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Theorem omssubaddlem 29200
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 11364 . . . . . 6  |-  ( ph  ->  E  e.  RR )
41, 3readdcld 9688 . . . . 5  |-  ( ph  ->  ( ( M `  A )  +  E
)  e.  RR )
54rexrd 9708 . . . 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 29193 . . . . . . . . 9  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo ) )  ->  (toOMeas `  R ) : ~P U. dom  R --> ( 0 [,] +oo ) )
96, 7, 8syl2anc 673 . . . . . . . 8  |-  ( ph  ->  (toOMeas `  R ) : ~P U. dom  R --> ( 0 [,] +oo ) )
10 oms.m . . . . . . . . 9  |-  M  =  (toOMeas `  R )
1110feq1i 5730 . . . . . . . 8  |-  ( M : ~P U. dom  R --> ( 0 [,] +oo ) 
<->  (toOMeas `  R ) : ~P U. dom  R --> ( 0 [,] +oo ) )
129, 11sylibr 217 . . . . . . 7  |-  ( ph  ->  M : ~P U. dom  R --> ( 0 [,] +oo ) )
13 omssubaddlem.a . . . . . . . . 9  |-  ( ph  ->  A  C_  U. Q )
14 fdm 5745 . . . . . . . . . . 11  |-  ( R : Q --> ( 0 [,] +oo )  ->  dom  R  =  Q )
157, 14syl 17 . . . . . . . . . 10  |-  ( ph  ->  dom  R  =  Q )
1615unieqd 4200 . . . . . . . . 9  |-  ( ph  ->  U. dom  R  = 
U. Q )
1713, 16sseqtr4d 3455 . . . . . . . 8  |-  ( ph  ->  A  C_  U. dom  R
)
18 uniexg 6607 . . . . . . . . . . 11  |-  ( Q  e.  V  ->  U. Q  e.  _V )
196, 18syl 17 . . . . . . . . . 10  |-  ( ph  ->  U. Q  e.  _V )
2013, 19jca 541 . . . . . . . . 9  |-  ( ph  ->  ( A  C_  U. Q  /\  U. Q  e.  _V ) )
21 ssexg 4542 . . . . . . . . 9  |-  ( ( A  C_  U. Q  /\  U. Q  e.  _V )  ->  A  e.  _V )
22 elpwg 3950 . . . . . . . . 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 240 . . . . . . 7  |-  ( ph  ->  A  e.  ~P U. dom  R )
2512, 24ffvelrnd 6038 . . . . . 6  |-  ( ph  ->  ( M `  A
)  e.  ( 0 [,] +oo ) )
26 elxrge0 11767 . . . . . . 7  |-  ( ( M `  A )  e.  ( 0 [,] +oo )  <->  ( ( M `
 A )  e. 
RR*  /\  0  <_  ( M `  A ) ) )
2726simprbi 471 . . . . . 6  |-  ( ( M `  A )  e.  ( 0 [,] +oo )  ->  0  <_ 
( M `  A
) )
2825, 27syl 17 . . . . 5  |-  ( ph  ->  0  <_  ( M `  A ) )
292rpge0d 11368 . . . . 5  |-  ( ph  ->  0  <_  E )
301, 3, 28, 29addge0d 10210 . . . 4  |-  ( ph  ->  0  <_  ( ( M `  A )  +  E ) )
31 elxrge0 11767 . . . 4  |-  ( ( ( M `  A
)  +  E )  e.  ( 0 [,] +oo )  <->  ( ( ( M `  A )  +  E )  e. 
RR*  /\  0  <_  ( ( M `  A
)  +  E ) ) )
325, 30, 31sylanbrc 677 . . 3  |-  ( ph  ->  ( ( M `  A )  +  E
)  e.  ( 0 [,] +oo ) )
3310fveq1i 5880 . . . . 5  |-  ( M `
 A )  =  ( (toOMeas `  R
) `  A )
34 omsfval 29191 . . . . . 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 1292 . . . . 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 2518 . . . 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 11394 . . . 4  |-  ( ph  ->  ( M `  A
)  <  ( ( M `  A )  +  E ) )
3836, 37eqbrtrd 4416 . . 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 11742 . . . . . 6  |-  ( 0 [,] +oo )  C_  RR*
40 xrltso 11463 . . . . . 6  |-  <  Or  RR*
41 soss 4778 . . . . . 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 29192 . . . . . 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 1292 . . . . 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 28415 . . . . 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 8024 . . 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 693 . 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 2471 . . . . . . . 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 28924 . . . . . . . 8  |- Σ* w  e.  x
( R `  w
)  e.  _V
5250, 51elrnmpti 5091 . . . . . . 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 709 . . . . . 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 2928 . . . . . 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 260 . . . . 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 1726 . . . 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 2762 . . . 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 3053 . . . 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 285 . . 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 4398 . . . . . 6  |-  ( u  = Σ* w  e.  x ( R `  w )  ->  ( u  < 
( ( M `  A )  +  E
)  <-> Σ* w  e.  x ( R `  w )  <  ( ( M `  A )  +  E
) ) )
6160biimpa 492 . . . . 5  |-  ( ( u  = Σ* w  e.  x
( R `  w
)  /\  u  <  ( ( M `  A
)  +  E ) )  -> Σ* w  e.  x
( R `  w
)  <  ( ( M `  A )  +  E ) )
6261exlimiv 1784 . . . 4  |-  ( E. u ( u  = Σ* w  e.  x ( R `
 w )  /\  u  <  ( ( M `
 A )  +  E ) )  -> Σ* w  e.  x ( R `  w )  <  (
( M `  A
)  +  E ) )
6362reximi 2852 . . 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 200 . 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 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031    C_ wss 3390   ~Pcpw 3942   U.cuni 4190   class class class wbr 4395    |-> cmpt 4454    Or wor 4759   dom cdm 4839   ran crn 4840   -->wf 5585   ` cfv 5589  (class class class)co 6308   omcom 6711    ~<_ cdom 7585  infcinf 7973   RRcr 9556   0cc0 9557    + caddc 9560   +oocpnf 9690   RR*cxr 9692    < clt 9693    <_ cle 9694   RR+crp 11325   [,]cicc 11663  Σ*cesum 28922  toOMeascoms 29186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xadd 11433  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-tset 15287  df-ple 15288  df-ds 15290  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-ordt 15477  df-xrs 15478  df-mre 15570  df-mrc 15571  df-acs 15573  df-ps 16524  df-tsr 16525  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-cntz 17049  df-cmn 17510  df-fbas 19044  df-fg 19045  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-ntr 20112  df-nei 20191  df-cn 20320  df-haus 20408  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-tsms 21219  df-esum 28923  df-oms 29188
This theorem is referenced by: (None)
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