MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovoliun Unicode version

Theorem ovoliun 19268
Description: The Lebesgue outer measure function is countably sub-additive. (Many books allow  +oo as a value for one of the sets in the sum, but in our setup we can't do arithmetic on infinity, and in any case the volume of a union containing an infinitely large set is already infinitely large by monotonicity ovolss 19248, so we need not consider this case here, although we do allow the sum itself to be infinite.) (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 )
Assertion
Ref Expression
ovoliun  |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
Distinct variable group:    ph, n
Allowed substitution hints:    A( n)    T( n)    G( n)

Proof of Theorem ovoliun
Dummy variables  k  m  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnuz 10453 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
2 1z 10243 . . . . . . . . . . 11  |-  1  e.  ZZ
32a1i 11 . . . . . . . . . 10  |-  ( ph  ->  1  e.  ZZ )
4 ovoliun.v . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( vol
* `  A )  e.  RR )
5 ovoliun.g . . . . . . . . . . . 12  |-  G  =  ( n  e.  NN  |->  ( vol * `  A
) )
64, 5fmptd 5832 . . . . . . . . . . 11  |-  ( ph  ->  G : NN --> RR )
76ffvelrnda 5809 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  e.  RR )
81, 3, 7serfre 11279 . . . . . . . . 9  |-  ( ph  ->  seq  1 (  +  ,  G ) : NN --> RR )
9 ovoliun.t . . . . . . . . . 10  |-  T  =  seq  1 (  +  ,  G )
109feq1i 5525 . . . . . . . . 9  |-  ( T : NN --> RR  <->  seq  1
(  +  ,  G
) : NN --> RR )
118, 10sylibr 204 . . . . . . . 8  |-  ( ph  ->  T : NN --> RR )
12 frn 5537 . . . . . . . 8  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
1311, 12syl 16 . . . . . . 7  |-  ( ph  ->  ran  T  C_  RR )
14 ressxr 9062 . . . . . . 7  |-  RR  C_  RR*
1513, 14syl6ss 3303 . . . . . 6  |-  ( ph  ->  ran  T  C_  RR* )
16 supxrcl 10825 . . . . . 6  |-  ( ran 
T  C_  RR*  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
1715, 16syl 16 . . . . 5  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
18 xrrebnd 10688 . . . . 5  |-  ( sup ( ran  T ,  RR* ,  <  )  e. 
RR*  ->  ( sup ( ran  T ,  RR* ,  <  )  e.  RR  <->  (  -oo  <  sup ( ran  T ,  RR* ,  <  )  /\  sup ( ran  T ,  RR* ,  <  )  <  +oo ) ) )
1917, 18syl 16 . . . 4  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  e.  RR  <->  (  -oo  <  sup ( ran  T ,  RR* ,  <  )  /\  sup ( ran  T ,  RR* ,  <  )  <  +oo ) ) )
20 mnfxr 10646 . . . . . . 7  |-  -oo  e.  RR*
2120a1i 11 . . . . . 6  |-  ( ph  ->  -oo  e.  RR* )
22 1nn 9943 . . . . . . . 8  |-  1  e.  NN
23 ffvelrn 5807 . . . . . . . 8  |-  ( ( T : NN --> RR  /\  1  e.  NN )  ->  ( T `  1
)  e.  RR )
2411, 22, 23sylancl 644 . . . . . . 7  |-  ( ph  ->  ( T `  1
)  e.  RR )
2524rexrd 9067 . . . . . 6  |-  ( ph  ->  ( T `  1
)  e.  RR* )
26 mnflt 10654 . . . . . . 7  |-  ( ( T `  1 )  e.  RR  ->  -oo  <  ( T `  1 ) )
2724, 26syl 16 . . . . . 6  |-  ( ph  ->  -oo  <  ( T `  1 ) )
28 ffn 5531 . . . . . . . . 9  |-  ( T : NN --> RR  ->  T  Fn  NN )
2911, 28syl 16 . . . . . . . 8  |-  ( ph  ->  T  Fn  NN )
30 fnfvelrn 5806 . . . . . . . 8  |-  ( ( T  Fn  NN  /\  1  e.  NN )  ->  ( T `  1
)  e.  ran  T
)
3129, 22, 30sylancl 644 . . . . . . 7  |-  ( ph  ->  ( T `  1
)  e.  ran  T
)
32 supxrub 10835 . . . . . . 7  |-  ( ( ran  T  C_  RR*  /\  ( T `  1 )  e.  ran  T )  -> 
( T `  1
)  <_  sup ( ran  T ,  RR* ,  <  ) )
3315, 31, 32syl2anc 643 . . . . . 6  |-  ( ph  ->  ( T `  1
)  <_  sup ( ran  T ,  RR* ,  <  ) )
3421, 25, 17, 27, 33xrltletrd 10683 . . . . 5  |-  ( ph  ->  -oo  <  sup ( ran  T ,  RR* ,  <  ) )
