Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sge0sn Structured version   Visualization version   Unicode version

Theorem sge0sn 38335
Description: A sum of a nonnegative extended real is the term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0sn.1  |-  ( ph  ->  A  e.  V )
sge0sn.2  |-  ( ph  ->  F : { A }
--> ( 0 [,] +oo ) )
Assertion
Ref Expression
sge0sn  |-  ( ph  ->  (Σ^ `  F )  =  ( F `  A ) )

Proof of Theorem sge0sn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 4641 . . . . 5  |-  { A }  e.  _V
21a1i 11 . . . 4  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  { A }  e.  _V )
3 sge0sn.2 . . . . 5  |-  ( ph  ->  F : { A }
--> ( 0 [,] +oo ) )
43adantr 472 . . . 4  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  F : { A } --> ( 0 [,] +oo ) )
5 id 22 . . . . . . 7  |-  ( ( F `  A )  = +oo  ->  ( F `  A )  = +oo )
65eqcomd 2477 . . . . . 6  |-  ( ( F `  A )  = +oo  -> +oo  =  ( F `  A ) )
76adantl 473 . . . . 5  |-  ( (
ph  /\  ( F `  A )  = +oo )  -> +oo  =  ( F `  A )
)
8 ffun 5742 . . . . . . . 8  |-  ( F : { A } --> ( 0 [,] +oo )  ->  Fun  F )
93, 8syl 17 . . . . . . 7  |-  ( ph  ->  Fun  F )
109adantr 472 . . . . . 6  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  Fun  F )
11 sge0sn.1 . . . . . . . . 9  |-  ( ph  ->  A  e.  V )
12 snidg 3986 . . . . . . . . 9  |-  ( A  e.  V  ->  A  e.  { A } )
1311, 12syl 17 . . . . . . . 8  |-  ( ph  ->  A  e.  { A } )
14 fdm 5745 . . . . . . . . . 10  |-  ( F : { A } --> ( 0 [,] +oo )  ->  dom  F  =  { A } )
153, 14syl 17 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  { A } )
1615eqcomd 2477 . . . . . . . 8  |-  ( ph  ->  { A }  =  dom  F )
1713, 16eleqtrd 2551 . . . . . . 7  |-  ( ph  ->  A  e.  dom  F
)
1817adantr 472 . . . . . 6  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  A  e.  dom  F )
19 fvelrn 6030 . . . . . 6  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( F `  A
)  e.  ran  F
)
2010, 18, 19syl2anc 673 . . . . 5  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  ( F `  A )  e.  ran  F )
217, 20eqeltrd 2549 . . . 4  |-  ( (
ph  /\  ( F `  A )  = +oo )  -> +oo  e.  ran  F )
222, 4, 21sge0pnfval 38329 . . 3  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  (Σ^ `  F )  = +oo )
23 simpr 468 . . 3  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  ( F `  A )  = +oo )
2422, 23eqtr4d 2508 . 2  |-  ( (
ph  /\  ( F `  A )  = +oo )  ->  (Σ^ `  F )  =  ( F `  A ) )
251a1i 11 . . . 4  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  { A }  e.  _V )
263adantr 472 . . . . 5  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  F : { A } --> ( 0 [,] +oo ) )
27 elsni 3985 . . . . . . . . 9  |-  ( +oo  e.  { ( F `  A ) }  -> +oo  =  ( F `  A ) )
2827eqcomd 2477 . . . . . . . 8  |-  ( +oo  e.  { ( F `  A ) }  ->  ( F `  A )  = +oo )
2928con3i 142 . . . . . . 7  |-  ( -.  ( F `  A
)  = +oo  ->  -. +oo  e.  { ( F `
 A ) } )
3029adantl 473 . . . . . 6  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  -. +oo  e.  { ( F `
 A ) } )
3111, 3rnsnf 37529 . . . . . . . 8  |-  ( ph  ->  ran  F  =  {
( F `  A
) } )
3231eqcomd 2477 . . . . . . 7  |-  ( ph  ->  { ( F `  A ) }  =  ran  F )
3332adantr 472 . . . . . 6  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  { ( F `  A ) }  =  ran  F
)
