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Theorem omssubaddlem 29120
Description: For any small margin  E, we can find a covering approaching the outer measure of a set  A by that margin. (Contributed by Thierry Arnoux, 18-Sep-2019.) (Revised by AV, 4-Oct-2020.)
Hypotheses
Ref Expression
oms.m  |-  M  =  (toOMeas `  R )
oms.o  |-  ( ph  ->  Q  e.  V )
oms.r  |-  ( ph  ->  R : Q --> ( 0 [,] +oo ) )
omssubaddlem.a  |-  ( ph  ->  A  C_  U. Q )
omssubaddlem.m  |-  ( ph  ->  ( M `  A
)  e.  RR )
omssubaddlem.e  |-  ( ph  ->  E  e.  RR+ )
Assertion
Ref Expression
omssubaddlem  |-  ( ph  ->  E. x  e.  {
z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }Σ* w  e.  x ( R `  w )  <  ( ( M `
 A )  +  E ) )
Distinct variable groups:    x, Q, z    x, R, z    x, V, z    ph, x, z   
w, A, x, z   
x, E    x, M    w, Q    w, R    w, V
Allowed substitution hints:    ph( w)    E( z, w)    M( z, w)

Proof of Theorem omssubaddlem
Dummy variables  e 
t  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omssubaddlem.m . . . . . 6  |-  ( ph  ->  ( M `  A
)  e.  RR )
2 omssubaddlem.e . . . . . . 7  |-  ( ph  ->  E  e.  RR+ )
32rpred 11338 . . . . . 6  |-  ( ph  ->  E  e.  RR )
41, 3readdcld 9667 . . . . 5  |-  ( ph  ->  ( ( M `  A )  +  E
)  e.  RR )
54rexrd 9687 . . . 4  |-  ( ph  ->  ( ( M `  A )  +  E
)  e.  RR* )
6 oms.o . . . . . . . . 9  |-  ( ph  ->  Q  e.  V )
7 oms.r . . . . . . . . 9  |-  ( ph  ->  R : Q --> ( 0 [,] +oo ) )
8 omsf 29113 . . . . . . . . 9  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo ) )  ->  (toOMeas `  R ) : ~P U. dom  R --> ( 0 [,] +oo ) )
96, 7, 8syl2anc 666 . . . . . . . 8  |-  ( ph  ->  (toOMeas `  R ) : ~P U. dom  R --> ( 0 [,] +oo ) )
10 oms.m . . . . . . . . 9  |-  M  =  (toOMeas `  R )
1110feq1i 5718 . . . . . . . 8  |-  ( M : ~P U. dom  R --> ( 0 [,] +oo ) 
<->  (toOMeas `  R ) : ~P U. dom  R --> ( 0 [,] +oo ) )
129, 11sylibr 216 . . . . . . 7  |-  ( ph  ->  M : ~P U. dom  R --> ( 0 [,] +oo ) )
13 omssubaddlem.a . . . . . . . . 9  |-  ( ph  ->  A  C_  U. Q )
14 fdm 5731 . . . . . . . . . . 11  |-  ( R : Q --> ( 0 [,] +oo )  ->  dom  R  =  Q )
157, 14syl 17 . . . . . . . . . 10  |-  ( ph  ->  dom  R  =  Q )
1615unieqd 4207 . . . . . . . . 9  |-  ( ph  ->  U. dom  R  = 
U. Q )
1713, 16sseqtr4d 3468 . . . . . . . 8  |-  ( ph  ->  A  C_  U. dom  R
)
18 uniexg 6585 . . . . . . . . . . 11  |-  ( Q  e.  V  ->  U. Q  e.  _V )
196, 18syl 17 . . . . . . . . . 10  |-  ( ph  ->  U. Q  e.  _V )
2013, 19jca 535 . . . . . . . . 9  |-  ( ph  ->  ( A  C_  U. Q  /\  U. Q  e.  _V ) )
21 ssexg 4548 . . . . . . . . 9  |-  ( ( A  C_  U. Q  /\  U. Q  e.  _V )  ->  A  e.  _V )
22 elpwg 3958 . . . . . . . . 9  |-  ( A  e.  _V  ->  ( A  e.  ~P U. dom  R  <-> 
A  C_  U. dom  R
) )
2320, 21, 223syl 18 . . . . . . . 8  |-  ( ph  ->  ( A  e.  ~P U.
