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Theorem ovoliun2 22043
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 22042.) (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 22042 . 2  |-  ( ph  ->  ( vol* `  U_ n  e.  NN  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
6 nnuz 11141 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
7 1zzd 10916 . . . . . . . 8  |-  ( ph  ->  1  e.  ZZ )
8 fvex 5882 . . . . . . . . . . 11  |-  ( vol* `  [_ m  /  n ]_ A )  e. 
_V
9 nfcv 2619 . . . . . . . . . . . . . 14  |-  F/_ m
( vol* `  A )
10 nfcv 2619 . . . . . . . . . . . . . . 15  |-  F/_ n vol*
11 nfcsb1v 3446 . . . . . . . . . . . . . . 15  |-  F/_ n [_ m  /  n ]_ A
1210, 11nffv 5879 . . . . . . . . . . . . . 14  |-  F/_ n
( vol* `  [_ m  /  n ]_ A )
13 csbeq1a 3439 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
1413fveq2d 5876 . . . . . . . . . . . . . 14  |-  ( n  =  m  ->  ( vol* `  A )  =  ( vol* `  [_ m  /  n ]_ A ) )
159, 12, 14cbvmpt 4547 . . . . . . . . . . . . 13  |-  ( n  e.  NN  |->  ( vol* `  A )
)  =  ( m  e.  NN  |->  ( vol* `  [_ m  /  n ]_ A ) )
162, 15eqtri 2486 . . . . . . . . . . . 12  |-  G  =  ( m  e.  NN  |->  ( vol* `  [_ m  /  n ]_ A ) )
1716fvmpt2 5964 . . . . . . . . . . 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 2871 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( vol* `  A
)  e.  RR )
219nfel1 2635 . . . . . . . . . . . 12  |-  F/ m
( vol* `  A )  e.  RR
2212nfel1 2635 . . . . . . . . . . . 12  |-  F/ n
( vol* `  [_ m  /  n ]_ A )  e.  RR
2314eleq1d 2526 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( vol* `  A )  e.  RR  <->  ( vol* `  [_ m  /  n ]_ A )  e.  RR ) )
2421, 22, 23cbvral 3080 . . . . . . . . . . 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 2826 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
2719, 26eqeltrd 2545 . . . . . . . 8  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
286, 7, 27serfre 12139 . . . . . . 7  |-  ( ph  ->  seq 1 (  +  ,  G ) : NN --> RR )
291feq1i 5729 . . . . . . 7  |-  ( T : NN --> RR  <->  seq 1
(  +  ,  G
) : NN --> RR )
3028, 29sylibr 212 . . . . . 6  |-  ( ph  ->  T : NN --> RR )
31 frn 5743 . . . . . 6  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
3230, 31syl 16 . . . . 5  |-  ( ph  ->  ran  T  C_  RR )
33 1nn 10567 . . . . . . . 8  |-  1  e.  NN
34 fdm 5741 . . . . . . . . 9  |-  ( T : NN --> RR  ->  dom 
T  =  NN )
3530, 34syl 16 . . . . . . . 8  |-  ( ph  ->  dom  T  =  NN )
3633, 35syl5eleqr 2552 . . . . . . 7  |-  ( ph  ->  1  e.  dom  T
)
37 ne0i 3799 . . . . . . 7  |-  ( 1  e.  dom  T  ->  dom  T  =/=  (/) )
3836, 37syl 16 . . . . . 6  |-  ( ph  ->  dom  T  =/=  (/) )
39 dm0rn0 5229 . . . . . . 7  |-  ( dom 
T  =  (/)  <->  ran  T  =  (/) )
4039necon3bii 2725 . . . . . 6  |-  ( dom 
T  =/=  (/)  <->  ran  T  =/=  (/) )
4138, 40sylib 196 . . . . 5  |-  ( ph  ->  ran  T  =/=  (/) )
42 ovoliun2.t . . . . . . . . 9  |-  ( ph  ->  T  e.  dom  ~~>  )
431, 42syl5eqelr 2550 . . . . . . . 8  |-  ( ph  ->  seq 1 (  +  ,  G )  e. 