3430, 33neleqtrd 2570 . . . . 5  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  -. +oo  e.  ran  F )
3526, 34fge0iccico 38326 . . . 4  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  F : { A } --> ( 0 [,) +oo ) )
3625, 35sge0reval 38328 . . 3  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  (Σ^ `  F
)  =  sup ( ran  ( x  e.  ( ~P { A }  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) , 
RR* ,  <  ) )
37 sum0 13864 . . . . . . . 8  |-  sum_ y  e.  (/)  ( F `  y )  =  0
3837eqcomi 2480 . . . . . . 7  |-  0  =  sum_ y  e.  (/)  ( F `  y )
3938a1i 11 . . . . . 6  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  0  =  sum_ y  e.  (/)  ( F `  y ) )
40 nfcvd 2613 . . . . . . . 8  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  F/_ y
( F `  A
) )
41 nfv 1769 . . . . . . . 8  |-  F/ y ( ph  /\  -.  ( F `  A )  = +oo )
42 fveq2 5879 . . . . . . . . 9  |-  ( y  =  A  ->  ( F `  y )  =  ( F `  A ) )
4342adantl 473 . . . . . . . 8  |-  ( ( ( ph  /\  -.  ( F `  A )  = +oo )  /\  y  =  A )  ->  ( F `  y
)  =  ( F `
 A ) )
4411adantr 472 . . . . . . . 8  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  A  e.  V )
45 rge0ssre 11766 . . . . . . . . . 10  |-  ( 0 [,) +oo )  C_  RR
46 ax-resscn 9614 . . . . . . . . . 10  |-  RR  C_  CC
4745, 46sstri 3427 . . . . . . . . 9  |-  ( 0 [,) +oo )  C_  CC
4844, 12syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  A  e.  { A } )
4935, 48ffvelrnd 6038 . . . . . . . . 9  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  ( F `  A )  e.  ( 0 [,) +oo ) )
5047, 49sseldi 3416 . . . . . . . 8  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  ( F `  A )  e.  CC )
5140, 41, 43, 44, 50sumsnd 37410 . . . . . . 7  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  sum_ y  e.  { A }  ( F `  y )  =  ( F `  A ) )
5251eqcomd 2477 . . . . . 6  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  ( F `  A )  =  sum_ y  e.  { A }  ( F `  y ) )
5339, 52preq12d 4050 . . . . 5  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  { 0 ,  ( F `  A ) }  =  { sum_ y  e.  (/)  ( F `  y ) ,  sum_ y  e.  { A }  ( F `  y ) } )
5453supeq1d 7978 . . . 4  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  sup ( { 0 ,  ( F `  A ) } ,  RR* ,  <  )  =  sup ( {
sum_ y  e.  (/)  ( F `  y ) ,  sum_ y  e.  { A }  ( F `  y ) } ,  RR* ,  <  ) )
55 xrltso 11463 . . . . . . . 8  |-  <  Or  RR*
5655a1i 11 . . . . . . 7  |-  ( ph  ->  <  Or  RR* )
57 0xr 9705 . . . . . . . 8  |-  0  e.  RR*
5857a1i 11 . . . . . . 7  |-  ( ph  ->  0  e.  RR* )
59 iccssxr 11742 . . . . . . . 8  |-  ( 0 [,] +oo )  C_  RR*
603, 13ffvelrnd 6038 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  e.  ( 0 [,] +oo ) )
6159, 60sseldi 3416 . . . . . . 7  |-  ( ph  ->  ( F `  A
)  e.  RR* )
62 suppr 8005 . . . . . . 7  |-  ( (  <  Or  RR*  /\  0  e.  RR*  /\  ( F `
 A )  e. 
RR* )  ->  sup ( { 0 ,  ( F `  A ) } ,  RR* ,  <  )  =  if ( ( F `  A )  <  0 ,  0 ,  ( F `  A ) ) )
6356, 58, 61, 62syl3anc 1292 . . . . . 6  |-  ( ph  ->  sup ( { 0 ,  ( F `  A ) } ,  RR* ,  <  )  =  if ( ( F `
 A )  <  0 ,  0 ,  ( F `  A
) ) )
64 pnfxr 11435 . . . . . . . . . . 11  |- +oo  e.  RR*
6564a1i 11 . . . . . . . . . 10  |-  ( ph  -> +oo  e.  RR* )
6658, 65, 603jca 1210 . . . . . . . . 9  |-  ( ph  ->  ( 0  e.  RR*  /\ +oo  e.  RR*  /\  ( F `  A )  e.  ( 0 [,] +oo ) ) )