3534biantrurd 495 . . . 4  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  <  +oo  <->  (  -oo  <  sup ( ran  T ,  RR* ,  <  )  /\  sup ( ran  T ,  RR* ,  <  )  <  +oo ) ) )
3619, 35bitr4d 248 . . 3  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  e.  RR  <->  sup ( ran  T ,  RR* ,  <  )  <  +oo ) )
37 nfcv 2523 . . . . . . . . 9  |-  F/_ m A
38 nfcsb1v 3226 . . . . . . . . 9  |-  F/_ n [_ m  /  n ]_ A
39 csbeq1a 3202 . . . . . . . . 9  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
4037, 38, 39cbviun 4069 . . . . . . . 8  |-  U_ n  e.  NN  A  =  U_ m  e.  NN  [_ m  /  n ]_ A
4140fveq2i 5671 . . . . . . 7  |-  ( vol
* `  U_ n  e.  NN  A )  =  ( vol * `  U_ m  e.  NN  [_ m  /  n ]_ A
)
42 nfcv 2523 . . . . . . . . . 10  |-  F/_ m
( vol * `  A )
43 nfcv 2523 . . . . . . . . . . 11  |-  F/_ n vol *
4443, 38nffv 5675 . . . . . . . . . 10  |-  F/_ n
( vol * `  [_ m  /  n ]_ A )
4539fveq2d 5672 . . . . . . . . . 10  |-  ( n  =  m  ->  ( vol * `  A )  =  ( vol * `  [_ m  /  n ]_ A ) )
4642, 44, 45cbvmpt 4240 . . . . . . . . 9  |-  ( n  e.  NN  |->  ( vol
* `  A )
)  =  ( m  e.  NN  |->  ( vol
* `  [_ m  /  n ]_ A ) )
475, 46eqtri 2407 . . . . . . . 8  |-  G  =  ( m  e.  NN  |->  ( vol * `  [_ m  /  n ]_ A ) )
48 ovoliun.a . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  A  C_  RR )
4948ralrimiva 2732 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  A  C_  RR )
50 nfv 1626 . . . . . . . . . . . 12  |-  F/ m  A  C_  RR
51 nfcv 2523 . . . . . . . . . . . . 13  |-  F/_ n RR
5238, 51nfss 3284 . . . . . . . . . . . 12  |-  F/ n [_ m  /  n ]_ A  C_  RR
5339sseq1d 3318 . . . . . . . . . . . 12  |-  ( n  =  m  ->  ( A  C_  RR  <->  [_ m  /  n ]_ A  C_  RR ) )
5450, 52, 53cbvral 2871 . . . . . . . . . . 11  |-  ( A. n  e.  NN  A  C_  RR  <->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
5549, 54sylib 189 . . . . . . . . . 10  |-  ( ph  ->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
5655ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
5756r19.21bi 2747 . . . . . . . 8  |-  ( ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  /\  m  e.  NN )  ->  [_ m  /  n ]_ A  C_  RR )
584ralrimiva 2732 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( vol * `  A
)  e.  RR )
5942nfel1 2533 . . . . . . . . . . . 12  |-  F/ m
( vol * `  A )  e.  RR
6044nfel1 2533 . . . . . . . . . . . 12  |-  F/ n
( vol * `  [_ m  /  n ]_ A )  e.  RR
6145eleq1d 2453 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( vol * `  A )  e.  RR  <->  ( vol * `  [_ m  /  n ]_ A )  e.  RR ) )
6259, 60, 61cbvral 2871 . . . . . . . . . . 11  |-  ( A. n  e.  NN  ( vol * `  A )  e.  RR  <->  A. m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
6358, 62sylib 189 . . . . . . . . . 10  |-  ( ph  ->  A. m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
6463ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  A. m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
6564r19.21bi 2747 . . . . . . . 8  |-  ( ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  /\  m  e.  NN )  ->  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
66 simplr 732 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
67 simpr 448 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  x  e.  RR+ )
689, 47, 57, 65, 66, 67ovoliunlem3 19267 . . . . . . 7  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  ( vol * `  U_ m  e.  NN  [_ m  /  n ]_ A )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  x ) )
6941, 68syl5eqbr 4186 . . . . . 6  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  ( vol * `  U_ n  e.  NN  A )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  x ) )
