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Theorem ovoliun 18864
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 18844, 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 10263 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
2 1z 10053 . . . . . . . . . . 11  |-  1  e.  ZZ
32a1i 10 . . . . . . . . . 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 5684 . . . . . . . . . . 11  |-  ( ph  ->  G : NN --> RR )
7 ffvelrn 5663 . . . . . . . . . . 11  |-  ( ( G : NN --> RR  /\  k  e.  NN )  ->  ( G `  k
)  e.  RR )
86, 7sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  e.  RR )
91, 3, 8serfre 11075 . . . . . . . . 9  |-  ( ph  ->  seq  1 (  +  ,  G ) : NN --> RR )
10 ovoliun.t . . . . . . . . . 10  |-  T  =  seq  1 (  +  ,  G )
1110feq1i 5383 . . . . . . . . 9  |-  ( T : NN --> RR  <->  seq  1
(  +  ,  G
) : NN --> RR )
129, 11sylibr 203 . . . . . . . 8  |-  ( ph  ->  T : NN --> RR )
13 frn 5395 . . . . . . . 8  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
1412, 13syl 15 . . . . . . 7  |-  ( ph  ->  ran  T  C_  RR )
15 ressxr 8876 . . . . . . 7  |-  RR  C_  RR*
1614, 15syl6ss 3191 . . . . . 6  |-  ( ph  ->  ran  T  C_  RR* )
17 supxrcl 10633 . . . . . 6  |-  ( ran 
T  C_  RR*  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
1816, 17syl 15 . . . . 5  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
19 xrrebnd 10497 . . . . 5  |-  ( sup ( ran  T ,  RR* ,  <  )  e. 
RR*  ->  ( sup ( ran  T ,  RR* ,  <  )  e.  RR  <->  (  -oo  <  sup ( ran  T ,  RR* ,  <  )  /\  sup ( ran  T ,  RR* ,  <  )  <  +oo ) ) )
2018, 19syl 15 . . . 4  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  e.  RR  <->  (  -oo  <  sup ( ran  T ,  RR* ,  <  )  /\  sup ( ran  T ,  RR* ,  <  )  <  +oo ) ) )
21 mnfxr 10456 . . . . . . 7  |-  -oo  e.  RR*
2221a1i 10 . . . . . 6  |-  ( ph  ->  -oo  e.  RR* )
23 1nn 9757 . . . . . . . 8  |-  1  e.  NN
24 ffvelrn 5663 . . . . . . . 8  |-  ( ( T : NN --> RR  /\  1  e.  NN )  ->  ( T `  1
)  e.  RR )
2512, 23, 24sylancl 643 . . . . . . 7  |-  ( ph  ->  ( T `  1
)  e.  RR )
2625rexrd 8881 . . . . . 6  |-  ( ph  ->  ( T `  1
)  e.  RR* )
27 mnflt 10464 . . . . . . 7  |-  ( ( T `  1 )  e.  RR  ->  -oo  <  ( T `  1 ) )
2825, 27syl 15 . . . . . 6  |-  ( ph  ->  -oo  <  ( T `  1 ) )
29 ffn 5389 . . . . . . . . 9  |-  ( T : NN --> RR  ->  T  Fn  NN )
3012, 29syl 15 . . . . . . . 8  |-  ( ph  ->  T  Fn  NN )
31 fnfvelrn 5662 . . . . . . . 8  |-  ( ( T  Fn  NN  /\  1  e.  NN )  ->  ( T `  1
)  e.  ran  T
)
3230, 23, 31sylancl 643 . . . . . . 7  |-  ( ph  ->  ( T `  1
)  e.  ran  T
)
33 supxrub 10643 . . . . . . 7  |-  ( ( ran  T  C_  RR*  /\  ( T `  1 )  e.  ran  T )  -> 
( T `  1
)  <_  sup ( ran  T ,  RR* ,  <  ) )
3416, 32, 33syl2anc 642 . . . . . 6  |-  ( ph  ->  ( T `  1
)  <_  sup ( ran  T ,  RR* ,  <  ) )
3522, 26, 18, 28, 34xrltletrd 10492 . . . . 5  |-  ( ph  ->  -oo  <  sup ( ran  T ,  RR* ,  <  ) )
3635biantrurd 494 . . . 4  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  <  +oo  <->  (  -oo  <  sup ( ran  T ,  RR* ,  <  )  /\  sup ( ran  T ,  RR* ,  <  )  <  +oo ) ) )
3720, 36bitr4d 247 . . 3  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  e.  RR  <->  sup ( ran  T ,  RR* ,  <  )  <  +oo ) )