dom  R  <->  A  C_  U. dom  R ) )
2417, 23mpbird 236 . . . . . . 7  |-  ( ph  ->  A  e.  ~P U. dom  R )
2512, 24ffvelrnd 6021 . . . . . 6  |-  ( ph  ->  ( M `  A
)  e.  ( 0 [,] +oo ) )
26 elxrge0 11738 . . . . . . 7  |-  ( ( M `  A )  e.  ( 0 [,] +oo )  <->  ( ( M `
 A )  e. 
RR*  /\  0  <_  ( M `  A ) ) )
2726simprbi 466 . . . . . 6  |-  ( ( M `  A )  e.  ( 0 [,] +oo )  ->  0  <_ 
( M `  A
) )
2825, 27syl 17 . . . . 5  |-  ( ph  ->  0  <_  ( M `  A ) )
292rpge0d 11342 . . . . 5  |-  ( ph  ->  0  <_  E )
301, 3, 28, 29addge0d 10186 . . . 4  |-  ( ph  ->  0  <_  ( ( M `  A )  +  E ) )
31 elxrge0 11738 . . . 4  |-  ( ( ( M `  A
)  +  E )  e.  ( 0 [,] +oo )  <->  ( ( ( M `  A )  +  E )  e. 
RR*  /\  0  <_  ( ( M `  A
)  +  E ) ) )
325, 30, 31sylanbrc 669 . . 3  |-  ( ph  ->  ( ( M `  A )  +  E
)  e.  ( 0 [,] +oo ) )
3310fveq1i 5864 . . . . 5  |-  ( M `
 A )  =  ( (toOMeas `  R
) `  A )
34 omsfval 29111 . . . . . 6  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  ( (toOMeas `  R
) `  A )  = inf ( ran  ( x  e.  { z  e. 
~P dom  R  | 
( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ*
w  e.  x ( R `  w ) ) ,  ( 0 [,] +oo ) ,  <  ) )
356, 7, 13, 34syl3anc 1267 . . . . 5  |-  ( ph  ->  ( (toOMeas `  R
) `  A )  = inf ( ran  ( x  e.  { z  e. 
~P dom  R  | 
( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ*
w  e.  x ( R `  w ) ) ,  ( 0 [,] +oo ) ,  <  ) )
3633, 35syl5req 2497 . . . 4  |-  ( ph  -> inf ( ran  ( x  e.  { z  e. 
~P dom  R  | 
( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ*
w  e.  x ( R `  w ) ) ,  ( 0 [,] +oo ) ,  <  )  =  ( M `  A ) )
371, 2ltaddrpd 11368 . . . 4  |-  ( ph  ->  ( M `  A
)  <  ( ( M `  A )  +  E ) )
3836, 37eqbrtrd 4422 . . 3  |-  ( ph  -> inf ( ran  ( x  e.  { z  e. 
~P dom  R  | 
( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ*
w  e.  x ( R `  w ) ) ,  ( 0 [,] +oo ) ,  <  )  <  (
( M `  A
)  +  E ) )
39 iccssxr 11714 . . . . . 6  |-  ( 0 [,] +oo )  C_  RR*
40 xrltso 11437 . . . . . 6  |-  <  Or  RR*
41 soss 4772 . . . . . 6  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  (  <  Or  RR* 
->  <  Or  ( 0 [,] +oo ) ) )
4239, 40, 41mp2 9 . . . . 5  |-  <  Or  ( 0 [,] +oo )
4342a1i 11 . . . 4  |-  ( ph  ->  <  Or  ( 0 [,] +oo ) )
44 omscl 29112 . . . . . 6  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  e.  ~P U. dom  R )  ->  ran  ( x  e.  { z  e. 
