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Theorem ovoliun 20947
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 20927, 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 10892 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 10673 . . . . . . . . . 10  |-  ( ph  ->  1  e.  ZZ )
3 ovoliun.v . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  ( vol* `  A )  e.  RR )
4 ovoliun.g . . . . . . . . . . . 12  |-  G  =  ( n  e.  NN  |->  ( vol* `  A
) )
53, 4fmptd 5864 . . . . . . . . . . 11  |-  ( ph  ->  G : NN --> RR )
65ffvelrnda 5840 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  e.  RR )
71, 2, 6serfre 11831 . . . . . . . . 9  |-  ( ph  ->  seq 1 (  +  ,  G ) : NN --> RR )
8 ovoliun.t . . . . . . . . . 10  |-  T  =  seq 1 (  +  ,  G )
98feq1i 5548 . . . . . . . . 9  |-  ( T : NN --> RR  <->  seq 1
(  +  ,  G
) : NN --> RR )
107, 9sylibr 212 . . . . . . . 8  |-  ( ph  ->  T : NN --> RR )
11 frn 5562 . . . . . . . 8  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
1210, 11syl 16 . . . . . . 7  |-  ( ph  ->  ran  T  C_  RR )
13 ressxr 9423 . . . . . . 7  |-  RR  C_  RR*
1412, 13syl6ss 3365 . . . . . 6  |-  ( ph  ->  ran  T  C_  RR* )
15 supxrcl 11273 . . . . . 6  |-  ( ran 
T  C_  RR*  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
1614, 15syl 16 . . . . 5  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
17 xrrebnd 11136 . . . . 5  |-  ( sup ( ran  T ,  RR* ,  <  )  e. 
RR*  ->  ( sup ( ran  T ,  RR* ,  <  )  e.  RR  <->  ( -oo  <  sup ( ran  T ,  RR* ,  <  )  /\  sup ( ran  T ,  RR* ,  <  )  < +oo ) ) )
1816, 17syl 16 . . . 4  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  e.  RR  <->  ( -oo  <  sup ( ran  T ,  RR* ,  <  )  /\  sup ( ran  T ,  RR* ,  <  )  < +oo ) ) )
19 mnfxr 11090 . . . . . . 7  |- -oo  e.  RR*
2019a1i 11 . . . . . 6  |-  ( ph  -> -oo  e.  RR* )
21 1nn 10329 . . . . . . . 8  |-  1  e.  NN
22 ffvelrn 5838 . . . . . . . 8  |-  ( ( T : NN --> RR  /\  1  e.  NN )  ->  ( T `  1
)  e.  RR )
2310, 21, 22sylancl 657 . . . . . . 7  |-  ( ph  ->  ( T `  1
)  e.  RR )
2423rexrd 9429 . . . . . 6  |-  ( ph  ->  ( T `  1
)  e.  RR* )
25 mnflt 11100 . . . . . . 7  |-  ( ( T `  1 )  e.  RR  -> -oo  <  ( T `  1 ) )
2623, 25syl 16 . . . . . 6  |-  ( ph  -> -oo  <  ( T `  1 ) )
27 ffn 5556 . . . . . . . . 9  |-  ( T : NN --> RR  ->  T  Fn  NN )
2810, 27syl 16 . . . . . . . 8  |-  ( ph  ->  T  Fn  NN )
29 fnfvelrn 5837 . . . . . . . 8  |-  ( ( T  Fn  NN  /\  1  e.  NN )  ->  ( T `  1
)  e.  ran  T
)
3028, 21, 29sylancl 657 . . . . . . 7  |-  ( ph  ->  ( T `  1
)  e.  ran  T
)
31 supxrub 11283 . . . . . . 7  |-  ( ( ran  T  C_  RR*  /\  ( T `  1 )  e.  ran  T )  -> 
( T `  1
)  <_  sup ( ran  T ,  RR* ,  <  ) )
3214, 30, 31syl2anc 656 . . . . . 6  |-  ( ph  ->  ( T `  1
)  <_  sup ( ran  T ,  RR* ,  <  ) )
3320, 24, 16, 26, 32xrltletrd 11131 . . . . 5  |-  ( ph  -> -oo  <  sup ( ran  T ,  RR* ,  <  ) )
3433biantrurd 505 . . . 4  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  < +oo  <->  ( -oo  <  sup ( ran  T ,  RR* ,  <  )  /\  sup ( ran  T ,  RR* ,  <  )  < +oo ) ) )
3518, 34bitr4d 256 . . 3  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  e.  RR  <->  sup ( ran  T ,  RR* ,  <  )  < +oo ) )
36 nfcv 2577 . . . . . . . . 9  |-  F/_ m A
37 nfcsb1v 3301 . . . . . . . . 9  |-  F/_ n [_ m  /  n ]_ A
38 csbeq1a 3294 . . . . . . . . 9  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
3936, 37, 38cbviun 4204 . . . . . . . 8  |-  U_ n  e.  NN  A  =  U_ m  e.  NN  [_ m  /  n ]_ A
