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Theorem ovoliun2 19407
Description: The Lebesgue outer measure function is countably sub-additive. (This version is a little easier to read, but does not allow infinite values like ovoliun 19406.) (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 )
ovoliun2.t  |-  ( ph  ->  T  e.  dom  ~~>  )
Assertion
Ref Expression
ovoliun2  |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A
)  <_  sum_ n  e.  NN  ( vol * `  A ) )
Distinct variable group:    ph, n
Allowed substitution hints:    A( n)    T( n)    G( n)

Proof of Theorem ovoliun2
Dummy variables  k  m  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovoliun.t . . 3  |-  T  =  seq  1 (  +  ,  G )
2 ovoliun.g . . 3  |-  G  =  ( n  e.  NN  |->  ( vol * `  A
) )
3 ovoliun.a . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  A  C_  RR )
4 ovoliun.v . . 3  |-  ( (
ph  /\  n  e.  NN )  ->  ( vol
* `  A )  e.  RR )
51, 2, 3, 4ovoliun 19406 . 2  |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A
)  <_  sup ( ran  T ,  RR* ,  <  ) )
6 nnuz 10526 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
7 1z 10316 . . . . . . . . 9  |-  1  e.  ZZ
87a1i 11 . . . . . . . 8  |-  ( ph  ->  1  e.  ZZ )
9 fvex 5745 . . . . . . . . . . 11  |-  ( vol
* `  [_ m  /  n ]_ A )  e. 
_V
10 nfcv 2574 . . . . . . . . . . . . . 14  |-  F/_ m
( vol * `  A )
11 nfcv 2574 . . . . . . . . . . . . . . 15  |-  F/_ n vol *
12 nfcsb1v 3285 . . . . . . . . . . . . . . 15  |-  F/_ n [_ m  /  n ]_ A
1311, 12nffv 5738 . . . . . . . . . . . . . 14  |-  F/_ n
( vol * `  [_ m  /  n ]_ A )
14 csbeq1a 3261 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  A  =  [_ m  /  n ]_ A )
1514fveq2d 5735 . . . . . . . . . . . . . 14  |-  ( n  =  m  ->  ( vol * `  A )  =  ( vol * `  [_ m  /  n ]_ A ) )
1610, 13, 15cbvmpt 4302 . . . . . . . . . . . . 13  |-  ( n  e.  NN  |->  ( vol
* `  A )
)  =  ( m  e.  NN  |->  ( vol
* `  [_ m  /  n ]_ A ) )
172, 16eqtri 2458 . . . . . . . . . . . 12  |-  G  =  ( m  e.  NN  |->  ( vol * `  [_ m  /  n ]_ A ) )
1817fvmpt2 5815 . . . . . . . . . . 11  |-  ( ( m  e.  NN  /\  ( vol * `  [_ m  /  n ]_ A )  e.  _V )  -> 
( G `  m
)  =  ( vol
* `  [_ m  /  n ]_ A ) )
199, 18mpan2 654 . . . . . . . . . 10  |-  ( m  e.  NN  ->  ( G `  m )  =  ( vol * `  [_ m  /  n ]_ A ) )
2019adantl 454 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  =  ( vol * `  [_ m  /  n ]_ A ) )
214ralrimiva 2791 . . . . . . . . . . 11  |-  ( ph  ->  A. n  e.  NN  ( vol * `  A
)  e.  RR )
2210nfel1 2584 . . . . . . . . . . . 12  |-  F/ m
( vol * `  A )  e.  RR
2313nfel1 2584 . . . . . . . . . . . 12  |-  F/ n
( vol * `  [_ m  /  n ]_ A )  e.  RR
2415eleq1d 2504 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
( vol * `  A )  e.  RR  <->  ( vol * `  [_ m  /  n ]_ A )  e.  RR ) )
2522, 23, 24cbvral 2930 . . . . . . . . . . 11  |-  ( A. n  e.  NN  ( vol * `  A )  e.  RR  <->  A. m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
2621, 25sylib 190 . . . . . . . . . 10  |-  ( ph  ->  A. m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
2726r19.21bi 2806 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  NN )  ->  ( vol
* `  [_ m  /  n ]_ A )  e.  RR )
2820, 27eqeltrd 2512 . . . . . . . 8  |-  ( (
ph  /\  m  e.  NN )  ->  ( G `
 m )  e.  RR )
296, 8, 28serfre 11357 . . . . . . 7  |-  ( ph  ->  seq  1 (  +  ,  G ) : NN --> RR )
301feq1i 5588 . . . . . . 7  |-  ( T : NN --> RR  <->  seq  1
(  +  ,  G
) : NN --> RR )
3129, 30sylibr 205 . . . . . 6  |-  ( ph  ->  T : NN --> RR )
32 frn 5600 . . . . . 6  |-  ( T : NN --> RR  ->  ran 
T  C_  RR )
