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Theorem omssubaddlemOLD 29131
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.) Obsolete version of omssubaddlem 29127 as of 4-Oct-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
omsOLD.m  |-  M  =  (toOMeas `  R )
omsOLD.o  |-  ( ph  ->  Q  e.  V )
omsOLD.r  |-  ( ph  ->  R : Q --> ( 0 [,] +oo ) )
omssubaddlemOLD.a  |-  ( ph  ->  A  C_  U. Q )
omssubaddlemOLD.m  |-  ( ph  ->  ( M `  A
)  e.  RR )
omssubaddlemOLD.e  |-  ( ph  ->  E  e.  RR+ )
Assertion
Ref Expression
omssubaddlemOLD  |-  ( 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 omssubaddlemOLD
Dummy variables  e 
t  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omssubaddlemOLD.m . . . . . . . 8  |-  ( ph  ->  ( M `  A
)  e.  RR )
2 omssubaddlemOLD.e . . . . . . . . 9  |-  ( ph  ->  E  e.  RR+ )
32rpred 11341 . . . . . . . 8  |-  ( ph  ->  E  e.  RR )
41, 3readdcld 9670 . . . . . . 7  |-  ( ph  ->  ( ( M `  A )  +  E
)  e.  RR )
54rexrd 9690 . . . . . 6  |-  ( ph  ->  ( ( M `  A )  +  E
)  e.  RR* )
6 omsOLD.o . . . . . . . . . . 11  |-  ( ph  ->  Q  e.  V )
7 omsOLD.r . . . . . . . . . . 11  |-  ( ph  ->  R : Q --> ( 0 [,] +oo ) )
8 omsfOLD 29124 . . . . . . . . . . 11  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo ) )  ->  (toOMeas `  R ) : ~P U. dom  R --> ( 0 [,] +oo ) )
96, 7, 8syl2anc 667 . . . . . . . . . 10  |-  ( ph  ->  (toOMeas `  R ) : ~P U. dom  R --> ( 0 [,] +oo ) )
10 omsOLD.m . . . . . . . . . . 11  |-  M  =  (toOMeas `  R )
1110feq1i 5720 . . . . . . . . . 10  |-  ( M : ~P U. dom  R --> ( 0 [,] +oo ) 
<->  (toOMeas `  R ) : ~P U. dom  R --> ( 0 [,] +oo ) )
129, 11sylibr 216 . . . . . . . . 9  |-  ( ph  ->  M : ~P U. dom  R --> ( 0 [,] +oo ) )
13 omssubaddlemOLD.a . . . . . . . . . . 11  |-  ( ph  ->  A  C_  U. Q )
14 fdm 5733 . . . . . . . . . . . . 13  |-  ( R : Q --> ( 0 [,] +oo )  ->  dom  R  =  Q )
157, 14syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  dom  R  =  Q )
1615unieqd 4208 . . . . . . . . . . 11  |-  ( ph  ->  U. dom  R  = 
U. Q )
1713, 16sseqtr4d 3469 . . . . . . . . . 10  |-  ( ph  ->  A  C_  U. dom  R
)
18 uniexg 6588 . . . . . . . . . . . . 13  |-  ( Q  e.  V  ->  U. Q  e.  _V )
196, 18syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  U. Q  e.  _V )
2013, 19jca 535 . . . . . . . . . . 11  |-  ( ph  ->  ( A  C_  U. Q  /\  U. Q  e.  _V ) )
21 ssexg 4549 . . . . . . . . . . 11  |-  ( ( A  C_  U. Q  /\  U. Q  e.  _V )  ->  A  e.  _V )
22 elpwg 3959 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  ( A  e.  ~P U. dom  R  <-> 
A  C_  U. dom  R
) )
2320, 21, 223syl 18 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  ~P U.
dom  R  <->  A  C_  U. dom  R ) )
2417, 23mpbird 236 . . . . . . . . 9  |-  ( ph  ->  A  e.  ~P U. dom  R )
2512, 24ffvelrnd 6023 . . . . . . . 8  |-  ( ph  ->  ( M `  A
)  e.  ( 0 [,] +oo ) )
26 elxrge0 11741 . . . . . . . . 9  |-  ( ( M `  A )  e.  ( 0 [,] +oo )  <->  ( ( M `
 A )  e. 
