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Theorem ovoliun 19393
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 19373, 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 10513 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
2 1z 10303 . . . . . . . . . . 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 5885 . . . . . . . . . . 11  |-  ( ph  ->  G : NN --> RR )
76ffvelrnda 5862 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  e.  RR )
81, 3, 7serfre 11344 . . . . . . . . 9  |-  ( ph  ->  seq  1 (  +  ,  G ) : NN --> RR )
9 ovoliun.t . . . . . . . . . 10  |-  T  =  seq  1 (  +  ,  G )
109feq1i 5577 . . . . . . . . 9  |-  ( T : NN --> RR  <->  seq  1
(  +  ,  G
) : NN --> RR )
118, 10sylibr 204 . . . . . . . 8  |-  ( ph  ->  T : NN --> RR )
12 frn 5589 . . . . . . . 8  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
1311, 12syl 16 . . . . . . 7  |-  ( ph  ->  ran  T  C_  RR )
14 ressxr 9121 . . . . . . 7  |-  RR  C_  RR*
1513, 14syl6ss 3352 . . . . . 6  |-  ( ph  ->  ran  T  C_  RR* )
16 supxrcl 10885 . . . . . 6  |-  ( ran 
T  C_  RR*  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
1715, 16syl 16 . . . . 5  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
18 xrrebnd 10748 . . . . 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 10706 . . . . . . 7  |-  -oo  e.  RR*
2120a1i 11 . . . . . 6  |-  ( ph  ->  -oo  e.  RR* )
22 1nn 10003 . . . . . . . 8  |-  1  e.  NN
23 ffvelrn 5860 . . . . . . . 8  |-  ( ( T : NN --> RR  /\  1  e.  NN )  ->  ( T `  1
)  e.  RR )
2411, 22, 23sylancl 644 . . . . . . 7  |-  ( ph  ->  ( T `  1
)  e.  RR )
2524rexrd 9126 . . . . . 6  |-  ( ph  ->  ( T `  1
)  e.  RR* )
26 mnflt 10714 . . . . . . 7  |-  ( ( T `  1 )  e.  RR  ->  -oo  <  ( T `  1 ) )
2724, 26syl 16 . . . . . 6  |-  ( ph  ->  -oo  <  ( T `  1 ) )
28 ffn 5583 . . . . . . . . 9  |-  ( T : NN --> RR  ->  T  Fn  NN )
2911, 28syl 16 . . . . . . . 8  |-  ( ph  ->  T  Fn  NN )
30 fnfvelrn 5859 . . . . . . . 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 10895 . . . . . . 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 10743 . . . . 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 2571 . . . . . . . . 9  |-  F/_ m A
38 nfcsb1v 3275 . . . . . . . . 9  |-  F/_ n [_ m  /  n ]_ A
39 csbeq1a 3251 . . . . . . . . 9  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
4037, 38, 39cbviun 4120 . . . . . . . 8  |-  U_ n  e.  NN  A  =  U_ m  e.  NN  [_ m  /  n ]_ A
4140fveq2i 5723 . . . . . . 7  |-  ( vol
* `  U_ n  e.  NN  A )  =  ( vol * `  U_ m  e.  NN  [_ m  /  n ]_ A
)
42 nfcv 2571 . . . . . . . . . 10  |-  F/_ m
( vol * `  A )
43 nfcv 2571 . . . . . . . . . . 11  |-  F/_ n vol *
4443, 38nffv 5727 . . . . . . . . . 10  |-  F/_ n
( vol * `  [_ m  /  n ]_ A )
4539fveq2d 5724 . . . . . . . . . 10  |-  ( n  =  m  ->  ( vol * `  A )  =  ( vol * `  [_ m  /  n ]_ A ) )
4642, 44, 45cbvmpt 4291 . . . . . . . . 9  |-  ( n  e.  NN  |->  ( vol
* `  A )
)  =  ( m  e.  NN  |->  ( vol
* `  [_ m  /  n ]_ A ) )
475, 46eqtri 2455 . . . . . . . 8  |-  G  =  ( m  e.  NN  |->  ( vol * `  [_ m  /  n ]_ A ) )
48 ovoliun.a . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  A  C_  RR )
4948ralrimiva 2781 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  A  C_  RR )
50 nfv 1629 . . . . . . . . . . . 12  |-  F/ m  A  C_  RR
51 nfcv 2571 . . . . . . . . . . . . 13  |-  F/_ n RR
5238, 51nfss 3333 . . . . . . . . . . . 12  |-  F/ n [_ m  /  n ]_ A  C_  RR
5339sseq1d 3367 . . . . . . . . . . . 12  |-  ( n  =  m  ->  ( A  C_  RR  <->  [_ m  /  n ]_ A  C_  RR ) )
5450, 52, 53cbvral 2920 . . . . . . . . . . 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 2796 . . . . . . . 8  |-  ( ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  /\  m  e.  NN )  ->  [_ m  /  n ]_ A  C_  RR )
584ralrimiva 2781 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( vol * `  A
)  e.  RR )
5942nfel1 2581 . . . . . . . . . . . 12  |-  F/ m
( vol * `  A )  e.  RR
6044nfel1 2581 . . . . . . . . . . . 12  |-  F/ n
( vol * `  [_ m  /  n ]_ A )  e.  RR
6145eleq1d 2501 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( vol * `  A )  e.  RR  <->  ( vol * `  [_ m  /  n ]_ A )  e.  RR ) )
6259, 60, 61cbvral 2920 . . . . . . . . . . 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 2796 . . . . . . . 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 19392 . . . . . . 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 4237 . . . . . 6  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  ( vol * `  U_ n  e.  NN  A )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  x ) )
7069ralrimiva 2781 . . . . 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 4124 . . . . . . . 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 19366 . . . . . . 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 10783 . . . . . 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 10746 . . . 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 10719 . . . . 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 4208 . . . 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 1652    e. wcel 1725   A.wral 2697   [_csb 3243    C_ wss 3312   U_ciun 4085   class class class wbr 4204    e. cmpt 4258   ran crn 4871    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   supcsup 7437   RRcr 8981   1c1 8983    + caddc 8985    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111    < clt 9112    <_ cle 9113   NNcn 9992   ZZcz 10274   RR+crp 10604    seq cseq 11315   vol
*covol 19351
This theorem is referenced by:  ovoliun2  19394  voliunlem2  19437  voliunlem3  19438  ex-ovoliunnfl  26239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cc 8307  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-ioo 10912  df-ico 10914  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-ovol 19353
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