3331, 32syl 16 . . . . 5  |-  ( ph  ->  ran  T  C_  RR )
34 1nn 10016 . . . . . . . 8  |-  1  e.  NN
35 fdm 5598 . . . . . . . . 9  |-  ( T : NN --> RR  ->  dom 
T  =  NN )
3631, 35syl 16 . . . . . . . 8  |-  ( ph  ->  dom  T  =  NN )
3734, 36syl5eleqr 2525 . . . . . . 7  |-  ( ph  ->  1  e.  dom  T
)
38 ne0i 3636 . . . . . . 7  |-  ( 1  e.  dom  T  ->  dom  T  =/=  (/) )
3937, 38syl 16 . . . . . 6  |-  ( ph  ->  dom  T  =/=  (/) )
40 dm0rn0 5089 . . . . . . 7  |-  ( dom 
T  =  (/)  <->  ran  T  =  (/) )
4140necon3bii 2635 . . . . . 6  |-  ( dom 
T  =/=  (/)  <->  ran  T  =/=  (/) )
4239, 41sylib 190 . . . . 5  |-  ( ph  ->  ran  T  =/=  (/) )
43 ovoliun2.t . . . . . . . . 9  |-  ( ph  ->  T  e.  dom  ~~>  )
441, 43syl5eqelr 2523 . . . . . . . 8  |-  ( ph  ->  seq  1 (  +  ,  G )  e. 
dom 
~~>  )
456, 8, 20, 27, 44isumrecl 12554 . . . . . . 7  |-  ( ph  -> 
sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
46 elfznn 11085 . . . . . . . . . . . . 13  |-  ( m  e.  ( 1 ... k )  ->  m  e.  NN )
4746adantl 454 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  m  e.  NN )
4847, 19syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( G `  m )  =  ( vol * `  [_ m  /  n ]_ A ) )
49 simpr 449 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  NN )
5049, 6syl6eleq 2528 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  NN )  ->  k  e.  ( ZZ>= `  1 )
)
51 simpl 445 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  NN )  ->  ph )
5251, 46, 27syl2an 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( vol * `  [_ m  /  n ]_ A )  e.  RR )
5352recnd 9119 . . . . . . . . . . 11  |-  ( ( ( ph  /\  k  e.  NN )  /\  m  e.  ( 1 ... k
) )  ->  ( vol * `  [_ m  /  n ]_ A )  e.  CC )
5448, 50, 53fsumser 12529 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol * `  [_ m  /  n ]_ A )  =  (  seq  1 (  +  ,  G ) `  k ) )
551fveq1i 5732 . . . . . . . . . 10  |-  ( T `
 k )  =  (  seq  1 (  +  ,  G ) `
 k )
5654, 55syl6eqr 2488 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol * `  [_ m  /  n ]_ A )  =  ( T `  k ) )
57 fzfid 11317 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... k
)  e.  Fin )
58 elfznn 11085 . . . . . . . . . . . . 13  |-  ( n  e.  ( 1 ... k )  ->  n  e.  NN )
5958ssriv 3354 . . . . . . . . . . . 12  |-  ( 1 ... k )  C_  NN
6059a1i 11 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... k
)  C_  NN )
613ralrimiva 2791 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. n  e.  NN  A  C_  RR )
62 nfv 1630 . . . . . . . . . . . . . . 15  |-  F/ m  A  C_  RR
63 nfcv 2574 . . . . . . . . . . . . . . . 16  |-  F/_ n RR
6412, 63nfss 3343 . . . . . . . . . . . . . . 15  |-  F/ n [_ m  /  n ]_ A  C_  RR
6514sseq1d 3377 . . . . . . . . . . . . . . 15  |-  ( n  =  m  ->  ( A  C_  RR  <->  [_ m  /  n ]_ A  C_  RR ) )
6662, 64, 65cbvral 2930 . . . . . . . . . . . . . 14  |-  ( A. n  e.  NN  A  C_  RR  <->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
6761, 66sylib 190 . . . . . . . . . . . . 13  |-  ( ph  ->  A. m  e.  NN  [_ m  /  n ]_ A  C_  RR )
6867r19.21bi 2806 . . . . . . . . . . . 12  |-  ( (
ph  /\  m  e.  NN )  ->  [_ m  /  n ]_ A  C_  RR )
69 ovolge0 19382 . . . . . . . . . . . 12  |-  ( [_ m  /  n ]_ A  C_  RR  ->  0  <_  ( vol * `  [_ m  /  n ]_ A ) )
7068, 69syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  NN )  ->  0  <_ 
( vol * `  [_ m  /  n ]_ A ) )
716, 8, 57, 60, 20, 27, 70, 44isumless 12630 . . . . . . . . . 10  |-  ( ph  -> 
sum_ m  e.  (
1 ... k ) ( vol * `  [_ m  /  n ]_ A )  <_  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) )
7271adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  NN )  ->  sum_ m  e.  ( 1 ... k
