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Theorem ovoliun2 18881
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 18880.) (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 18880 . 2  |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
6 nnuz 10279 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
7 1z 10069 . . . . . . . . 9  |-  1  e.  ZZ
87a1i 10 . . . . . . . 8  |-  ( ph  ->  1  e.  ZZ )
9 fvex 5555 . . . . . . . . . . 11  |-  ( vol
* `  [_ m  /  n ]_ A )  e. 
_V
10 nfcv 2432 . . . . . . . . . . . . . 14  |-  F/_ m
( vol * `  A )
11 nfcv 2432 . . . . . . . . . . . . . . 15  |-  F/_ n vol *
12 nfcsb1v 3126 . . . . . . . . . . . . . . 15  |-  F/_ n [_ m  /  n ]_ A
1311, 12nffv 5548 . . . . . . . . . . . . . 14  |-  F/_ n
( vol * `  [_ m  /  n ]_ A )
14 csbeq1a 3102 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
1514fveq2d 5545 . . . . . . . . . . . . . 14  |-  ( n  =  m  ->  ( vol * `  A )  =  ( vol * `  [_ m  /  n ]_ A ) )
1610, 13, 15cbvmpt 4126 . . . . . . . . . . . . 13  |-  ( n  e.  NN  |->  ( vol
* `  A )
)  =  ( m  e.  NN  |->  ( vol
* `  [_ m  /  n ]_ A ) )
172, 16eqtri 2316 . . . . . . . . . . . 12  |-  G  =  ( m  e.  NN  |->  ( vol * `  [_ m  /  n ]_ A ) )
1817fvmpt2 5624 . . . . . . . . . . 11  |-  ( ( m  e.  NN  /\  ( vol * `  [_ m  /  n ]_ A )  e.  _V )  -> 
( G `  m
)  =  ( vol
* `  [_ m  /  n ]_ A ) )
199, 18mpan2 652 . . . . . . . . . 10  |-  ( m  e.  NN  ->  ( G `  m )  =  ( vol * `  [_ m  /  n ]_ A ) )
2019adantl 452 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  =  ( vol * `  [_ m  /  n ]_ A ) )
214ralrimiva 2639 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( vol * `  A
)  e.  RR )
2210nfel1 2442 . . . . . . . . . . . 12  |-  F/ m
( vol * `  A )  e.  RR
2313nfel1 2442 . . . . . . . . . . . 12  |-  F/ n
( vol * `  [_ m  /  n ]_ A )  e.  RR
2415eleq1d 2362 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( vol * `  A )  e.  RR  <->  ( vol * `  [_ m  /  n ]_ A )  e.  RR ) )
2522, 23, 24cbvral 2773 . . . . . . . . . . 11  |-  ( A. n  e.  NN  ( vol * `  A )  e.  RR  <->  A. m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
2621, 25sylib 188 . . . . . . . . . 10  |-  ( ph  ->  A. m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
2726r19.21bi 2654 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol
* `  [_ m  /  n ]_ A )  e.  RR )
2820, 27eqeltrd 2370 . . . . . . . 8  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
296, 8, 28serfre 11091 . . . . . . 7  |-  ( ph  ->  seq  1 (  +  ,  G ) : NN --> RR )
301feq1i 5399 . . . . . . 7  |-  ( T : NN --> RR  <->  seq  1
(  +  ,  G
) : NN --> RR )
3129, 30sylibr 203 . . . . . 6  |-  ( ph  ->  T : NN --> RR )
32 frn 5411 . . . . . 6  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
3331, 32syl 15 . . . . 5  |-  ( ph  ->  ran  T  C_  RR )
34 1nn 9773 . . . . . . . 8  |-  1  e.  NN
35 fdm 5409 . . . . . . . . 9  |-  ( T : NN --> RR  ->  dom 
T  =  NN )
3631, 35syl 15 . . . . . . . 8  |-  ( ph  ->  dom  T  =  NN )
3734, 36syl5eleqr 2383 . . . . . . 7  |-  ( ph  ->  1  e.  dom  T
)
38 ne0i 3474 . . . . . . 7  |-  ( 1  e.  dom  T  ->  dom  T  =/=  (/) )
3937, 38syl 15 . . . . . 6  |-  ( ph  ->  dom  T  =/=  (/) )
40 dm0rn0 4911 . . . . . . 7  |-  ( dom 
T  =  (/)  <->  ran  T  =  (/) )
4140necon3bii 2491 . . . . . 6  |-  ( dom 
T  =/=  (/)  <->  ran  T  =/=  (/) )
4239, 41sylib 188 . . . . 5  |-  ( ph  ->  ran  T  =/=  (/) )
43 ovoliun2.t . . . . . . . . 9  |-  ( ph  ->  T  e.  dom  ~~>  )
441, 43syl5eqelr 2381 . . . . . . . 8  |-  ( ph  ->  seq  1 (  +  ,  G )  e. 