dom 
~~>  )
446, 7, 19, 26, 43isumrecl 13592 . . . . . . 7  |-  ( ph  -> 
sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
45 elfznn 11739 . . . . . . . . . . . . 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 2555 . . . . . . . . . . 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 9639 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( vol* `  [_ m  /  n ]_ A )  e.  CC )
5347, 49, 52fsumser 13564 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol* `  [_ m  /  n ]_ A )  =  (  seq 1 (  +  ,  G ) `  k ) )
541fveq1i 5873 . . . . . . . . . 10  |-  ( T `
 k )  =  (  seq 1 (  +  ,  G ) `
 k )
5553, 54syl6eqr 2516 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol* `  [_ m  /  n ]_ A )  =  ( T `  k ) )
56 fzfid 12086 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... k
)  e.  Fin )
57 elfznn 11739 . . . . . . . . . . . . 13  |-  ( n  e.  ( 1 ... k )  ->  n  e.  NN )
5857ssriv 3503 . . . . . . . . . . . 12  |-  ( 1 ... k )  C_  NN
5958a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... k
)  C_  NN )
603ralrimiva 2871 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. n  e.  NN  A  C_  RR )
61 nfv 1708 . . . . . . . . . . . . . . 15  |-  F/ m  A  C_  RR
62 nfcv 2619 . . . . . . . . . . . . . . . 16  |-  F/_ n RR
6311, 62nfss 3492 . . . . . . . . . . . . . . 15  |-  F/ n [_ m  /  n ]_ A  C_  RR
6413sseq1d 3526 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  ( A  C_  RR  <->  [_ m  /  n ]_ A  C_  RR ) )
6561, 63, 64cbvral 3080 . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  [_ m  /  n ]_ A  C_  RR )
68 ovolge0 22018 . . . . . . . . . . . 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 13669 . . . . . . . . . 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 4478 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( T `
 k )  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
7372ralrimiva 2871 . . . . . . 7  |-  ( ph  ->  A. k  e.  NN  ( T `  k )  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
74 breq2 4460 . . . . . . . . 9  |-  ( x  =  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  ->  ( ( T `
 k )  <_  x 
<->  ( T `  k
)  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) ) )
7574ralbidv 2896 . . . . . . . 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 3210 . . . . . . 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 5737 . . . . . . . . 9  |-  ( T : NN --> RR  ->  T  Fn  NN )
7930, 78syl 16 . . . . . . . 8  |-  ( ph  ->  T  Fn  NN )
80 breq1 4459 . . . . . . . . 9  |-  ( z  =  ( T `  k )  ->  (
z  <_  x  <->  ( T `  k )  <_  x
) )
8180ralrn 6035 . . . . . . . 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 2968 . . . . . 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 11544 . . . . 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 13671 . . . 4  |-  ( ph  -> 
sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  =  sup ( ran 
T ,  RR ,  <  ) )
8886, 87eqtr4d 2501 . . 3  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
899, 12, 14cbvsumi 13531 . . 3  |-  sum_ n  e.  NN  ( vol* `  A )  =  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )
9088, 89syl6eqr 2516 . 2  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sum_ n  e.  NN  ( vol* `  A
) )
915, 90breqtrd 4480 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 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808   _Vcvv 3109   [_csb 3430    C_ wss 3471   (/)c0 3793   U_ciun 4332   class class class wbr 4456    |-> cmpt 4515   dom cdm 5008   ran crn 5009    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   supcsup 7918   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512   RR*cxr 9644    < clt 9645    <_ cle 9646   NNcn 10556   ZZ>=cuz 11106   ...cfz 11697    seqcseq 12110    ~~> cli 13319   sum_csu 13520   vol*covol 22000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cc 8832  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-ioo 11558  df-ico 11560  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-rlim 13324  df-sum 13521  df-ovol 22002
This theorem is referenced by:  ovoliunnul  22044  vitalilem5  22147
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