) ( vol * `  [_ m  /  n ]_ A )  <_  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) )
7356, 72eqbrtrrd 4237 . . . . . . . 8  |-  ( (
ph  /\  k  e.  NN )  ->  ( T `
 k )  <_  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) )
7473ralrimiva 2791 . . . . . . 7  |-  ( ph  ->  A. k  e.  NN  ( T `  k )  <_  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) )
75 breq2 4219 . . . . . . . . 9  |-  ( x  =  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  ->  ( ( T `
 k )  <_  x 
<->  ( T `  k
)  <_  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) ) )
7675ralbidv 2727 . . . . . . . 8  |-  ( x  =  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  ->  ( A. k  e.  NN  ( T `  k )  <_  x  <->  A. k  e.  NN  ( T `  k )  <_ 
sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) ) )
7776rspcev 3054 . . . . . . 7  |-  ( (
sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  e.  RR  /\  A. k  e.  NN  ( T `  k )  <_ 
sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) )  ->  E. x  e.  RR  A. k  e.  NN  ( T `  k )  <_  x
)
7845, 74, 77syl2anc 644 . . . . . 6  |-  ( ph  ->  E. x  e.  RR  A. k  e.  NN  ( T `  k )  <_  x )
79 ffn 5594 . . . . . . . . 9  |-  ( T : NN --> RR  ->  T  Fn  NN )
8031, 79syl 16 . . . . . . . 8  |-  ( ph  ->  T  Fn  NN )
81 breq1 4218 . . . . . . . . 9  |-  ( z  =  ( T `  k )  ->  (
z  <_  x  <->  ( T `  k )  <_  x
) )
8281ralrn 5876 . . . . . . . 8  |-  ( T  Fn  NN  ->  ( A. z  e.  ran  T  z  <_  x  <->  A. k  e.  NN  ( T `  k )  <_  x
) )
8380, 82syl 16 . . . . . . 7  |-  ( ph  ->  ( A. z  e. 
ran  T  z  <_  x  <->  A. k  e.  NN  ( T `  k )  <_  x ) )
8483rexbidv 2728 . . . . . 6  |-  ( ph  ->  ( E. x  e.  RR  A. z  e. 
ran  T  z  <_  x  <->  E. x  e.  RR  A. k  e.  NN  ( T `  k )  <_  x ) )
8578, 84mpbird 225 . . . . 5  |-  ( ph  ->  E. x  e.  RR  A. z  e.  ran  T  z  <_  x )
86 supxrre 10911 . . . . 5  |-  ( ( ran  T  C_  RR  /\ 
ran  T  =/=  (/)  /\  E. x  e.  RR  A. z  e.  ran  T  z  <_  x )  ->  sup ( ran  T ,  RR* ,  <  )  =  sup ( ran  T ,  RR ,  <  ) )
8733, 42, 85, 86syl3anc 1185 . . . 4  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sup ( ran  T ,  RR ,  <  )
)
886, 1, 8, 20, 27, 70, 78isumsup 12632 . . . 4  |-  ( ph  -> 
sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A )  =  sup ( ran 
T ,  RR ,  <  ) )
8987, 88eqtr4d 2473 . . 3  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sum_ m  e.  NN  ( vol * `  [_ m  /  n ]_ A ) )
9010, 13, 15cbvsumi 12496 . . 3  |-  sum_ n  e.  NN  ( vol * `  A )  =  sum_ m  e.  NN  ( vol
* `  [_ m  /  n ]_ A )
9189, 90syl6eqr 2488 . 2  |-  ( ph  ->  sup ( ran  T ,  RR* ,  <  )  =  sum_ n  e.  NN  ( vol * `  A
) )
925, 91breqtrd 4239 1  |-  ( ph  ->  ( vol * `  U_ n  e.  NN  A
)  <_  sum_ n  e.  NN  ( vol * `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   _Vcvv 2958   [_csb 3253    C_ wss 3322   (/)c0 3630   U_ciun 4095   class class class wbr 4215    e. cmpt 4269   dom cdm 4881   ran crn 4882    Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084   supcsup 7448   RRcr 8994   0cc0 8995   1c1 8996    + caddc 8998   RR*cxr 9124    < clt 9125    <_ cle 9126   NNcn 10005   ZZcz 10287   ZZ>=cuz 10493   ...cfz 11048    seq cseq 11328    ~~> cli 12283   sum_csu 12484   vol *covol 19364
This theorem is referenced by:  ovoliunnul  19408  vitalilem5  19509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cc 8320  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-q 10580  df-rp 10618  df-ioo 10925  df-ico 10927  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485  df-ovol 19366
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