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Theorem sge0le 38363
Description: If all of the terms of sums compare, so do the sums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0le.x  |-  ( ph  ->  X  e.  V )
sge0le.F  |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
sge0le.g  |-  ( ph  ->  G : X --> ( 0 [,] +oo ) )
sge0le.le  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  <_  ( G `  x
) )
Assertion
Ref Expression
sge0le  |-  ( ph  ->  (Σ^ `  F )  <_  (Σ^ `  G
) )
Distinct variable groups:    x, F    x, G    x, X    ph, x
Allowed substitution hint:    V( x)

Proof of Theorem sge0le
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 sge0le.x . . . . . 6  |-  ( ph  ->  X  e.  V )
2 sge0le.F . . . . . 6  |-  ( ph  ->  F : X --> ( 0 [,] +oo ) )
31, 2sge0xrcl 38341 . . . . 5  |-  ( ph  ->  (Σ^ `  F )  e.  RR* )
4 pnfge 11455 . . . . 5  |-  ( (Σ^ `  F
)  e.  RR*  ->  (Σ^ `  F
)  <_ +oo )
53, 4syl 17 . . . 4  |-  ( ph  ->  (Σ^ `  F )  <_ +oo )
65adantr 472 . . 3  |-  ( (
ph  /\  (Σ^ `  G )  = +oo )  ->  (Σ^ `  F )  <_ +oo )
7 id 22 . . . . 5  |-  ( (Σ^ `  G
)  = +oo  ->  (Σ^ `  G
)  = +oo )
87eqcomd 2477 . . . 4  |-  ( (Σ^ `  G
)  = +oo  -> +oo  =  (Σ^ `  G ) )
98adantl 473 . . 3  |-  ( (
ph  /\  (Σ^ `  G )  = +oo )  -> +oo  =  (Σ^ `  G
) )
106, 9breqtrd 4420 . 2  |-  ( (
ph  /\  (Σ^ `  G )  = +oo )  ->  (Σ^ `  F )  <_  (Σ^ `  G
) )
11 elinel2 3611 . . . . . . . 8  |-  ( y  e.  ( ~P X  i^i  Fin )  ->  y  e.  Fin )
1211adantl 473 . . . . . . 7  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  y  e.  Fin )
132adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  -.  (Σ^ `  G
)  = +oo )  ->  F : X --> ( 0 [,] +oo ) )
141adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  X  e.  V )
15 sge0le.g . . . . . . . . . . . . . 14  |-  ( ph  ->  G : X --> ( 0 [,] +oo ) )
1615adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  G : X
--> ( 0 [,] +oo ) )
17 simpr 468 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\ +oo  e.  ran  F )  -> +oo  e.  ran  F )
182ffnd 5740 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  F  Fn  X )
19 fvelrnb 5926 . . . . . . . . . . . . . . . . 17  |-  ( F  Fn  X  ->  ( +oo  e.  ran  F  <->  E. x  e.  X  ( F `  x )  = +oo ) )
2018, 19syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( +oo  e.  ran  F  <->  E. x  e.  X  ( F `  x )  = +oo ) )
2120adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  ( +oo  e.  ran  F  <->  E. x  e.  X  ( F `  x )  = +oo ) )
2217, 21mpbid 215 . . . . . . . . . . . . . 14  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  E. x  e.  X  ( F `  x )  = +oo )
23 iccssxr 11742 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 0 [,] +oo )  C_  RR*
2415ffvelrnda 6037 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  ( 0 [,] +oo ) )
2523, 24sseldi 3416 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  RR* )
2625adantr 472 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  x  e.  X )  /\  ( F `  x )  = +oo )  ->  ( G `  x )  e.  RR* )
27 id 22 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( F `  x )  = +oo  ->  ( F `  x )  = +oo )
2827eqcomd 2477 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( F `  x )  = +oo  -> +oo  =  ( F `  x ) )
2928adantl 473 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  x  e.  X )  /\  ( F `  x )  = +oo )  -> +oo  =  ( F `  x ) )
30 sge0le.le . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  <_  ( G `  x
) )
3130adantr 472 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  x  e.  X )  /\  ( F `  x )  = +oo )  ->  ( F `  x )  <_  ( G `  x
) )
3229, 31eqbrtrd 4416 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  x  e.  X )  /\  ( F `  x )  = +oo )  -> +oo  <_  ( G `  x ) )
3326, 32xrgepnfd 37641 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  x  e.  X )  /\  ( F `  x )  = +oo )  ->  ( G `  x )  = +oo )
3433eqcomd 2477 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  x  e.  X )  /\  ( F `  x )  = +oo )  -> +oo  =  ( G `  x ) )