67 iccgelb 11716 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR*  /\  ( F `
 A )  e.  ( 0 [,] +oo ) )  ->  0  <_  ( F `  A
) )
6866, 67syl 17 . . . . . . . 8  |-  ( ph  ->  0  <_  ( F `  A ) )
6958, 61xrlenltd 9718 . . . . . . . 8  |-  ( ph  ->  ( 0  <_  ( F `  A )  <->  -.  ( F `  A
)  <  0 ) )
7068, 69mpbid 215 . . . . . . 7  |-  ( ph  ->  -.  ( F `  A )  <  0
)
7170iffalsed 3883 . . . . . 6  |-  ( ph  ->  if ( ( F `
 A )  <  0 ,  0 ,  ( F `  A
) )  =  ( F `  A ) )
7263, 71eqtr2d 2506 . . . . 5  |-  ( ph  ->  ( F `  A
)  =  sup ( { 0 ,  ( F `  A ) } ,  RR* ,  <  ) )
7372adantr 472 . . . 4  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  ( F `  A )  =  sup ( { 0 ,  ( F `  A ) } ,  RR* ,  <  ) )
74 pwsn 4184 . . . . . . . . . . . 12  |-  ~P { A }  =  { (/)
,  { A } }
7574ineq1i 3621 . . . . . . . . . . 11  |-  ( ~P { A }  i^i  Fin )  =  ( {
(/) ,  { A } }  i^i  Fin )
76 0fin 7817 . . . . . . . . . . . . 13  |-  (/)  e.  Fin
77 snfi 7668 . . . . . . . . . . . . 13  |-  { A }  e.  Fin
78 prssi 4119 . . . . . . . . . . . . 13  |-  ( (
(/)  e.  Fin  /\  { A }  e.  Fin )  ->  { (/) ,  { A } }  C_  Fin )
7976, 77, 78mp2an 686 . . . . . . . . . . . 12  |-  { (/) ,  { A } }  C_ 
Fin
80 df-ss 3404 . . . . . . . . . . . . 13  |-  ( {
(/) ,  { A } }  C_  Fin  <->  ( { (/)
,  { A } }  i^i  Fin )  =  { (/) ,  { A } } )
8180biimpi 199 . . . . . . . . . . . 12  |-  ( {
(/) ,  { A } }  C_  Fin  ->  ( { (/) ,  { A } }  i^i  Fin )  =  { (/) ,  { A } } )
8279, 81ax-mp 5 . . . . . . . . . . 11  |-  ( {
(/) ,  { A } }  i^i  Fin )  =  { (/) ,  { A } }
8375, 82eqtri 2493 . . . . . . . . . 10  |-  ( ~P { A }  i^i  Fin )  =  { (/) ,  { A } }
84 mpteq1 4476 . . . . . . . . . 10  |-  ( ( ~P { A }  i^i  Fin )  =  { (/)
,  { A } }  ->  ( x  e.  ( ~P { A }  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) )  =  ( x  e.  { (/) ,  { A } }  |->  sum_ y  e.  x  ( F `  y ) ) )
8583, 84ax-mp 5 . . . . . . . . 9  |-  ( x  e.  ( ~P { A }  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) )  =  ( x  e.  { (/) ,  { A } }  |->  sum_ y  e.  x  ( F `  y ) )
86 0ex 4528 . . . . . . . . . . . . 13  |-  (/)  e.  _V
8786a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  (/)  e.  _V )
881a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  { A }  e.  _V )
89 sumex 13831 . . . . . . . . . . . . 13  |-  sum_ y  e.  (/)  ( F `  y )  e.  _V
9089a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  sum_ y  e.  (/)  ( F `  y )  e.  _V )
91 sumex 13831 . . . . . . . . . . . . 13  |-  sum_ y  e.  { A }  ( F `  y )  e.  _V
9291a1i 11 . . . . . . . . . . . 12  |-  ( T. 
->  sum_ y  e.  { A }  ( F `  y )  e.  _V )
93 sumeq1 13832 . . . . . . . . . . . . 13  |-  ( x  =  (/)  ->  sum_ y  e.  x  ( F `  y )  =  sum_ y  e.  (/)  ( F `
 y ) )
9493adantl 473 . . . . . . . . . . . 12  |-  ( ( T.  /\  x  =  (/) )  ->  sum_ y  e.  x  ( F `  y )  =  sum_ y  e.  (/)  ( F `
 y ) )
95 sumeq1 13832 . . . . . . . . . . . . 13  |-  ( x  =  { A }  -> 
sum_ y  e.  x  ( F `  y )  =  sum_ y  e.  { A }  ( F `  y ) )
9695adantl 473 . . . . . . . . . . . 12  |-  ( ( T.  /\  x  =  { A } )  ->  sum_ y  e.  x  ( F `  y )  =  sum_ y  e.  { A }  ( F `  y ) )
9787, 88, 90, 92, 94, 96fmptpr 6105 . . . . . . . . . . 11  |-  ( T. 
