->  n  e.  NN )
5857ssriv 3508 . . . . . . . . . . . 12  |-  ( 1 ... k )  C_  NN
5958a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... k
)  C_  NN )
603ralrimiva 2878 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. n  e.  NN  A  C_  RR )
61 nfv 1683 . . . . . . . . . . . . . . 15  |-  F/ m  A  C_  RR
62 nfcv 2629 . . . . . . . . . . . . . . . 16  |-  F/_ n RR
6311, 62nfss 3497 . . . . . . . . . . . . . . 15  |-  F/ n [_ m  /  n ]_ A  C_  RR
6413sseq1d 3531 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  ( A  C_  RR  <->  [_ m  /  n ]_ A  C_  RR ) )
6561, 63, 64cbvral 3084 . . . . . . . . . . . . . 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 2833 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  [_ m  /  n ]_ A  C_  RR )
68 ovolge0 21627 . . . . . . . . . . . 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 13616 . . . . . . . . . 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 4469 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( T `
 k )  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
7372ralrimiva 2878 . . . . . . 7  |-  ( ph  ->  A. k  e.  NN  ( T `  k )  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
74 breq2 4451 . . . . . . . . 9  |-  ( x  =  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  ->  ( ( T `
 k )  <_  x 
<->  ( T `  k
)  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) ) )
7574ralbidv 2903 . . . . . . . 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 3214 . . . . . . 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 5729 . . . . . . . . 9  |-  ( T : NN --> RR  ->  T  Fn  NN )
7930, 78syl 16 . . . . . . . 8  |-  ( ph  ->  T  Fn  NN )
80 breq1 4450 . . . . . . . . 9  |-  ( z  =  ( T `  k )  ->  (
z  <_  x  <->  ( T `  k )  <_  x
) )
8180ralrn 6022 . . . . . . . 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 2973 . . . . . 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 11515 . . . . 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 1228 . . . 4  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sup ( ran  T ,  RR ,  <  )
)
876, 1, 7, 19, 26, 69, 77isumsup 13618 . . . 4  |-  ( ph  -> 
sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  =  sup ( ran 
T ,  RR ,  <  ) )
8886, 87eqtr4d 2511 . . 3  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
899, 12, 14cbvsumi 13478 . . 3  |-  sum_ n  e.  NN  ( vol* `  A )  =  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )
9088, 89syl6eqr 2526 . 2  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sum_ n  e.  NN  ( vol* `  A
) )
915, 90breqtrd 4471 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113   [_csb 3435    C_ wss 3476   (/)c0 3785   U_ciun 4325   class class class wbr 4447    |-> cmpt 4505   dom cdm 4999   ran crn 5000    Fn wfn 5581   -->wf 5582   ` cfv 5586  (class class class)co 6282   supcsup 7896   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491   RR*cxr 9623    < clt 9624    <_ cle 9625   NNcn 10532   ZZ>=cuz 11078   ...cfz 11668    seqcseq 12071    ~~> cli 13266   sum_csu 13467   vol*covol 21609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cc 8811  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179  df-rp 11217  df-ioo 11529  df-ico 11531  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-rlim 13271  df-sum 13468  df-ovol 21611
This theorem is referenced by:  ovoliunnul  21653  vitalilem5  21756
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