dom 
~~>  )
456, 8, 20, 27, 44isumrecl 12244 . . . . . . 7  |-  ( ph  -> 
sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
46 elfznn 10835 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... k )  ->  m  e.  NN )
4746adantl 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  m  e.  NN )
4847, 19syl 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( G `  m )  =  ( vol * `  [_ m  /  n ]_ A ) )
49 simpr 447 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  NN )
5049, 6syl6eleq 2386 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  ( ZZ>= `  1 )
)
51 simpl 443 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  NN )  ->  ph )
5251, 46, 27syl2an 463 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
5352recnd 8877 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( vol * `  [_ m  /  n ]_ A )  e.  CC )
5448, 50, 53fsumser 12219 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol * `  [_ m  /  n ]_ A )  =  (  seq  1 (  +  ,  G ) `  k ) )
551fveq1i 5542 . . . . . . . . . 10  |-  ( T `
 k )  =  (  seq  1 (  +  ,  G ) `
 k )
5654, 55syl6eqr 2346 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol * `  [_ m  /  n ]_ A )  =  ( T `  k ) )
57 fzfid 11051 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... k
)  e.  Fin )
58 elfznn 10835 . . . . . . . . . . . . 13  |-  ( n  e.  ( 1 ... k )  ->  n  e.  NN )
5958ssriv 3197 . . . . . . . . . . . 12  |-  ( 1 ... k )  C_  NN
6059a1i 10 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... k
)  C_  NN )
613ralrimiva 2639 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. n  e.  NN  A  C_  RR )
62 nfv 1609 . . . . . . . . . . . . . . 15  |-  F/ m  A  C_  RR
63 nfcv 2432 . . . . . . . . . . . . . . . 16  |-  F/_ n RR
6412, 63nfss 3186 . . . . . . . . . . . . . . 15  |-  F/ n [_ m  /  n ]_ A  C_  RR
6514sseq1d 3218 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  ( A  C_  RR  <->  [_ m  /  n ]_ A  C_  RR ) )
6662, 64, 65cbvral 2773 . . . . . . . . . . . . . 14  |-  ( A. n  e.  NN  A  C_  RR  <->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
6761, 66sylib 188 . . . . . . . . . . . . 13  |-  ( ph  ->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
6867r19.21bi 2654 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  [_ m  /  n ]_ A  C_  RR )
69 ovolge0 18856 . . . . . . . . . . . 12  |-  ( [_ m  /  n ]_ A  C_  RR  ->  0  <_  ( vol * `  [_ m  /  n ]_ A ) )
7068, 69syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  0  <_ 
( vol * `  [_ m  /  n ]_ A ) )
716, 8, 57, 60, 20, 27, 70, 44isumless 12320 . . . . . . . . . 10  |-  ( ph  -> 
sum_ m  e.  (
1 ... k ) ( vol * `  [_ m  /  n ]_ A )  <_  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) )
7271adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol * `  [_ m  /  n ]_ A )  <_  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) )
7356, 72eqbrtrrd 4061 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( T `
 k )  <_  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) )
7473ralrimiva 2639 . . . . . . 7  |-  ( ph  ->  A. k  e.  NN  ( T `  k )  <_  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) )
75 breq2 4043 . . . . . . . . 9  |-  ( x  =  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  ->  ( ( T `
 k )  <_  x 
<->  ( T `  k
)  <_  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) ) )
7675ralbidv 2576 . . . . . . . 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 ) ) )
7776rspcev 2897 . . . . . . 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
)
7845, 74, 77syl2anc 642 . . . . . 6  |-  ( ph  ->  E. x  e.  RR  A. k  e.  NN  ( T `  k )  <_  x )
79 ffn 5405 . . . . . . . . 9  |-  ( T : NN --> RR  ->  T  Fn  NN )
8031, 79syl 15 . . . . . . . 8  |-  ( ph  ->  T  Fn  NN )
81 breq1 4042 . . . . . . . . 9  |-  ( z  =  ( T `  k )  ->  (
z  <_  x  <->  ( T `  k )  <_  x
) )
8281ralrn 5684 . . . . . . . 8  |-  ( T  Fn  NN  ->  ( A. z  e.  ran  T  z  <_  x  <->  A. k  e.  NN  ( T `  k )  <_  x
) )
8380, 82syl 15 . . . . . . 7  |-  ( ph  ->  ( A. z  e. 
ran  T  z  <_  x  <->  A. k  e.  NN  ( T `  k )  <_  x ) )
8483rexbidv 2577 . . . . . 6  |-  ( ph  ->  ( E. x  e.  RR  A. z  e. 
ran  T  z  <_  x  <->  E. x  e.  RR  A. k  e.  NN  ( T `  k )  <_  x ) )
8578, 84mpbird 223 . . . . 5  |-  ( ph  ->  E. x  e.  RR  A. z  e.  ran  T  z  <_  x )
86 supxrre 10662 . . . . 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 ,  <  ) )
8733, 42, 85, 86syl3anc 1182 . . . 4  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sup ( ran  T ,  RR ,  <  )
)
886, 1, 8, 20, 27, 70, 78isumsup 12322 . . . 4  |-  ( ph  -> 
sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  =  sup ( ran 
T ,  RR ,  <  ) )
8987, 88eqtr4d 2331 . . 3  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) )
9010, 13, 15cbvsumi 12186 . . 3  |-  sum_ n  e.  NN  ( vol * `  A )  =  sum_ m  e.  NN  ( vol
* `  [_ m  /  n ]_ A )
9189, 90syl6eqr 2346 . 2  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sum_ n  e.  NN  ( vol * `  A
) )
925, 91breqtrd 4063 1  |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A
)  <_  sum_ n  e.  NN  ( vol * `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801   [_csb 3094    C_ wss 3165   (/)c0 3468   U_ciun 3921   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756   RR*cxr 8882    < clt 8883    <_ cle 8884   NNcn 9762   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062    ~~> cli 11974   sum_csu 12174   vol *covol 18838
This theorem is referenced by:  ovoliunnul  18882  vitalilem5  18983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cc 8077  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-ioo 10676  df-ico 10678  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-ovol 18840
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