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Theorem ovoliun2 20958
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 20957.) (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 20957 . 2  |-  ( ph  ->  ( vol* `  U_ n  e.  NN  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
6 nnuz 10888 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
7 1zzd 10669 . . . . . . . 8  |-  ( ph  ->  1  e.  ZZ )
8 fvex 5694 . . . . . . . . . . 11  |-  ( vol* `  [_ m  /  n ]_ A )  e. 
_V
9 nfcv 2573 . . . . . . . . . . . . . 14  |-  F/_ m
( vol* `  A )
10 nfcv 2573 . . . . . . . . . . . . . . 15  |-  F/_ n vol*
11 nfcsb1v 3297 . . . . . . . . . . . . . . 15  |-  F/_ n [_ m  /  n ]_ A
1210, 11nffv 5691 . . . . . . . . . . . . . 14  |-  F/_ n
( vol* `  [_ m  /  n ]_ A )
13 csbeq1a 3290 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
1413fveq2d 5688 . . . . . . . . . . . . . 14  |-  ( n  =  m  ->  ( vol* `  A )  =  ( vol* `  [_ m  /  n ]_ A ) )
159, 12, 14cbvmpt 4375 . . . . . . . . . . . . 13  |-  ( n  e.  NN  |->  ( vol* `  A )
)  =  ( m  e.  NN  |->  ( vol* `  [_ m  /  n ]_ A ) )
162, 15eqtri 2457 . . . . . . . . . . . 12  |-  G  =  ( m  e.  NN  |->  ( vol* `  [_ m  /  n ]_ A ) )
1716fvmpt2 5774 . . . . . . . . . . 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 2793 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( vol* `  A
)  e.  RR )
219nfel1 2583 . . . . . . . . . . . 12  |-  F/ m
( vol* `  A )  e.  RR
2212nfel1 2583 . . . . . . . . . . . 12  |-  F/ n
( vol* `  [_ m  /  n ]_ A )  e.  RR
2314eleq1d 2503 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( vol* `  A )  e.  RR  <->  ( vol* `  [_ m  /  n ]_ A )  e.  RR ) )
2421, 22, 23cbvral 2937 . . . . . . . . . . 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 2808 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
2719, 26eqeltrd 2511 . . . . . . . 8  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
286, 7, 27serfre 11827 . . . . . . 7  |-  ( ph  ->  seq 1 (  +  ,  G ) : NN --> RR )
291feq1i 5544 . . . . . . 7  |-  ( T : NN --> RR  <->  seq 1
(  +  ,  G
) : NN --> RR )
3028, 29sylibr 212 . . . . . 6  |-  ( ph  ->  T : NN --> RR )
31 frn 5558 . . . . . 6  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
3230, 31syl 16 . . . . 5  |-  ( ph  ->  ran  T  C_  RR )
33 1nn 10325 . . . . . . . 8  |-  1  e.  NN
34 fdm 5556 . . . . . . . . 9  |-  ( T : NN --> RR  ->  dom 
T  =  NN )
3530, 34syl 16 . . . . . . . 8  |-  ( ph  ->  dom  T  =  NN )
3633, 35syl5eleqr 2524 . . . . . . 7  |-  ( ph  ->  1  e.  dom  T
)
37 ne0i 3636 . . . . . . 7  |-  ( 1  e.  dom  T  ->  dom  T  =/=  (/) )
3836, 37syl 16 . . . . . 6  |-  ( ph  ->  dom  T  =/=  (/) )
39 dm0rn0 5048 . . . . . . 7  |-  ( dom 
T  =  (/)  <->  ran  T  =  (/) )
4039necon3bii 2634 . . . . . 6  |-  ( dom 
T  =/=  (/)  <->  ran  T  =/=  (/) )
4138, 40sylib 196 . . . . 5  |-  ( ph  ->  ran  T  =/=  (/) )
42 ovoliun2.t . . . . . . . . 9  |-  ( ph  ->  T  e.  dom  ~~>  )
431, 42syl5eqelr 2522 . . . . . . . 8  |-  ( ph  ->  seq 1 (  +  ,  G )  e. 
