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Theorem ovoliun2 22537
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 22536.) (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 22536 . 2  |-  ( ph  ->  ( vol* `  U_ n  e.  NN  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
6 nnuz 11218 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
7 1zzd 10992 . . . . . . . 8  |-  ( ph  ->  1  e.  ZZ )
8 fvex 5889 . . . . . . . . . . 11  |-  ( vol* `  [_ m  /  n ]_ A )  e. 
_V
9 nfcv 2612 . . . . . . . . . . . . . 14  |-  F/_ m
( vol* `  A )
10 nfcv 2612 . . . . . . . . . . . . . . 15  |-  F/_ n vol*
11 nfcsb1v 3365 . . . . . . . . . . . . . . 15  |-  F/_ n [_ m  /  n ]_ A
1210, 11nffv 5886 . . . . . . . . . . . . . 14  |-  F/_ n
( vol* `  [_ m  /  n ]_ A )
13 csbeq1a 3358 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
1413fveq2d 5883 . . . . . . . . . . . . . 14  |-  ( n  =  m  ->  ( vol* `  A )  =  ( vol* `  [_ m  /  n ]_ A ) )
159, 12, 14cbvmpt 4487 . . . . . . . . . . . . 13  |-  ( n  e.  NN  |->  ( vol* `  A )
)  =  ( m  e.  NN  |->  ( vol* `  [_ m  /  n ]_ A ) )
162, 15eqtri 2493 . . . . . . . . . . . 12  |-  G  =  ( m  e.  NN  |->  ( vol* `  [_ m  /  n ]_ A ) )
1716fvmpt2 5972 . . . . . . . . . . 11  |-  ( ( m  e.  NN  /\  ( vol* `  [_ m  /  n ]_ A )  e.  _V )  -> 
( G `  m
)  =  ( vol* `  [_ m  /  n ]_ A ) )
188, 17mpan2 685 . . . . . . . . . 10  |-  ( m  e.  NN  ->  ( G `  m )  =  ( vol* `  [_ m  /  n ]_ A ) )
1918adantl 473 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  =  ( vol* `  [_ m  /  n ]_ A ) )
204ralrimiva 2809 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( vol* `  A
)  e.  RR )
219nfel1 2626 . . . . . . . . . . . 12  |-  F/ m
( vol* `  A )  e.  RR
2212nfel1 2626 . . . . . . . . . . . 12  |-  F/ n
( vol* `  [_ m  /  n ]_ A )  e.  RR
2314eleq1d 2533 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( vol* `  A )  e.  RR  <->  ( vol* `  [_ m  /  n ]_ A )  e.  RR ) )
2421, 22, 23cbvral 3001 . . . . . . . . . . 11  |-  ( A. n  e.  NN  ( vol* `  A )  e.  RR  <->  A. m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
2520, 24sylib 201 . . . . . . . . . 10  |-  ( ph  ->  A. m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
2625r19.21bi 2776 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
2719, 26eqeltrd 2549 . . . . . . . 8  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
286, 7, 27serfre 12280 . . . . . . 7  |-  ( ph  ->  seq 1 (  +  ,  G ) : NN --> RR )
291feq1i 5730 . . . . . . 7  |-  ( T : NN --> RR  <->  seq 1
(  +  ,  G
) : NN --> RR )
3028, 29sylibr 217 . . . . . 6  |-  ( ph  ->  T : NN --> RR )
31 frn 5747 . . . . . 6  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
3230, 31syl 17 . . . . 5  |-  ( ph  ->  ran  T  C_  RR )
33 1nn 10642 . . . . . . . 8  |-  1  e.  NN
34 fdm 5745 . . . . . . . . 9  |-  ( T : NN --> RR  ->  dom 
T  =  NN )
3530, 34syl 17 . . . . . . . 8  |-  ( ph  ->  dom  T  =  NN )
3633, 35syl5eleqr 2556 . . . . . . 7  |-  ( ph  ->  1  e.  dom  T
)
37 ne0i 3728 . . . . . . 7  |-  ( 1  e.  dom  T  ->  dom  T  =/=  (/) )
3836, 37syl 17 . . . . . 6  |-  ( ph  ->  dom  T  =/=  (/) )
39 dm0rn0 5057 . . . . . . 7  |-  ( dom 
T  =  (/)  <->  ran  T  =  (/) )
4039necon3bii 2695 . . . . . 6  |-  ( dom 
T  =/=  (/)  <->  ran  T  =/=  (/) )
4138, 40sylib 201 . . . . 5  |-  ( ph  ->  ran  T  =/=  (/) )
42 ovoliun2.t . . . . . . . . 9  |-  ( ph  ->  T  e.  dom  ~~>  )
431, 42syl5eqelr 2554 . . . . . . . 8  |-  ( ph  ->  seq 1 (  +  ,  G )  e. 