4039fveq2i 5691 . . . . . . 7  |-  ( vol* `  U_ n  e.  NN  A )  =  ( vol* `  U_ m  e.  NN  [_ m  /  n ]_ A
)
41 nfcv 2577 . . . . . . . . . 10  |-  F/_ m
( vol* `  A )
42 nfcv 2577 . . . . . . . . . . 11  |-  F/_ n vol*
4342, 37nffv 5695 . . . . . . . . . 10  |-  F/_ n
( vol* `  [_ m  /  n ]_ A )
4438fveq2d 5692 . . . . . . . . . 10  |-  ( n  =  m  ->  ( vol* `  A )  =  ( vol* `  [_ m  /  n ]_ A ) )
4541, 43, 44cbvmpt 4379 . . . . . . . . 9  |-  ( n  e.  NN  |->  ( vol* `  A )
)  =  ( m  e.  NN  |->  ( vol* `  [_ m  /  n ]_ A ) )
464, 45eqtri 2461 . . . . . . . 8  |-  G  =  ( m  e.  NN  |->  ( vol* `  [_ m  /  n ]_ A ) )
47 ovoliun.a . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  A  C_  RR )
4847ralrimiva 2797 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  A  C_  RR )
49 nfv 1678 . . . . . . . . . . . 12  |-  F/ m  A  C_  RR
50 nfcv 2577 . . . . . . . . . . . . 13  |-  F/_ n RR
5137, 50nfss 3346 . . . . . . . . . . . 12  |-  F/ n [_ m  /  n ]_ A  C_  RR
5238sseq1d 3380 . . . . . . . . . . . 12  |-  ( n  =  m  ->  ( A  C_  RR  <->  [_ m  /  n ]_ A  C_  RR ) )
5349, 51, 52cbvral 2941 . . . . . . . . . . 11  |-  ( A. n  e.  NN  A  C_  RR  <->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
5448, 53sylib 196 . . . . . . . . . 10  |-  ( ph  ->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
5554ad2antrr 720 . . . . . . . . 9  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
5655r19.21bi 2812 . . . . . . . 8  |-  ( ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  /\  m  e.  NN )  ->  [_ m  /  n ]_ A  C_  RR )
573ralrimiva 2797 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( vol* `  A
)  e.  RR )
5841nfel1 2587 . . . . . . . . . . . 12  |-  F/ m
( vol* `  A )  e.  RR
5943nfel1 2587 . . . . . . . . . . . 12  |-  F/ n
( vol* `  [_ m  /  n ]_ A )  e.  RR
6044eleq1d 2507 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( vol* `  A )  e.  RR  <->  ( vol* `  [_ m  /  n ]_ A )  e.  RR ) )
6158, 59, 60cbvral 2941 . . . . . . . . . . 11  |-  ( A. n  e.  NN  ( vol* `  A )  e.  RR  <->  A. m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
6257, 61sylib 196 . . . . . . . . . 10  |-  ( ph  ->  A. m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
6362ad2antrr 720 . . . . . . . . 9  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  A. m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
6463r19.21bi 2812 . . . . . . . 8  |-  ( ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  /\  m  e.  NN )  ->  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
65 simplr 749 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
66 simpr 458 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  x  e.  RR+ )
678, 46, 56, 64, 65, 66ovoliunlem3 20946 . . . . . . 7  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  ( vol* `  U_ m  e.  NN  [_ m  /  n ]_ A )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  x ) )
6840, 67syl5eqbr 4322 . . . . . 6  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  ( vol* `  U_ n  e.  NN  A )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  x ) )
6968ralrimiva 2797 . . . . 5  |-  ( (
ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  ->  A. x  e.  RR+  ( vol* `  U_ n  e.  NN  A )  <_ 
( sup ( ran 
T ,  RR* ,  <  )  +  x ) )
70 iunss 4208 . . . . . . . 8  |-  ( U_ n  e.  NN  A  C_  RR  <->  A. n  e.  NN  A  C_  RR )
7148, 70sylibr 212 . . . . . . 7  |-  ( ph  ->  U_ n  e.  NN  A  C_  RR )
72 ovolcl 20920 . . . . . . 7  |-  ( U_ n  e.  NN  A  C_  RR  ->  ( vol* `  U_ n  e.  NN  A )  e. 
