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Theorem ovoliun 22042
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 22022, 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 11141 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 10916 . . . . . . . . . 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 6056 . . . . . . . . . . 11  |-  ( ph  ->  G : NN --> RR )
65ffvelrnda 6032 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  e.  RR )
71, 2, 6serfre 12139 . . . . . . . . 9  |-  ( ph  ->  seq 1 (  +  ,  G ) : NN --> RR )
8 ovoliun.t . . . . . . . . . 10  |-  T  =  seq 1 (  +  ,  G )
98feq1i 5729 . . . . . . . . 9  |-  ( T : NN --> RR  <->  seq 1
(  +  ,  G
) : NN --> RR )
107, 9sylibr 212 . . . . . . . 8  |-  ( ph  ->  T : NN --> RR )
11 frn 5743 . . . . . . . 8  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
1210, 11syl 16 . . . . . . 7  |-  ( ph  ->  ran  T  C_  RR )
13 ressxr 9654 . . . . . . 7  |-  RR  C_  RR*
1412, 13syl6ss 3511 . . . . . 6  |-  ( ph  ->  ran  T  C_  RR* )
15 supxrcl 11531 . . . . . 6  |-  ( ran 
T  C_  RR*  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
1614, 15syl 16 . . . . 5  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
17 xrrebnd 11394 . . . . 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 11348 . . . . . . 7  |- -oo  e.  RR*
2019a1i 11 . . . . . 6  |-  ( ph  -> -oo  e.  RR* )
21 1nn 10567 . . . . . . . 8  |-  1  e.  NN
22 ffvelrn 6030 . . . . . . . 8  |-  ( ( T : NN --> RR  /\  1  e.  NN )  ->  ( T `  1
)  e.  RR )
2310, 21, 22sylancl 662 . . . . . . 7  |-  ( ph  ->  ( T `  1
)  e.  RR )
2423rexrd 9660 . . . . . 6  |-  ( ph  ->  ( T `  1
)  e.  RR* )
25 mnflt 11358 . . . . . . 7  |-  ( ( T `  1 )  e.  RR  -> -oo  <  ( T `  1 ) )
2623, 25syl 16 . . . . . 6  |-  ( ph  -> -oo  <  ( T `  1 ) )
27 ffn 5737 . . . . . . . . 9  |-  ( T : NN --> RR  ->  T  Fn  NN )
2810, 27syl 16 . . . . . . . 8  |-  ( ph  ->  T  Fn  NN )
29 fnfvelrn 6029 . . . . . . . 8  |-  ( ( T  Fn  NN  /\  1  e.  NN )  ->  ( T `  1
)  e.  ran  T
)
3028, 21, 29sylancl 662 . . . . . . 7  |-  ( ph  ->  ( T `  1
)  e.  ran  T
)
31 supxrub 11541 . . . . . . 7  |-  ( ( ran  T  C_  RR*  /\  ( T `  1 )  e.  ran  T )  -> 
( T `  1
)  <_  sup ( ran  T ,  RR* ,  <  ) )
3214, 30, 31syl2anc 661 . . . . . 6  |-  ( ph  ->  ( T `  1
)  <_  sup ( ran  T ,  RR* ,  <  ) )
3320, 24, 16, 26, 32xrltletrd 11389 . . . . 5  |-  ( ph  -> -oo  <  sup ( ran  T ,  RR* ,  <  ) )
3433biantrurd 508 . . . 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 2619 . . . . . . . . 9  |-  F/_ m A
37 nfcsb1v 3446 . . . . . . . . 9  |-  F/_ n [_ m  /  n ]_ A
38 csbeq1a 3439 . . . . . . . . 9  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
3936, 37, 38cbviun 4369 . . . . . . . 8  |-  U_ n  e.  NN  A  =  U_ m  e.  NN  [_ m  /  n ]_ A
4039fveq2i 5875 . . . . . . 7  |-  ( vol* `  U_ n  e.  NN  A )  =  ( vol* `  U_ m  e.  NN  [_ m  /  n ]_ A
)
41 nfcv 2619 . . . . . . . . . 10  |-  F/_ m
( vol* `  A )
42 nfcv 2619 . . . . . . . . . . 11  |-  F/_ n vol*
4342, 37nffv 5879 . . . . . . . . . 10  |-  F/_ n
( vol* `  [_ m  /  n ]_ A )
4438fveq2d 5876 . . . . . . . . . 10  |-  ( n  =  m  ->  ( vol* `  A )  =  ( vol* `  [_ m  /  n ]_ A ) )
4541, 43, 44cbvmpt 4547 . . . . . . . . 9  |-  ( n  e.  NN  |->  ( vol* `  A )
)  =  ( m  e.  NN  |->  ( vol* `  [_ m  /  n ]_ A ) )
464, 45eqtri 2486 . . . . . . . 8  |-  G  =  ( m  e.  NN  |->  ( vol* `  [_ m  /  n ]_ A ) )