~P dom  R  | 
( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ*
w  e.  x ( R `  w ) )  C_  ( 0 [,] +oo ) )
456, 7, 24, 44syl3anc 1267 . . . . 5  |-  ( ph  ->  ran  ( x  e. 
{ z  e.  ~P dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) )  C_  (
0 [,] +oo )
)
46 xrge0infss 28333 . . . . 5  |-  ( ran  ( x  e.  {
z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) )  C_  (
0 [,] +oo )  ->  E. e  e.  ( 0 [,] +oo )
( A. t  e. 
ran  ( x  e. 
{ z  e.  ~P dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) )  -.  t  <  e  /\  A. t  e.  ( 0 [,] +oo ) ( e  < 
t  ->  E. u  e.  ran  ( x  e. 
{ z  e.  ~P dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) ) u  < 
t ) ) )
4745, 46syl 17 . . . 4  |-  ( ph  ->  E. e  e.  ( 0 [,] +oo )
( A. t  e. 
ran  ( x  e. 
{ z  e.  ~P dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) )  -.  t  <  e  /\  A. t  e.  ( 0 [,] +oo ) ( e  < 
t  ->  E. u  e.  ran  ( x  e. 
{ z  e.  ~P dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) ) u  < 
t ) ) )
4843, 47infglb 8003 . . 3  |-  ( ph  ->  ( ( ( ( M `  A )  +  E )  e.  ( 0 [,] +oo )  /\ inf ( ran  (
x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) ) ,  ( 0 [,] +oo ) ,  <  )  <  (
( M `  A
)  +  E ) )  ->  E. u  e.  ran  ( x  e. 
{ z  e.  ~P dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) ) u  < 
( ( M `  A )  +  E
) ) )
4932, 38, 48mp2and 684 . 2  |-  ( ph  ->  E. u  e.  ran  ( x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) ) u  < 
( ( M `  A )  +  E
) )
50 eqid 2450 . . . . . . . 8  |-  ( x  e.  { z  e. 
~P dom  R  | 
( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ*
w  e.  x ( R `  w ) )  =  ( x  e.  { z  e. 
~P dom  R  | 
( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ*
w  e.  x ( R `  w ) )
51 esumex 28843 . . . . . . . 8  |- Σ* w  e.  x
( R `  w
)  e.  _V
5250, 51elrnmpti 5084 . . . . . . 7  |-  ( u  e.  ran  ( x  e.  { z  e. 
~P dom  R  | 
( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ*
w  e.  x ( R `  w ) )  <->  E. x  e.  {
z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) } u  = Σ* w  e.  x ( R `
 w ) )
5352anbi1i 700 . . . . . 6  |-  ( ( u  e.  ran  (
x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) )  /\  u  <  ( ( M `  A )  +  E
) )  <->  ( E. x  e.  { z  e.  ~P dom  R  | 
( A  C_  U. z  /\  z  ~<_  om ) } u  = Σ* w  e.  x
( R `  w
)  /\  u  <  ( ( M `  A
)  +  E ) ) )
54 r19.41v 2941 . . . . . 6  |-  ( E. x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  ( u  = Σ* w  e.  x ( R `  w )  /\  u  <  (
( M `  A
)  +  E ) )  <->  ( E. x  e.  { z  e.  ~P dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) } u  = Σ* w  e.  x ( R `
 w )  /\  u  <  ( ( M `
 A )  +  E ) ) )
5553, 54bitr4i 256 . . . . 5  |-  ( ( u  e.  ran  (
x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) )  /\  u  <  ( ( M `  A )  +  E
) )  <->  E. x  e.  { z  e.  ~P dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) }  ( u  = Σ* w  e.  x ( R `  w )  /\  u  <  (
( M `  A
)  +  E ) ) )
5655exbii 1717 . . . 4  |-  ( E. u ( u  e. 
ran  ( x  e. 
{ z  e.  ~P dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) )  /\  u  <  ( ( M `  A )  +  E
) )  <->  E. u E. x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  ( u  = Σ* w  e.  x ( R `  w )  /\  u  <  (
( M `  A
)  +  E ) ) )
57 df-rex 2742 . . . 4  |-  ( E. u  e.  ran  (
x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) ) u  < 
( ( M `  A )  +  E
)  <->  E. u ( u  e.  ran  ( x  e.  { z  e. 
