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Theorem ovoliun 21104
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 21084, 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 10997 . . . . . . . . . 10  |-  NN  =  ( ZZ>= `  1 )
2 1zzd 10778 . . . . . . . . . 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 5966 . . . . . . . . . . 11  |-  ( ph  ->  G : NN --> RR )
65ffvelrnda 5942 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  ( G `
 k )  e.  RR )
71, 2, 6serfre 11936 . . . . . . . . 9  |-  ( ph  ->  seq 1 (  +  ,  G ) : NN --> RR )
8 ovoliun.t . . . . . . . . . 10  |-  T  =  seq 1 (  +  ,  G )
98feq1i 5649 . . . . . . . . 9  |-  ( T : NN --> RR  <->  seq 1
(  +  ,  G
) : NN --> RR )
107, 9sylibr 212 . . . . . . . 8  |-  ( ph  ->  T : NN --> RR )
11 frn 5663 . . . . . . . 8  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
1210, 11syl 16 . . . . . . 7  |-  ( ph  ->  ran  T  C_  RR )
13 ressxr 9528 . . . . . . 7  |-  RR  C_  RR*
1412, 13syl6ss 3466 . . . . . 6  |-  ( ph  ->  ran  T  C_  RR* )
15 supxrcl 11378 . . . . . 6  |-  ( ran 
T  C_  RR*  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
1614, 15syl 16 . . . . 5  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  e.  RR* )
17 xrrebnd 11241 . . . . 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 11195 . . . . . . 7  |- -oo  e.  RR*
2019a1i 11 . . . . . 6  |-  ( ph  -> -oo  e.  RR* )
21 1nn 10434 . . . . . . . 8  |-  1  e.  NN
22 ffvelrn 5940 . . . . . . . 8  |-  ( ( T : NN --> RR  /\  1  e.  NN )  ->  ( T `  1
)  e.  RR )
2310, 21, 22sylancl 662 . . . . . . 7  |-  ( ph  ->  ( T `  1
)  e.  RR )
2423rexrd 9534 . . . . . 6  |-  ( ph  ->  ( T `  1
)  e.  RR* )
25 mnflt 11205 . . . . . . 7  |-  ( ( T `  1 )  e.  RR  -> -oo  <  ( T `  1 ) )
2623, 25syl 16 . . . . . 6  |-  ( ph  -> -oo  <  ( T `  1 ) )
27 ffn 5657 . . . . . . . . 9  |-  ( T : NN --> RR  ->  T  Fn  NN )
2810, 27syl 16 . . . . . . . 8  |-  ( ph  ->  T  Fn  NN )
29 fnfvelrn 5939 . . . . . . . 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 11388 . . . . . . 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 11236 . . . . 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 2613 . . . . . . . . 9  |-  F/_ m A
37 nfcsb1v 3402 . . . . . . . . 9  |-  F/_ n [_ m  /  n ]_ A
38 csbeq1a 3395 . . . . . . . . 9  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
3936, 37, 38cbviun 4305 . . . . . . . 8  |-  U_ n  e.  NN  A  =  U_ m  e.  NN  [_ m  /  n ]_ A
4039fveq2i 5792 . . . . . . 7  |-  ( vol* `  U_ n  e.  NN  A )  =  ( vol* `  U_ m  e.  NN  [_ m  /  n ]_ A
)
41 nfcv 2613 . . . . . . . . . 10  |-  F/_ m
( vol* `  A )
42 nfcv 2613 . . . . . . . . . . 11  |-  F/_ n vol*
4342, 37nffv 5796 . . . . . . . . . 10  |-  F/_ n
( vol* `  [_ m  /  n ]_ A )
4438fveq2d 5793 . . . . . . . . . 10  |-  ( n  =  m  ->  ( vol* `  A )  =  ( vol* `  [_ m  /  n ]_ A ) )
4541, 43, 44cbvmpt 4480 . . . . . . . . 9  |-  ( n  e.  NN  |->  ( vol* `  A )
)  =  ( m  e.  NN  |->  ( vol* `  [_ m  /  n ]_ A ) )
464, 45eqtri 2480 . . . . . . . 8  |-  G  =  ( m  e.  NN  |->  ( vol* `  [_ m  /  n ]_ A ) )
47 ovoliun.a . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  NN )  ->  A  C_  RR )
4847ralrimiva 2822 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  A  C_  RR )
49 nfv 1674 . . . . . . . . . . . 12  |-  F/ m  A  C_  RR
50 nfcv 2613 . . . . . . . . . . . . 13  |-  F/_ n RR
5137, 50nfss 3447 . . . . . . . . . . . 12  |-  F/ n [_ m  /  n ]_ A  C_  RR
5238sseq1d 3481 . . . . . . . . . . . 12  |-  ( n  =  m  ->  ( A  C_  RR  <->  [_ m  /  n ]_ A  C_  RR ) )
5349, 51, 52cbvral 3039 . . . . . . . . . . 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 2910 . . . . . . . 8  |-  ( ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  /\  m  e.  NN )  ->  [_ m  /  n ]_ A  C_  RR )
573ralrimiva 2822 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( vol* `  A
)  e.  RR )
5841nfel1 2628 . . . . . . . . . . . 12  |-  F/ m
( vol* `  A )  e.  RR
5943nfel1 2628 . . . . . . . . . . . 12  |-  F/ n
( vol* `  [_ m  /  n ]_ A )  e.  RR
6044eleq1d 2520 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( vol* `  A )  e.  RR  <->  ( vol* `  [_ m  /  n ]_ A )  e.  RR ) )
6158, 59, 60cbvral 3039 . . . . . . . . . . 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 2910 . . . . . . . 8  |-  ( ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  /\  m  e.  NN )  ->  ( vol* `  [_ m  /  n ]_ A )  e.  RR )
65 simplr 754 . . . . . . . 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 21103 . . . . . . 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 4423 . . . . . 6  |-  ( ( ( ph  /\  sup ( ran  T ,  RR* ,  <  )  e.  RR )  /\  x  e.  RR+ )  ->  ( vol* `  U_ n  e.  NN  A )  <_  ( sup ( ran  T ,  RR* ,  <  )  +  x ) )
6968ralrimiva 2822 . . . . 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 4309 . . . . . . . 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 21077 . . . . . . 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 11276 . . . . . 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 11239 . . . 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 11211 . . . . 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 4394 . . . 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 1370    e. wcel 1758   A.wral 2795   [_csb 3386    C_ wss 3426   U_ciun 4269   class class class wbr 4390    |-> cmpt 4448   ran crn 4939    Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190   supcsup 7791   RRcr 9382   1c1 9384    + caddc 9386   +oocpnf 9516   -oocmnf 9517   RR*cxr 9518    < clt 9519    <_ cle 9520   NNcn 10423   RR+crp 11092    seqcseq 11907   vol*covol 21062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cc 8705  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-sup 7792  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-q 11055  df-rp 11093  df-ioo 11405  df-ico 11407  df-fz 11539  df-fzo 11650  df-fl 11743  df-seq 11908  df-exp 11967  df-hash 12205  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-clim 13068  df-rlim 13069  df-sum 13266  df-ovol 21064
This theorem is referenced by:  ovoliun2  21105  voliunlem2  21148  voliunlem3  21149  ex-ovoliunnfl  28572
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