47 ovoliun.a . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  A  C_  RR )
4847ralrimiva 2871 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  A  C_  RR )
49 nfv 1708 . . . . . . . . . . . 12  |-  F/ m  A  C_  RR
50 nfcv 2619 . . . . . . . . . . . . 13  |-  F/_ n RR
5137, 50nfss 3492 . . . . . . . . . . . 12  |-  F/ n [_ m  /  n ]_ A  C_  RR
5238sseq1d 3526 . . . . . . . . . . . 12  |-  ( n  =  m  ->  ( A  C_  RR  <->  [_ m  /  n ]_ A  C_  RR ) )
5349, 51, 52cbvral 3080 . . . . . . . . . . 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 725 . . . . . . . . 9  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
5655r19.21bi 2826 . . . . . . . 8  |-  ( ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  /\  m  e.  NN )  ->  [_ m  /  n ]_ A  C_  RR )
573ralrimiva 2871 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( vol* `  A
)  e.  RR )
5841nfel1 2635 . . . . . . . . . . . 12  |-  F/ m
( vol* `  A )  e.  RR
5943nfel1 2635 . . . . . . . . . . . 12  |-  F/ n
( vol* `  [_ m  /  n ]_ A )  e.  RR
6044eleq1d 2526 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( vol* `  A )  e.  RR  <->  ( vol* `  [_ m  /  n ]_ A )  e.  RR ) )
6158, 59, 60cbvral 3080 . . . . . . . . . . 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 725 . . . . . . . . 9  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  A. m  e.  NN  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
6463r19.21bi 2826 . . . . . . . 8  |-  ( ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  /\  m  e.  NN )  ->  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
65 simplr 755 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR )
66 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  x  e.  RR+ )
678, 46, 56, 64, 65, 66ovoliunlem3 22041 . . . . . . 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 4489 . . . . . 6  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  ( vol* `  U_ n  e.  NN  A )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  x ) )
6968ralrimiva 2871 . . . . 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 4373 . . . . . . . 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 22015 . . . . . . 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 11429 . . . . . 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 471 . . . . 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 11392 . . . 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 11364 . . . . 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 4460 . . . 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 1395    e. wcel 1819   A.wral 2807   [_csb 3430    C_ wss 3471   U_ciun 4332   class class class wbr 4456    |-> cmpt 4515   ran crn 5009    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   supcsup 7918   RRcr 9508   1c1 9510    + caddc 9512   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644    < clt 9645    <_ cle 9646   NNcn 10556   RR+crp 11245    seqcseq 12110   vol*covol 22000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cc 8832  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-q 11208  df-rp 11246  df-ioo 11558  df-ico 11560  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-rlim 13324  df-sum 13521  df-ovol 22002
This theorem is referenced by:  ovoliun2  22043  voliunlem2  22087  voliunlem3  22088  ex-ovoliunnfl  30262
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