RR*  /\  0  <_  ( M `  A ) ) )
2726simprbi 466 . . . . . . . 8  |-  ( ( M `  A )  e.  ( 0 [,] +oo )  ->  0  <_ 
( M `  A
) )
2825, 27syl 17 . . . . . . 7  |-  ( ph  ->  0  <_  ( M `  A ) )
292rpge0d 11345 . . . . . . 7  |-  ( ph  ->  0  <_  E )
301, 3, 28, 29addge0d 10189 . . . . . 6  |-  ( ph  ->  0  <_  ( ( M `  A )  +  E ) )
315, 30jca 535 . . . . 5  |-  ( ph  ->  ( ( ( M `
 A )  +  E )  e.  RR*  /\  0  <_  ( ( M `  A )  +  E ) ) )
32 elxrge0 11741 . . . . 5  |-  ( ( ( M `  A
)  +  E )  e.  ( 0 [,] +oo )  <->  ( ( ( M `  A )  +  E )  e. 
RR*  /\  0  <_  ( ( M `  A
)  +  E ) ) )
3331, 32sylibr 216 . . . 4  |-  ( ph  ->  ( ( M `  A )  +  E
)  e.  ( 0 [,] +oo ) )
341, 2ltaddrpd 11371 . . . . . 6  |-  ( ph  ->  ( M `  A
)  <  ( ( M `  A )  +  E ) )
35 ovex 6318 . . . . . . 7  |-  ( ( M `  A )  +  E )  e. 
_V
36 fvex 5875 . . . . . . 7  |-  ( M `
 A )  e. 
_V
3735, 36brcnv 5017 . . . . . 6  |-  ( ( ( M `  A
)  +  E ) `'  <  ( M `  A )  <->  ( M `  A )  <  (
( M `  A
)  +  E ) )
3834, 37sylibr 216 . . . . 5  |-  ( ph  ->  ( ( M `  A )  +  E
) `'  <  ( M `  A )
)
3910fveq1i 5866 . . . . . 6  |-  ( M `
 A )  =  ( (toOMeas `  R
) `  A )
40 omsfvalOLD 29122 . . . . . . 7  |-  ( ( Q  e.  V  /\  R : Q --> ( 0 [,] +oo )  /\  A  C_  U. Q )  ->  ( (toOMeas `  R
) `  A )  =  sup ( ran  (
x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) ) ,  ( 0 [,] +oo ) ,  `'  <  ) )
416, 7, 13, 40syl3anc 1268 . . . . . 6  |-  ( ph  ->  ( (toOMeas `  R
) `  A )  =  sup ( ran  (
x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) ) ,  ( 0 [,] +oo ) ,  `'  <  ) )
4239, 41syl5eq 2497 . . . . 5  |-  ( ph  ->  ( M `  A
)  =  sup ( ran  ( x  e.  {
z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) ) ,  ( 0 [,] +oo ) ,  `'  <  ) )
4338, 42breqtrd 4427 . . . 4  |-  ( ph  ->  ( ( M `  A )  +  E
) `'  <  sup ( ran  ( x  e. 
{ z  e.  ~P dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) ) ,  ( 0 [,] +oo ) ,  `'  <  ) )
4433, 43jca 535 . . 3  |-  ( ph  ->  ( ( ( M `
 A )  +  E )  e.  ( 0 [,] +oo )  /\  ( ( M `  A )  +  E
) `'  <  sup ( ran  ( x  e. 