dom 
~~>  )
446, 7, 19, 26, 43isumrecl 13224 . . . . . . 7  |-  ( ph  -> 
sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
45 elfznn 11470 . . . . . . . . . . . . 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 2527 . . . . . . . . . . 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 9404 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( vol* `  [_ m  /  n ]_ A )  e.  CC )
5347, 49, 52fsumser 13199 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol* `  [_ m  /  n ]_ A )  =  (  seq 1 (  +  ,  G ) `  k ) )
541fveq1i 5685 . . . . . . . . . 10  |-  ( T `
 k )  =  (  seq 1 (  +  ,  G ) `
 k )
5553, 54syl6eqr 2487 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol* `  [_ m  /  n ]_ A )  =  ( T `  k ) )
56 fzfid 11787 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... k
)  e.  Fin )
57 elfznn 11470 . . . . . . . . . . . . 13  |-  ( n  e.  ( 1 ... k )  ->  n  e.  NN )
5857ssriv 3353 . . . . . . . . . . . 12  |-  ( 1 ... k )  C_  NN
5958a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... k
)  C_  NN )
603ralrimiva 2793 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. n  e.  NN  A  C_  RR )
61 nfv 1673 . . . . . . . . . . . . . . 15  |-  F/ m  A  C_  RR
62 nfcv 2573 . . . . . . . . . . . . . . . 16  |-  F/_ n RR
6311, 62nfss 3342 . . . . . . . . . . . . . . 15  |-  F/ n [_ m  /  n ]_ A  C_  RR
6413sseq1d 3376 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  ( A  C_  RR  <->  [_ m  /  n ]_ A  C_  RR ) )
6561, 63, 64cbvral 2937 . . . . . . . . . . . . . 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 2808 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  [_ m  /  n ]_ A  C_  RR )
68 ovolge0 20933 . . . . . . . . . . . 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 13300 . . . . . . . . . 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 4307 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( T `
 k )  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
7372ralrimiva 2793 . . . . . . 7  |-  ( ph  ->  A. k  e.  NN  ( T `  k )  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
74 breq2 4289 . . . . . . . . 9  |-  ( x  =  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  ->  ( ( T `
 k )  <_  x 
<->  ( T `  k
)  <_  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) ) )
7574ralbidv 2729 . . . . . . . 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 3066 . . . . . . 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 5552 . . . . . . . . 9  |-  ( T : NN --> RR  ->  T  Fn  NN )
7930, 78syl 16 . . . . . . . 8  |-  ( ph  ->  T  Fn  NN )
80 breq1 4288 . . . . . . . . 9  |-  ( z  =  ( T `  k )  ->  (
z  <_  x  <->  ( T `  k )  <_  x
) )
8180ralrn 5839 . . . . . . . 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 2730 . . . . . 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 11282 . . . . 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 13302 . . . 4  |-  ( ph  -> 
sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  =  sup ( ran 
T ,  RR ,  <  ) )
8886, 87eqtr4d 2472 . . 3  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A ) )
899, 12, 14cbvsumi 13166 . . 3  |-  sum_ n  e.  NN  ( vol* `  A )  =  sum_ m  e.  NN  ( vol* `  [_ m  /  n ]_ A )
9088, 89syl6eqr 2487 . 2  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sum_ n  e.  NN  ( vol* `  A
) )
915, 90breqtrd 4309 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 2600   A.wral 2709   E.wrex 2710   _Vcvv 2966   [_csb 3281    C_ wss 3321   (/)c0 3630   U_ciun 4164   class class class wbr 4285    e. cmpt 4343   dom cdm 4832   ran crn 4833    Fn wfn 5406   -->wf 5407   ` cfv 5411  (class class class)co 6086   supcsup 7682   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277   RR*cxr 9409    < clt 9410    <_ cle 9411   NNcn 10314   ZZ>=cuz 10853   ...cfz 11429    seqcseq 11798    ~~> cli 12954   sum_csu 13155   vol*covol 20915
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 2418  ax-rep 4396  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367  ax-inf2 7839  ax-cc 8596  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
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 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-tp 3875  df-op 3877  df-uni 4085  df-int 4122  df-iun 4166  df-br 4286  df-opab 4344  df-mpt 4345  df-tr 4379  df-eprel 4624  df-id 4628  df-po 4633  df-so 4634  df-fr 4671  df-se 4672  df-we 4673  df-ord 4714  df-on 4715  df-lim 4716  df-suc 4717  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-rn 4843  df-res 4844  df-ima 4845  df-iota 5374  df-fun 5413  df-fn 5414  df-f 5415  df-f1 5416  df-fo 5417  df-f1o 5418  df-fv 5419  df-isom 5420  df-riota 6045  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-ioo 11296  df-ico 11298  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-ovol 20917
This theorem is referenced by:  ovoliunnul  20959  vitalilem5  21061
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