3515ffnd 5740 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  G  Fn  X )
3635adantr 472 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  X )  ->  G  Fn  X )
37 simpr 468 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
38 fnfvelrn 6034 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( G  Fn  X  /\  x  e.  X )  ->  ( G `  x
)  e.  ran  G
)
3936, 37, 38syl2anc 673 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  ran  G )
4039adantr 472 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  x  e.  X )  /\  ( F `  x )  = +oo )  ->  ( G `  x )  e.  ran  G )
4134, 40eqeltrd 2549 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  x  e.  X )  /\  ( F `  x )  = +oo )  -> +oo  e.  ran  G )
4241ex 441 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  X )  ->  (
( F `  x
)  = +oo  -> +oo  e.  ran  G ) )
4342adantlr 729 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\ +oo  e.  ran  F )  /\  x  e.  X )  ->  ( ( F `  x )  = +oo  -> +oo  e.  ran  G
) )
4443rexlimdva 2871 . . . . . . . . . . . . . 14  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  ( E. x  e.  X  ( F `  x )  = +oo  -> +oo  e.  ran  G ) )
4522, 44mpd 15 . . . . . . . . . . . . 13  |-  ( (
ph  /\ +oo  e.  ran  F )  -> +oo  e.  ran  G )
4614, 16, 45sge0pnfval 38329 . . . . . . . . . . . 12  |-  ( (
ph  /\ +oo  e.  ran  F )  ->  (Σ^ `  G )  = +oo )
4746adantlr 729 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\ +oo  e.  ran  F )  ->  (Σ^ `  G )  = +oo )
48 simplr 770 . . . . . . . . . . 11  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\ +oo  e.  ran  F )  ->  -.  (Σ^ `  G
)  = +oo )
4947, 48pm2.65da 586 . . . . . . . . . 10  |-  ( (
ph  /\  -.  (Σ^ `  G
)  = +oo )  ->  -. +oo  e.  ran  F )
5013, 49fge0iccico 38326 . . . . . . . . 9  |-  ( (
ph  /\  -.  (Σ^ `  G
)  = +oo )  ->  F : X --> ( 0 [,) +oo ) )
5150adantr 472 . . . . . . . 8  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  F : X --> ( 0 [,) +oo ) )
52 elpwinss 37446 . . . . . . . . 9  |-  ( y  e.  ( ~P X  i^i  Fin )  ->  y  C_  X )
5352adantl 473 . . . . . . . 8  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  y  C_  X )
5451, 53fssresd 5762 . . . . . . 7  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  ( F  |`  y ) : y --> ( 0 [,) +oo ) )
5512, 54sge0fsum 38343 . . . . . 6  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( F  |`  y ) )  =  sum_ x  e.  y  ( ( F  |`  y ) `  x
) )
56 rge0ssre 11766 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  RR
5754ffvelrnda 6037 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  y )  ->  (
( F  |`  y
) `  x )  e.  ( 0 [,) +oo ) )
5856, 57sseldi 3416 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  y )  ->  (
( F  |`  y
) `  x )  e.  RR )
5912, 58fsumrecl 13877 . . . . . 6  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  sum_ x  e.  y  ( ( F  |`  y ) `  x )  e.  RR )
6055, 59eqeltrd 2549 . . . . 5  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( F  |`  y ) )  e.  RR )
6115adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  -.  (Σ^ `  G
)  = +oo )  ->  G : X --> ( 0 [,] +oo ) )
621adantr 472 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  (Σ^ `  G
)  = +oo )  ->  X  e.  V )
63 simpr 468 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  (Σ^ `  G
)  = +oo )  ->  -.  (Σ^ `  G )  = +oo )
6462, 61sge0repnf 38342 . . . . . . . . . . . 12  |-  ( (
ph  /\  -.  (Σ^ `  G
)  = +oo )  ->  ( (Σ^ `  G )  e.  RR  <->  -.  (Σ^ `  G )  = +oo ) )
6563, 64mpbird 240 . . . . . . . . . . 11  |-  ( (
ph  /\  -.  (Σ^ `  G
)  = +oo )  ->  (Σ^ `  G )  e.  RR )
6662, 61, 65sge0rern 38344 . . . . . . . . . 10  |-  ( (
ph  /\  -.  (Σ^ `  G
)  = +oo )  ->  -. +oo  e.  ran  G )
6761, 66fge0iccico 38326 . . . . . . . . 9  |-  ( (
ph  /\  -.  (Σ^ `  G
)  = +oo )  ->  G : X --> ( 0 [,) +oo ) )
6867adantr 472 . . . . . . . 8  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  G : X --> ( 0 [,) +oo ) )