{ z  e.  ~P dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) ) ,  ( 0 [,] +oo ) ,  `'  <  ) ) )
45 iccssxr 11717 . . . . . . 7  |-  ( 0 [,] +oo )  C_  RR*
46 xrltso 11440 . . . . . . 7  |-  <  Or  RR*
47 soss 4773 . . . . . . 7  |-  ( ( 0 [,] +oo )  C_ 
RR*  ->  (  <  Or  RR* 
->  <  Or  ( 0 [,] +oo ) ) )
4845, 46, 47mp2 9 . . . . . 6  |-  <  Or  ( 0 [,] +oo )
49 cnvso 5375 . . . . . 6  |-  (  < 
Or  ( 0 [,] +oo )  <->  `'  <  Or  (
0 [,] +oo )
)
5048, 49mpbi 212 . . . . 5  |-  `'  <  Or  ( 0 [,] +oo )
5150a1i 11 . . . 4  |-  ( ph  ->  `'  <  Or  ( 0 [,] +oo ) )
52 omsclOLD 29123 . . . . . 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 ) )
536, 7, 24, 52syl3anc 1268 . . . . 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 )
)
54 xrge0infssOLD 28341 . . . . 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
) )  -.  e `'  <  t  /\  A. t  e.  ( 0 [,] +oo ) ( t `'  <  e  ->  E. u  e.  ran  ( x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) ) t `'  <  u ) ) )
5553, 54syl 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
) )  -.  e `'  <  t  /\  A. t  e.  ( 0 [,] +oo ) ( t `'  <  e  ->  E. u  e.  ran  ( x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) ) t `'  <  u ) ) )
5651, 55suplub 7974 . . 3  |-  ( ph  ->  ( ( ( ( M `  A )  +  E )  e.  ( 0 [,] +oo )  /\  ( ( M `
 A )  +  E ) `'  <  sup ( ran  ( x  e.  { z  e. 
~P dom  R  | 
( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ*
w  e.  x ( R `  w ) ) ,  ( 0 [,] +oo ) ,  `'  <  ) )  ->  E. u  e.  ran  ( x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) ) ( ( M `  A )  +  E ) `'  <  u ) )
5744, 56mpd 15 . 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
) ) ( ( M `  A )  +  E ) `'  <  u )
58 eqid 2451 . . . . . . . 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 ) )
59 esumex 28850 . . . . . . . 8  |- Σ* w  e.  x
( R `  w
)  e.  _V
6058, 59elrnmpti 5085 . . . . . . 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 ) )
6160anbi1i 701 . . . . . 6  |-  ( ( u  e.  ran  (
x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) )  /\  (
( M `  A
)  +  E ) `'  <  u )  <->  ( E. x  e.  { z  e.  ~P dom  R  | 
( A  C_  U. z  /\  z  ~<_  om ) } u  = Σ* w  e.  x
( R `  w
)  /\  ( ( M `  A )  +  E ) `'  <  u ) )
62 r19.41v 2942 . . . . . 6  |-  ( E. x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  ( u  = Σ* w  e.  x ( R `  w )  /\  ( ( M `
 A )  +  E ) `'  <  u )  <->  ( E. x  e.  { z  e.  ~P dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) } u  = Σ* w  e.  x ( R `
 w )  /\  ( ( M `  A )  +  E
) `'  <  u
) )
6361, 62bitr4i 256 . . . . 5  |-  ( ( u  e.  ran  (
x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) )  /\  (
( M `  A
)  +  E ) `'  <  u )  <->  E. x  e.  { z  e.  ~P dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) }  ( u  = Σ* w  e.  x ( R `  w )  /\  ( ( M `
 A )  +  E ) `'  <  u ) )
6463exbii 1718 . . . 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
) )  /\  (
( M `  A
)  +  E ) `'  <  u )  <->  E. u E. x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  ( u  = Σ* w  e.  x ( R `  w )  /\  ( ( M `
 A )  +  E ) `'  <  u ) )
65 df-rex 2743 . . . 4  |-  ( E. u  e.  ran  (
x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) ) ( ( M `  A )  +  E ) `'  <  u  <->  E. u
( u  e.  ran  ( x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  |-> Σ* w  e.  x
( R `  w