RR* )
7371, 72syl 16 . . . . . 6  |-  ( ph  ->  ( vol* `  U_ n  e.  NN  A
)  e.  RR* )
74 xralrple 11171 . . . . . 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 ) ) )
7573, 74sylan 468 . . . . 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 ) ) )
7669, 75mpbird 232 . . . 4  |-  ( (
ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  -> 
( vol* `  U_ n  e.  NN  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
7776ex 434 . . 3  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  e.  RR  ->  ( vol* `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR* ,  <  ) ) )
7835, 77sylbird 235 . 2  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  < +oo  ->  ( vol* `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR* ,  <  ) ) )
79 nltpnft 11134 . . . 4  |-  ( sup ( ran  T ,  RR* ,  <  )  e. 
RR*  ->  ( sup ( ran  T ,  RR* ,  <  )  = +oo  <->  -.  sup ( ran  T ,  RR* ,  <  )  < +oo ) )
8016, 79syl 16 . . 3  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  = +oo  <->  -.  sup ( ran  T ,  RR* ,  <  )  < +oo ) )
81 pnfge 11106 . . . . 5  |-  ( ( vol* `  U_ n  e.  NN  A )  e. 
RR*  ->  ( vol* `  U_ n  e.  NN  A )  <_ +oo )
8273, 81syl 16 . . . 4  |-  ( ph  ->  ( vol* `  U_ n  e.  NN  A
)  <_ +oo )
83 breq2 4293 . . . 4  |-  ( sup ( ran  T ,  RR* ,  <  )  = +oo  ->  ( ( vol* `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR* ,  <  )  <->  ( vol* `  U_ n  e.  NN  A )  <_ +oo ) )
8482, 83syl5ibrcom 222 . . 3  |-  ( ph  ->  ( sup ( ran 
T ,  RR* ,  <  )  = +oo  ->  ( vol* `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR* ,  <  ) ) )
8580, 84sylbird 235 . 2  |-  ( ph  ->  ( -.  sup ( ran  T ,  RR* ,  <  )  < +oo  ->  ( vol* `  U_ n  e.  NN  A )  <_  sup ( ran  T ,  RR* ,  <  ) ) )
8678, 85pm2.61d 158 1  |-  ( ph  ->  ( vol* `  U_ n  e.  NN  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   [_csb 3285    C_ wss 3325   U_ciun 4168   class class class wbr 4289    e. cmpt 4347   ran crn 4837    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   supcsup 7686   RRcr 9277   1c1 9279    + caddc 9281   +oocpnf 9411   -oocmnf 9412   RR*cxr 9413    < clt 9414    <_ cle 9415   NNcn 10318   RR+crp 10987    seqcseq 11802   vol*covol 20905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cc 8600  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-ioo 11300  df-ico 11302  df-fz 11434  df-fzo 11545  df-fl 11638  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-clim 12962  df-rlim 12963  df-sum 13160  df-ovol 20907
This theorem is referenced by:  ovoliun2  20948  voliunlem2  20991  voliunlem3  20992  ex-ovoliunnfl  28359
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