->  { <. (/) ,  sum_ y  e.  (/)  ( F `  y ) >. ,  <. { A } ,  sum_ y  e.  { A }  ( F `  y ) >. }  =  ( x  e.  { (/) ,  { A } }  |-> 
sum_ y  e.  x  ( F `  y ) ) )
9897trud 1461 . . . . . . . . . 10  |-  { <. (/)
,  sum_ y  e.  (/)  ( F `  y )
>. ,  <. { A } ,  sum_ y  e. 
{ A }  ( F `  y ) >. }  =  ( x  e.  { (/) ,  { A } }  |->  sum_ y  e.  x  ( F `  y ) )
9998eqcomi 2480 . . . . . . . . 9  |-  ( x  e.  { (/) ,  { A } }  |->  sum_ y  e.  x  ( F `  y ) )  =  { <. (/) ,  sum_ y  e.  (/)  ( F `  y ) >. ,  <. { A } ,  sum_ y  e.  { A }  ( F `  y ) >. }
10085, 99eqtri 2493 . . . . . . . 8  |-  ( x  e.  ( ~P { A }  i^i  Fin )  |-> 
sum_ y  e.  x  ( F `  y ) )  =  { <. (/)
,  sum_ y  e.  (/)  ( F `  y )
>. ,  <. { A } ,  sum_ y  e. 
{ A }  ( F `  y ) >. }
101100rneqi 5067 . . . . . . 7  |-  ran  (
x  e.  ( ~P { A }  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) )  =  ran  { <.
(/) ,  sum_ y  e.  (/)  ( F `  y
) >. ,  <. { A } ,  sum_ y  e. 
{ A }  ( F `  y ) >. }
102 rnpropg 5323 . . . . . . . 8  |-  ( (
(/)  e.  _V  /\  { A }  e.  _V )  ->  ran  { <. (/) ,  sum_ y  e.  (/)  ( F `
 y ) >. ,  <. { A } ,  sum_ y  e.  { A }  ( F `  y ) >. }  =  { sum_ y  e.  (/)  ( F `  y ) ,  sum_ y  e.  { A }  ( F `  y ) } )
10386, 1, 102mp2an 686 . . . . . . 7  |-  ran  { <.
(/) ,  sum_ y  e.  (/)  ( F `  y
) >. ,  <. { A } ,  sum_ y  e. 
{ A }  ( F `  y ) >. }  =  { sum_ y  e.  (/)  ( F `
 y ) , 
sum_ y  e.  { A }  ( F `  y ) }
104101, 103eqtri 2493 . . . . . 6  |-  ran  (
x  e.  ( ~P { A }  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) )  =  { sum_ y  e.  (/)  ( F `
 y ) , 
sum_ y  e.  { A }  ( F `  y ) }
105104supeq1i 7979 . . . . 5  |-  sup ( ran  ( x  e.  ( ~P { A }  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) , 
RR* ,  <  )  =  sup ( { sum_ y  e.  (/)  ( F `
 y ) , 
sum_ y  e.  { A }  ( F `  y ) } ,  RR* ,  <  )
106105a1i 11 . . . 4  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  sup ( ran  ( x  e.  ( ~P { A }  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) ,  RR* ,  <  )  =  sup ( {
sum_ y  e.  (/)  ( F `  y ) ,  sum_ y  e.  { A }  ( F `  y ) } ,  RR* ,  <  ) )
10754, 73, 1063eqtr4d 2515 . . 3  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  ( F `  A )  =  sup ( ran  (
x  e.  ( ~P { A }  i^i  Fin )  |->  sum_ y  e.  x  ( F `  y ) ) ,  RR* ,  <  ) )
10836, 107eqtr4d 2508 . 2  |-  ( (
ph  /\  -.  ( F `  A )  = +oo )  ->  (Σ^ `  F
)  =  ( F `
 A ) )
10924, 108pm2.61dan 808 1  |-  ( ph  ->  (Σ^ `  F )  =  ( F `  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452   T. wtru 1453    e. wcel 1904   _Vcvv 3031    i^i cin 3389    C_ wss 3390   (/)c0 3722   ifcif 3872   ~Pcpw 3942   {csn 3959   {cpr 3961   <.cop 3965   class class class wbr 4395    |-> cmpt 4454    Or wor 4759   dom cdm 4839   ran crn 4840   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   supcsup 7972   CCcc 9555   RRcr 9556   0cc0 9557   +oocpnf 9690   RR*cxr 9692    < clt 9693    <_ cle 9694   [,)cico 11662   [,]cicc 11663   sum_csu 13829  Σ^csumge0 38318
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-inf2 8164  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-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-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  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-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-clim 13629  df-sum 13830  df-sumge0 38319
This theorem is referenced by:  sge0snmpt  38339  sge0sup  38347  sge0snmptf  38393  caratheodorylem1  38466
  Copyright terms: Public domain W3C validator