~P dom  R  | 
( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ*
w  e.  x ( R `  w ) )  /\  u  < 
( ( M `  A )  +  E
) ) )
58 rexcom4 3066 . . . 4  |-  ( E. x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) } E. u
( u  = Σ* w  e.  x ( R `  w )  /\  u  <  ( ( M `  A )  +  E
) )  <->  E. u E. x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  ( u  = Σ* w  e.  x ( R `  w )  /\  u  <  (
( M `  A
)  +  E ) ) )
5956, 57, 583bitr4i 281 . . 3  |-  ( E. u  e.  ran  (
x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) ) u  < 
( ( M `  A )  +  E
)  <->  E. x  e.  {
z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) } E. u
( u  = Σ* w  e.  x ( R `  w )  /\  u  <  ( ( M `  A )  +  E
) ) )
60 breq1 4404 . . . . . 6  |-  ( u  = Σ* w  e.  x ( R `  w )  ->  ( u  < 
( ( M `  A )  +  E
)  <-> Σ* w  e.  x ( R `  w )  <  ( ( M `  A )  +  E
) ) )
6160biimpa 487 . . . . 5  |-  ( ( u  = Σ* w  e.  x
( R `  w
)  /\  u  <  ( ( M `  A
)  +  E ) )  -> Σ* w  e.  x
( R `  w
)  <  ( ( M `  A )  +  E ) )
6261exlimiv 1775 . . . 4  |-  ( E. u ( u  = Σ* w  e.  x ( R `
 w )  /\  u  <  ( ( M `
 A )  +  E ) )  -> Σ* w  e.  x ( R `  w )  <  (
( M `  A
)  +  E ) )
6362reximi 2854 . . 3  |-  ( E. x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) } E. u
( u  = Σ* w  e.  x ( R `  w )  /\  u  <  ( ( M `  A )  +  E
) )  ->  E. x  e.  { z  e.  ~P dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) }Σ* w  e.  x ( R `  w )  <  ( ( M `
 A )  +  E ) )
6459, 63sylbi 199 . 2  |-  ( E. u  e.  ran  (
x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) ) u  < 
( ( M `  A )  +  E
)  ->  E. x  e.  { z  e.  ~P dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) }Σ* w  e.  x ( R `  w )  <  ( ( M `
 A )  +  E ) )
6549, 64syl 17 1  |-  ( ph  ->  E. x  e.  {
z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }Σ* w  e.  x ( R `  w )  <  ( ( M `
 A )  +  E ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443   E.wex 1662    e. wcel 1886   A.wral 2736   E.wrex 2737   {crab 2740   _Vcvv 3044    C_ wss 3403   ~Pcpw 3950   U.cuni 4197   class class class wbr 4401    |-> cmpt 4460    Or wor 4753   dom cdm 4833   ran crn 4834   -->wf 5577   ` cfv 5581  (class class class)co 6288   omcom 6689    ~<_ cdom 7564  infcinf 7952   RRcr 9535   0cc0 9536    + caddc 9539   +oocpnf 9669   RR*cxr 9671    < clt 9672    <_ cle 9673   RR+crp 11299   [,]cicc 11635  Σ*cesum 28841  toOMeascoms 29106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xadd 11407  df-ioo 11636  df-ioc 11637  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-seq 12211  df-hash 12513  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-tset 15202  df-ple 15203  df-ds 15205  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-ordt 15392  df-xrs 15393  df-mre 15485  df-mrc 15486  df-acs 15488  df-ps 16439  df-tsr 16440  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-cntz 16964  df-cmn 17425  df-fbas 18960  df-fg 18961  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-ntr 20028  df-nei 20107  df-cn 20236  df-haus 20324  df-fil 20854  df-fm 20946  df-flim 20947  df-flf 20948  df-tsms 21134  df-esum 28842  df-oms 29108
This theorem is referenced by: (None)
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