6968, 53fssresd 5762 . . . . . . 7  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  ( G  |`  y ) : y --> ( 0 [,) +oo ) )
7012, 69sge0fsum 38343 . . . . . 6  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( G  |`  y ) )  =  sum_ x  e.  y  ( ( G  |`  y ) `  x
) )
7169ffvelrnda 6037 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  y )  ->  (
( G  |`  y
) `  x )  e.  ( 0 [,) +oo ) )
7256, 71sseldi 3416 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  y )  ->  (
( G  |`  y
) `  x )  e.  RR )
7312, 72fsumrecl 13877 . . . . . 6  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  sum_ x  e.  y  ( ( G  |`  y ) `  x )  e.  RR )
7470, 73eqeltrd 2549 . . . . 5  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( G  |`  y ) )  e.  RR )
7565adantr 472 . . . . 5  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  G
)  e.  RR )
76 simplll 776 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  y )  ->  ph )
7753sselda 3418 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  y )  ->  x  e.  X )
7876, 77, 30syl2anc 673 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  y )  ->  ( F `  x )  <_  ( G `  x
) )
79 fvres 5893 . . . . . . . . . 10  |-  ( x  e.  y  ->  (
( F  |`  y
) `  x )  =  ( F `  x ) )
8079adantl 473 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  y )  ->  (
( F  |`  y
) `  x )  =  ( F `  x ) )
81 fvres 5893 . . . . . . . . . 10  |-  ( x  e.  y  ->  (
( G  |`  y
) `  x )  =  ( G `  x ) )
8281adantl 473 . . . . . . . . 9  |-  ( ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  y )  ->  (
( G  |`  y
) `  x )  =  ( G `  x ) )
8380, 82breq12d 4408 . . . . . . . 8  |-  ( ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  y )  ->  (
( ( F  |`  y ) `  x
)  <_  ( ( G  |`  y ) `  x )  <->  ( F `  x )  <_  ( G `  x )
) )
8478, 83mpbird 240 . . . . . . 7  |-  ( ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  /\  x  e.  y )  ->  (
( F  |`  y
) `  x )  <_  ( ( G  |`  y ) `  x
) )
8512, 58, 72, 84fsumle 13936 . . . . . 6  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  sum_ x  e.  y  ( ( F  |`  y ) `  x )  <_  sum_ x  e.  y  ( ( G  |`  y ) `  x ) )
8655, 70breq12d 4408 . . . . . 6  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  (
(Σ^ `  ( F  |`  y
) )  <_  (Σ^ `  ( G  |`  y ) )  <->  sum_ x  e.  y  ( ( F  |`  y
) `  x )  <_ 
sum_ x  e.  y 
( ( G  |`  y ) `  x
) ) )
8785, 86mpbird 240 . . . . 5  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( F  |`  y ) )  <_  (Σ^ `  ( G  |`  y
) ) )
881adantr 472 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  X  e.  V )
8915adantr 472 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  G : X --> ( 0 [,] +oo ) )
9088, 89sge0less 38348 . . . . . 6  |-  ( (
ph  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( G  |`  y ) )  <_  (Σ^ `  G ) )
9190adantlr 729 . . . . 5  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( G  |`  y ) )  <_  (Σ^ `  G ) )
9260, 74, 75, 87, 91letrd 9809 . . . 4  |-  ( ( ( ph  /\  -.  (Σ^ `  G )  = +oo )  /\  y  e.  ( ~P X  i^i  Fin ) )  ->  (Σ^ `  ( F  |`  y ) )  <_  (Σ^ `  G ) )
9392ralrimiva 2809 . . 3  |-  ( (
ph  /\  -.  (Σ^ `  G
)  = +oo )  ->  A. y  e.  ( ~P X  i^i  Fin ) (Σ^ `  ( F  |`  y
) )  <_  (Σ^ `  G
) )
9462, 61sge0xrcl 38341 . . . 4  |-  ( (
ph  /\  -.  (Σ^ `  G
)  = +oo )  ->  (Σ^ `  G )  e.  RR* )
9562, 13, 94sge0lefi 38354 . . 3  |-  ( (
ph  /\  -.  (Σ^ `  G
)  = +oo )  ->  ( (Σ^ `  F )  <_  (Σ^ `  G
)  <->  A. y  e.  ( ~P X  i^i  Fin ) (Σ^ `  ( F  |`  y
) )  <_  (Σ^ `  G
) ) )
9693, 95mpbird 240 . 2  |-  ( (
ph  /\  -.  (Σ^ `  G
)  = +oo )  ->  (Σ^ `  F )  <_  (Σ^ `  G
) )
9710, 96pm2.61dan 808 1  |-  ( ph  ->  (Σ^ `  F )  <_  (Σ^ `  G
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   class class class wbr 4395   ran crn 4840    |` cres 4841    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   Fincfn 7587   RRcr 9556   0cc0 9557   +oocpnf 9690   RR*cxr 9692    <_ 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:  sge0lempt  38366
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