) )  /\  (
( M `  A
)  +  E ) `'  <  u ) )
66 rexcom4 3067 . . . 4  |-  ( E. x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) } E. u
( u  = Σ* w  e.  x ( R `  w )  /\  (
( M `  A
)  +  E ) `'  <  u )  <->  E. u E. x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }  ( u  = Σ* w  e.  x ( R `  w )  /\  ( ( M `
 A )  +  E ) `'  <  u ) )
6764, 65, 663bitr4i 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
) ) ( ( M `  A )  +  E ) `'  <  u  <->  E. x  e.  { z  e.  ~P dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) } E. u
( u  = Σ* w  e.  x ( R `  w )  /\  (
( M `  A
)  +  E ) `'  <  u ) )
68 simpr 463 . . . . . . 7  |-  ( ( u  = Σ* w  e.  x
( R `  w
)  /\  ( ( M `  A )  +  E ) `'  <  u )  ->  ( ( M `  A )  +  E ) `'  <  u )
69 simpl 459 . . . . . . 7  |-  ( ( u  = Σ* w  e.  x
( R `  w
)  /\  ( ( M `  A )  +  E ) `'  <  u )  ->  u  = Σ* w  e.  x ( R `  w ) )
7068, 69breqtrd 4427 . . . . . 6  |-  ( ( u  = Σ* w  e.  x
( R `  w
)  /\  ( ( M `  A )  +  E ) `'  <  u )  ->  ( ( M `  A )  +  E ) `'  < Σ* w  e.  x ( R `  w ) )
7135, 59brcnv 5017 . . . . . 6  |-  ( ( ( M `  A
)  +  E ) `'  < Σ* w  e.  x ( R `  w )  <-> Σ* w  e.  x ( R `  w )  <  (
( M `  A
)  +  E ) )
7270, 71sylib 200 . . . . 5  |-  ( ( u  = Σ* w  e.  x
( R `  w
)  /\  ( ( M `  A )  +  E ) `'  <  u )  -> Σ* w  e.  x
( R `  w
)  <  ( ( M `  A )  +  E ) )
7372exlimiv 1776 . . . 4  |-  ( E. u ( u  = Σ* w  e.  x ( R `
 w )  /\  ( ( M `  A )  +  E
) `'  <  u
)  -> Σ* w  e.  x
( R `  w
)  <  ( ( M `  A )  +  E ) )
7473reximi 2855 . . 3  |-  ( E. x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) } E. u
( u  = Σ* w  e.  x ( R `  w )  /\  (
( M `  A
)  +  E ) `'  <  u )  ->  E. x  e.  { z  e.  ~P dom  R  |  ( A  C_  U. z  /\  z  ~<_  om ) }Σ* w  e.  x ( R `  w )  <  ( ( M `
 A )  +  E ) )
7567, 74sylbi 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
) ) ( ( M `  A )  +  E ) `'  <  u  ->  E. x  e.  { z  e.  ~P dom  R  |  ( A 
C_  U. z  /\  z  ~<_  om ) }Σ* w  e.  x ( R `  w )  <  ( ( M `
 A )  +  E ) )
7657, 75syl 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 1444   E.wex 1663    e. wcel 1887   A.wral 2737   E.wrex 2738   {crab 2741   _Vcvv 3045    C_ wss 3404   ~Pcpw 3951   U.cuni 4198   class class class wbr 4402    |-> cmpt 4461    Or wor 4754   `'ccnv 4833   dom cdm 4834   ran crn 4835   -->wf 5578   ` cfv 5582  (class class class)co 6290   omcom 6692    ~<_ cdom 7567   supcsup 7954   RRcr 9538   0cc0 9539    + caddc 9542   +oocpnf 9672   RR*cxr 9674    < clt 9675    <_ cle 9676   RR+crp 11302   [,]cicc 11638  Σ*cesum 28848  toOMeascomsold 29114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xadd 11410  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-seq 12214  df-hash 12516  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-tset 15209  df-ple 15210  df-ds 15212  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-ordt 15399  df-xrs 15400  df-mre 15492  df-mrc 15493  df-acs 15495  df-ps 16446  df-tsr 16447  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-cntz 16971  df-cmn 17432  df-fbas 18967  df-fg 18968  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-ntr 20035  df-nei 20114  df-cn 20243  df-haus 20331  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-tsms 21141  df-esum 28849  df-omsOLD 29116
This theorem is referenced by: (None)
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