dom 
~~>  )
446, 7, 19, 26, 43isumrecl 13903 . . . . . . 7  |-  ( ph  -> 
sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
45 elfznn 11854 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... k )  ->  m  e.  NN )
4645adantl 473 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  m  e.  NN )
4746, 18syl 17 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( G `  m )  =  ( vol* `  [_ m  /  n ]_ A ) )
48 simpr 468 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  NN )
4948, 6syl6eleq 2559 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  ( ZZ>= `  1 )
)
50 simpl 464 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  NN )  ->  ph )
5150, 45, 26syl2an 485 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
5251recnd 9687 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( vol* `  [_ m  /  n ]_ A )  e.  CC )
5347, 49, 52fsumser 13873 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol* `  [_ m  /  n ]_ A )  =  (  seq 1 (  +  ,  G ) `  k ) )
541fveq1i 5880 . . . . . . . . . 10  |-  ( T `
 k )  =  (  seq 1 (  +  ,  G ) `
 k )
5553, 54syl6eqr 2523 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol* `  [_ m  /  n ]_ A )  =  ( T `  k ) )
56 fzfid 12224 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... k
)  e.  Fin )
57 elfznn 11854 . . . . . . . . . . . . 13  |-  ( n  e.  ( 1 ... k )  ->  n  e.  NN )
5857ssriv 3422 . . . . . . . . . . . 12  |-  ( 1 ... k )  C_  NN
5958a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... k
)  C_  NN )
603ralrimiva 2809 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. n  e.  NN  A  C_  RR )
61 nfv 1769 . . . . . . . . . . . . . . 15  |-  F/ m  A  C_  RR
62 nfcv 2612 . . . . . . . . . . . . . . . 16  |-  F/_ n RR
6311, 62nfss 3411 . . . . . . . . . . . . . . 15  |-  F/ n [_ m  /  n ]_ A  C_  RR
6413sseq1d 3445 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  ( A  C_  RR  <->  [_ m  /  n ]_ A  C_  RR ) )
6561, 63, 64cbvral 3001 . . . . . . . . . . . . . 14  |-  ( A. n  e.  NN  A  C_  RR  <->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
6660, 65sylib 201 . . . . . . . . . . . . 13  |-  ( ph  ->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
6766r19.21bi 2776 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  [_ m  /  n ]_ A  C_  RR )
68 ovolge0 22512 . . . . . . . . . . . 12  |-  ( [_ m  /  n ]_ A  C_  RR  ->  0  <_  ( vol* `  [_ m  /  n ]_ A ) )
6967, 68syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  0  <_ 
( vol* `  [_ m  /  n ]_ A ) )
706, 7, 56, 59, 19, 26, 69, 43isumless 13980 . . . . . . . . . 10  |-  ( ph  -> 
sum_ m  e.  (
1 ... k ) ( vol* `  [_ m  /  n ]_ A )  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
7170adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol* `  [_ m  /  n ]_ A )  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
7255, 71eqbrtrrd 4418 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( T `
 k )  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
7372ralrimiva 2809 . . . . . . 7  |-  ( ph  ->  A. k  e.  NN  ( T `  k )  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
74 breq2 4399 . . . . . . . . 9  |-  ( x  =  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  ->  ( ( T `
 k )  <_  x 
<->  ( T `  k
)  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) ) )
7574ralbidv 2829 . . . . . . . 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 3136 . . . . . . 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 673 . . . . . 6  |-  ( ph  ->  E. x  e.  RR  A. k  e.  NN  ( T `  k )  <_  x )
78 ffn 5739 . . . . . . . . 9  |-  ( T : NN --> RR  ->  T  Fn  NN )
7930, 78syl 17 . . . . . . . 8  |-  ( ph  ->  T  Fn  NN )
80 breq1 4398 . . . . . . . . 9  |-  ( z  =  ( T `  k )  ->  (
z  <_  x  <->  ( T `  k )  <_  x
) )
8180ralrn 6040 . . . . . . . 8  |-  ( T  Fn  NN  ->  ( A. z  e.  ran  T  z  <_  x  <->  A. k  e.  NN  ( T `  k )  <_  x
) )
8279, 81syl 17 . . . . . . 7  |-  ( ph  ->  ( A. z  e. 
ran  T  z  <_  x  <->  A. k  e.  NN  ( T `  k )  <_  x ) )
8382rexbidv 2892 . . . . . 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 240 . . . . 5  |-  ( ph  ->  E. x  e.  RR  A. z  e.  ran  T  z  <_  x )
85 supxrre 11638 . . . . 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 1292 . . . 4  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sup ( ran  T ,  RR ,  <  )
)
876, 1, 7, 19, 26, 69, 77isumsup 13982 . . . 4  |-  ( ph  -> 
sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  =  sup ( ran 
T ,  RR ,  <  ) )
8886, 87eqtr4d 2508 . . 3  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
899, 12, 14cbvsumi 13840 . . 3  |-  sum_ n  e.  NN  ( vol* `  A )  =  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )
9088, 89syl6eqr 2523 . 2  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sum_ n  e.  NN  ( vol* `  A
) )
915, 90breqtrd 4420 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 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   _Vcvv 3031   [_csb 3349    C_ wss 3390   (/)c0 3722   U_ciun 4269   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   ran crn 4840    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   supcsup 7972   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560   RR*cxr 9692    < clt 9693    <_ cle 9694   NNcn 10631   ZZ>=cuz 11182   ...cfz 11810    seqcseq 12251    ~~> cli 13625   sum_csu 13829   vol*covol 22491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cc 8883  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-ioo 11664  df-ico 11666  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-rlim 13630  df-sum 13830  df-ovol 22494
This theorem is referenced by:  ovoliunnul  22538  vitalilem5  22649
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