38 nfcv 2419 . . . . . . . . 9  |-  F/_ m A
39 nfcsb1v 3113 . . . . . . . . 9  |-  F/_ n [_ m  /  n ]_ A
40 csbeq1a 3089 . . . . . . . . 9  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
4138, 39, 40cbviun 3939 . . . . . . . 8  |-  U_ n  e.  NN  A  =  U_ m  e.  NN  [_ m  /  n ]_ A
4241fveq2i 5528 . . . . . . 7  |-  ( vol
* `  U_ n  e.  NN  A )  =  ( vol * `  U_ m  e.  NN  [_ m  /  n ]_ A
)
43 nfcv 2419 . . . . . . . . . 10  |-  F/_ m
( vol * `  A )
44 nfcv 2419 . . . . . . . . . . 11  |-  F/_ n vol *
4544, 39nffv 5532 . . . . . . . . . 10  |-  F/_ n
( vol * `  [_ m  /  n ]_ A )
4640fveq2d 5529 . . . . . . . . . 10  |-  ( n  =  m  ->  ( vol * `  A )  =  ( vol * `  [_ m  /  n ]_ A ) )
4743, 45, 46cbvmpt 4110 . . . . . . . . 9  |-  ( n  e.  NN  |->  ( vol
* `  A )
)  =  ( m  e.  NN  |->  ( vol
* `  [_ m  /  n ]_ A ) )
485, 47eqtri 2303 . . . . . . . 8  |-  G  =  ( m  e.  NN  |->  ( vol * `  [_ m  /  n ]_ A ) )
49 ovoliun.a . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  A  C_  RR )
5049ralrimiva 2626 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  A  C_  RR )
51 nfv 1605 . . . . . . . . . . . 12  |-  F/ m  A  C_  RR
52 nfcv 2419 . . . . . . . . . . . . 13  |-  F/_ n RR
5339, 52nfss 3173 . . . . . . . . . . . 12  |-  F/ n [_ m  /  n ]_ A  C_  RR
5440sseq1d 3205 . . . . . . . . . . . 12  |-  ( n  =  m  ->  ( A  C_  RR  <->  [_ m  /  n ]_ A  C_  RR ) )
5551, 53, 54cbvral 2760 . . . . . . . . . . 11  |-  ( A. n  e.  NN  A  C_  RR  <->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
5650, 55sylib 188 . . . . . . . . . 10  |-  ( ph  ->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
5756ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
5857r19.21bi 2641 . . . . . . . 8  |-  ( ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  /\  m  e.  NN )  ->  [_ m  /  n ]_ A  C_  RR )
594ralrimiva 2626 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( vol * `  A
)  e.  RR )
6043nfel1 2429 . . . . . . . . . . . 12  |-  F/ m
( vol * `  A )  e.  RR
6145nfel1 2429 . . . . . . . . . . . 12  |-  F/ n
( vol * `  [_ m  /  n ]_ A )  e.  RR
6246eleq1d 2349 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( vol * `  A )  e.  RR  <->  ( vol * `  [_ m  /  n ]_ A )  e.  RR ) )
6360, 61, 62cbvral 2760 . . . . . . . . . . 11  |-  ( A. n  e.  NN  ( vol * `  A )  e.  RR  <->  A. m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
6459, 63sylib 188 . . . . . . . . . 10  |-  ( ph  ->  A. m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
6564ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  A. m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
6665r19.21bi 2641 . . . . . . . 8  |-  ( ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  /\  m  e.  NN )  ->  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
67 simplr 731 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
68 simpr 447 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  x  e.  RR+ )
6910, 48, 58, 66, 67, 68ovoliunlem3 18863 . . . . . . 7  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  ( vol * `  U_ m  e.  NN  [_ m  /  n ]_ A )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  x ) )