7069ralrimiva 2732 . . . . 5  |-  ( (
ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  ->  A. x  e.  RR+  ( vol * `  U_ n  e.  NN  A )  <_ 
( sup ( ran 
T ,  RR* ,  <  )  +  x ) )
71 iunss 4073 . . . . . . . 8  |-  ( U_ n  e.  NN  A  C_  RR  <->  A. n  e.  NN  A  C_  RR )
7249, 71sylibr 204 . . . . . . 7  |-  ( ph  ->  U_ n  e.  NN  A  C_  RR )
73 ovolcl 19241 . . . . . . 7  |-  ( U_ n  e.  NN  A  C_  RR  ->  ( vol * `
 U_ n  e.  NN  A )  e.  RR* )
7472, 73syl 16 . . . . . 6  |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A
)  e.  RR* )
75 xralrple 10723 . . . . . 6  |-  ( ( ( vol * `  U_ n  e.  NN  A
)  e.  RR*  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  ->  ( ( vol
* `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR* ,  <  )  <->  A. x  e.  RR+  ( vol * `  U_ n  e.  NN  A )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  x ) ) )
7674, 75sylan 458 . . . . 5  |-  ( (
ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  -> 
( ( vol * `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR* ,  <  )  <->  A. x  e.  RR+  ( vol * `  U_ n  e.  NN  A )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  x ) ) )
7770, 76mpbird 224 . . . 4  |-  ( (
ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  -> 
( vol * `  U_ n  e.  NN  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
7877ex 424 . . 3  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  e.  RR  ->  ( vol * `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR* ,  <  ) ) )
7936, 78sylbird 227 . 2  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  <  +oo  ->  ( vol
* `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR* ,  <  ) ) )
80 nltpnft 10686 . . . 4  |-  ( sup ( ran  T ,  RR* ,  <  )  e. 
RR*  ->  ( sup ( ran  T ,  RR* ,  <  )  =  +oo  <->  -.  sup ( ran  T ,  RR* ,  <  )  <  +oo ) )
8117, 80syl 16 . . 3  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  =  +oo  <->  -.  sup ( ran  T ,  RR* ,  <  )  <  +oo ) )
82 pnfge 10659 . . . . 5  |-  ( ( vol * `  U_ n  e.  NN  A )  e. 
RR*  ->  ( vol * `  U_ n  e.  NN  A )  <_  +oo )
8374, 82syl 16 . . . 4  |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A
)  <_  +oo )
84 breq2 4157 . . . 4  |-  ( sup ( ran  T ,  RR* ,  <  )  = 
+oo  ->  ( ( vol
* `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR* ,  <  )  <->  ( vol * `
 U_ n  e.  NN  A )  <_  +oo )
)
8583, 84syl5ibrcom 214 . . 3  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  =  +oo  ->  ( vol * `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR* ,  <  ) ) )
8681, 85sylbird 227 . 2  |-  ( ph  ->  ( -.  sup ( ran  T ,  RR* ,  <  )  <  +oo  ->  ( vol
* `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR* ,  <  ) ) )
8779, 86pm2.61d 152 1  |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   [_csb 3194    C_ wss 3263   U_ciun 4035   class class class wbr 4153    e. cmpt 4207   ran crn 4819    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   supcsup 7380   RRcr 8922   1c1 8924    + caddc 8926    +oocpnf 9050    -oocmnf 9051   RR*cxr 9052    < clt 9053    <_ cle 9054   NNcn 9932   ZZcz 10214   RR+crp 10544    seq cseq 11250   vol
*covol 19226
This theorem is referenced by:  ovoliun2  19269  voliunlem2  19312  voliunlem3  19313  ex-ovoliunnfl  25954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cc 8248  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-q 10507  df-rp 10545  df-ioo 10852  df-ico 10854  df-fz 10976  df-fzo 11066  df-fl 11129  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-rlim 12210  df-sum 12407  df-ovol 19228
  Copyright terms: Public domain W3C validator