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Theorem ovoliun 18880
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 18860, 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 10279 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
2 1z 10069 . . . . . . . . . . 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 5700 . . . . . . . . . . 11  |-  ( ph  ->  G : NN --> RR )
7 ffvelrn 5679 . . . . . . . . . . 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 11091 . . . . . . . . 9  |-  ( ph  ->  seq  1 (  +  ,  G ) : NN --> RR )
10 ovoliun.t . . . . . . . . . 10  |-  T  =  seq  1 (  +  ,  G )
1110feq1i 5399 . . . . . . . . 9  |-  ( T : NN --> RR  <->  seq  1
(  +  ,  G
) : NN --> RR )
129, 11sylibr 203 . . . . . . . 8  |-  ( ph  ->  T : NN --> RR )
13 frn 5411 . . . . . . . 8  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
1412, 13syl 15 . . . . . . 7  |-  ( ph  ->  ran  T  C_  RR )
15 ressxr 8892 . . . . . . 7  |-  RR  C_  RR*
1614, 15syl6ss 3204 . . . . . 6  |-  ( ph  ->  ran  T  C_  RR* )
17 supxrcl 10649 . . . . . 6  |-  ( ran 
T  C_  RR*  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
1816, 17syl 15 . . . . 5  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
19 xrrebnd 10513 . . . . 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 10472 . . . . . . 7  |-  -oo  e.  RR*
2221a1i 10 . . . . . 6  |-  ( ph  ->  -oo  e.  RR* )
23 1nn 9773 . . . . . . . 8  |-  1  e.  NN
24 ffvelrn 5679 . . . . . . . 8  |-  ( ( T : NN --> RR  /\  1  e.  NN )  ->  ( T `  1
)  e.  RR )
2512, 23, 24sylancl 643 . . . . . . 7  |-  ( ph  ->  ( T `  1
)  e.  RR )
2625rexrd 8897 . . . . . 6  |-  ( ph  ->  ( T `  1
)  e.  RR* )
27 mnflt 10480 . . . . . . 7  |-  ( ( T `  1 )  e.  RR  ->  -oo  <  ( T `  1 ) )
2825, 27syl 15 . . . . . 6  |-  ( ph  ->  -oo  <  ( T `  1 ) )
29 ffn 5405 . . . . . . . . 9  |-  ( T : NN --> RR  ->  T  Fn  NN )
3012, 29syl 15 . . . . . . . 8  |-  ( ph  ->  T  Fn  NN )
31 fnfvelrn 5678 . . . . . . . 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 10659 . . . . . . 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 10508 . . . . 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 2432 . . . . . . . . 9  |-  F/_ m A
39 nfcsb1v 3126 . . . . . . . . 9  |-  F/_ n [_ m  /  n ]_ A
40 csbeq1a 3102 . . . . . . . . 9  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
4138, 39, 40cbviun 3955 . . . . . . . 8  |-  U_ n  e.  NN  A  =  U_ m  e.  NN  [_ m  /  n ]_ A
4241fveq2i 5544 . . . . . . 7  |-  ( vol
* `  U_ n  e.  NN  A )  =  ( vol * `  U_ m  e.  NN  [_ m  /  n ]_ A
)
43 nfcv 2432 . . . . . . . . . 10  |-  F/_ m
( vol * `  A )
44 nfcv 2432 . . . . . . . . . . 11  |-  F/_ n vol *
4544, 39nffv 5548 . . . . . . . . . 10  |-  F/_ n
( vol * `  [_ m  /  n ]_ A )
4640fveq2d 5545 . . . . . . . . . 10  |-  ( n  =  m  ->  ( vol * `  A )  =  ( vol * `  [_ m  /  n ]_ A ) )
4743, 45, 46cbvmpt 4126 . . . . . . . . 9  |-  ( n  e.  NN  |->  ( vol
* `  A )
)  =  ( m  e.  NN  |->  ( vol
* `  [_ m  /  n ]_ A ) )
485, 47eqtri 2316 . . . . . . . 8  |-  G  =  ( m  e.  NN  |->  ( vol * `  [_ m  /  n ]_ A ) )
49 ovoliun.a . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  A  C_  RR )
5049ralrimiva 2639 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  A  C_  RR )
51 nfv 1609 . . . . . . . . . . . 12  |-  F/ m  A  C_  RR
52 nfcv 2432 . . . . . . . . . . . . 13  |-  F/_ n RR
5339, 52nfss 3186 . . . . . . . . . . . 12  |-  F/ n [_ m  /  n ]_ A  C_  RR
5440sseq1d 3218 . . . . . . . . . . . 12  |-  ( n  =  m  ->  ( A  C_  RR  <->  [_ m  /  n ]_ A  C_  RR ) )
5551, 53, 54cbvral 2773 . . . . . . . . . . 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 2654 . . . . . . . 8  |-  ( ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  /\  m  e.  NN )  ->  [_ m  /  n ]_ A  C_  RR )
594ralrimiva 2639 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( vol * `  A
)  e.  RR )
6043nfel1 2442 . . . . . . . . . . . 12  |-  F/ m
( vol * `  A )  e.  RR
6145nfel1 2442 . . . . . . . . . . . 12  |-  F/ n
( vol * `  [_ m  /  n ]_ A )  e.  RR
6246eleq1d 2362 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( vol * `  A )  e.  RR  <->  ( vol * `  [_ m  /  n ]_ A )  e.  RR ) )
6360, 61, 62cbvral 2773 . . . . . . . . . . 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 2654 . . . . . . . 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 18879 . . . . . . 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 4072 . . . . . 6  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  ( vol * `  U_ n  e.  NN  A )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  x ) )
7170ralrimiva 2639 . . . . 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 3959 . . . . . . . 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 18853 . . . . . . 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 10548 . . . . . 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 10511 . . . 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 10485 . . . . 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 4043 . . . 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 1632    e. wcel 1696   A.wral 2556   [_csb 3094    C_ wss 3165   U_ciun 3921   class class class wbr 4039    e. cmpt 4093   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752   1c1 8754    + caddc 8756    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882    < clt 8883    <_ cle 8884   NNcn 9762   ZZcz 10040   RR+crp 10370    seq cseq 11062   vol
*covol 18838
This theorem is referenced by:  ovoliun2  18881  voliunlem2  18924  voliunlem3  18925  ex-ovoliunnfl  25002
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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