7042, 69syl5eqbr 4056 . . . . . 6  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  ( vol * `  U_ n  e.  NN  A )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  x ) )
7170ralrimiva 2626 . . . . 5  |-  ( (
ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  ->  A. x  e.  RR+  ( vol * `  U_ n  e.  NN  A )  <_ 
( sup ( ran 
T ,  RR* ,  <  )  +  x ) )
72 iunss 3943 . . . . . . . 8  |-  ( U_ n  e.  NN  A  C_  RR  <->  A. n  e.  NN  A  C_  RR )
7350, 72sylibr 203 . . . . . . 7  |-  ( ph  ->  U_ n  e.  NN  A  C_  RR )
74 ovolcl 18837 . . . . . . 7  |-  ( U_ n  e.  NN  A  C_  RR  ->  ( vol * `
 U_ n  e.  NN  A )  e.  RR* )
7573, 74syl 15 . . . . . 6  |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A
)  e.  RR* )
76 xralrple 10532 . . . . . 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 ) ) )
7775, 76sylan 457 . . . . 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 ) ) )
7871, 77mpbird 223 . . . 4  |-  ( (
ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  -> 
( vol * `  U_ n  e.  NN  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
7978ex 423 . . 3  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  e.  RR  ->  ( vol * `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR* ,  <  ) ) )
8037, 79sylbird 226 . 2  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  <  +oo  ->  ( vol
* `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR* ,  <  ) ) )
81 nltpnft 10495 . . . 4  |-  ( sup ( ran  T ,  RR* ,  <  )  e. 
RR*  ->  ( sup ( ran  T ,  RR* ,  <  )  =  +oo  <->  -.  sup ( ran  T ,  RR* ,  <  )  <  +oo ) )
8218, 81syl 15 . . 3  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  =  +oo  <->  -.  sup ( ran  T ,  RR* ,  <  )  <  +oo ) )
83 pnfge 10469 . . . . 5  |-  ( ( vol * `  U_ n  e.  NN  A )  e. 
RR*  ->  ( vol * `  U_ n  e.  NN  A )  <_  +oo )
8475, 83syl 15 . . . 4  |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A
)  <_  +oo )
85 breq2 4027 . . . 4  |-  ( sup ( ran  T ,  RR* ,  <  )  = 
+oo  ->  ( ( vol
* `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR* ,  <  )  <->  ( vol * `
 U_ n  e.  NN  A )  <_  +oo )
)
8684, 85syl5ibrcom 213 . . 3  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  =  +oo  ->  ( vol * `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR* ,  <  ) ) )
8782, 86sylbird 226 . 2  |-  ( ph  ->  ( -.  sup ( ran  T ,  RR* ,  <  )  <  +oo  ->  ( vol
* `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR* ,  <  ) ) )
8880, 87pm2.61d 150 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   [_csb 3081    C_ wss 3152   U_ciun 3905   class class class wbr 4023    e. cmpt 4077   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   1c1 8738    + caddc 8740    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867    <_ cle 8868   NNcn 9746   ZZcz 10024   RR+crp 10354    seq cseq 11046   vol
*covol 18822
This theorem is referenced by:  ovoliun2  18865  voliunlem2  18908  voliunlem3  18909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cc 8061  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-ioo 10660  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-ovol 18824
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