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Theorem sge0sn 40596
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 4908 . . . . 5 |- { A } e. _V
21a1i 11 . . . 4 |- ( ( ph /\ ( F ` A ) = +oo ) -> { A } e. _V )
3 sge0sn.2 . . . . 5 |- ( ph -> F : { A } --> ( 0 [,] +oo ) )
43adantr 481 . . . 4 |- ( ( ph /\ ( F ` A ) = +oo ) -> F : { A } --> ( 0 [,] +oo ) )
5 id 22 . . . . . . 7 |- ( ( F ` A ) = +oo -> ( F ` A ) = +oo )
65eqcomd 2628 . . . . . 6 |- ( ( F ` A ) = +oo -> +oo = ( F ` A ) )
76adantl 482 . . . . 5 |- ( ( ph /\ ( F ` A ) = +oo ) -> +oo = ( F ` A ) )
8 ffun 6048 . . . . . . . 8 |- ( F : { A } --> ( 0 [,] +oo ) -> Fun F )
93, 8syl 17 . . . . . . 7 |- ( ph -> Fun F )
109adantr 481 . . . . . 6 |- ( ( ph /\ ( F ` A ) = +oo ) -> Fun F )
11 sge0sn.1 . . . . . . . . 9 |- ( ph -> A e. V )
12 snidg 4206 . . . . . . . . 9 |- ( A e. V -> A e. { A } )
1311, 12syl 17 . . . . . . . 8 |- ( ph -> A e. { A } )
14 fdm 6051 . . . . . . . . . 10 |- ( F : { A } --> ( 0 [,] +oo ) -> dom F = { A } )
153, 14syl 17 . . . . . . . . 9 |- ( ph -> dom F = { A } )
1615eqcomd 2628 . . . . . . . 8 |- ( ph -> { A } = dom F )
1713, 16eleqtrd 2703 . . . . . . 7 |- ( ph -> A e. dom F )
1817adantr 481 . . . . . 6 |- ( ( ph /\ ( F ` A ) = +oo ) -> A e. dom F )
19 fvelrn 6352 . . . . . 6 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F )
2010, 18, 19syl2anc 693 . . . . 5 |- ( ( ph /\ ( F ` A ) = +oo ) -> ( F ` A ) e. ran F )
217, 20eqeltrd 2701 . . . 4 |- ( ( ph /\ ( F ` A ) = +oo ) -> +oo e. ran F )
222, 4, 21sge0pnfval 40590 . . 3 |- ( ( ph /\ ( F ` A ) = +oo ) -> (Σ^ ` F ) = +oo )
23 simpr 477 . . 3 |- ( ( ph /\ ( F ` A ) = +oo ) -> ( F ` A ) = +oo )
2422, 23eqtr4d 2659 . 2 |- ( ( ph /\ ( F ` A ) = +oo ) -> (Σ^ ` F ) = ( F ` A ) )
251a1i 11 . . . 4 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> { A } e. _V )
263adantr 481 . . . . 5 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> F : { A } --> ( 0 [,] +oo ) )
27 elsni 4194 . . . . . . . . 9 |- ( +oo e. { ( F ` A ) } -> +oo = ( F ` A ) )
2827eqcomd 2628 . . . . . . . 8 |- ( +oo e. { ( F ` A ) } -> ( F ` A ) = +oo )
2928con3i 150 . . . . . . 7 |- ( -. ( F ` A ) = +oo -> -. +oo e. { ( F ` A ) } )
3029adantl 482 . . . . . 6 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> -. +oo e. { ( F ` A ) } )
3111, 3rnsnf 39370 . . . . . . . 8 |- ( ph -> ran F = { ( F ` A ) } )
3231eqcomd 2628 . . . . . . 7 |- ( ph -> { ( F ` A ) } = ran F )
3332adantr 481 . . . . . 6 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> { ( F ` A ) } = ran F )
3430, 33neleqtrd 2722 . . . . 5 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> -. +oo e. ran F )
3526, 34fge0iccico 40587 . . . 4 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> F : { A } --> ( 0 [,) +oo ) )
3625, 35sge0reval 40589 . . 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 14452 . . . . . . . 8 |- sum_ y e. (/) ( F ` y ) = 0
3837eqcomi 2631 . . . . . . 7 |- 0 = sum_ y e. (/) ( F ` y )
3938a1i 11 . . . . . 6 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> 0 = sum_ y e. (/) ( F ` y ) )
40 nfcvd 2765 . . . . . . . 8 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> F/_ y ( F ` A ) )
41 nfv 1843 . . . . . . . 8 |- F/ y ( ph /\ -. ( F ` A ) = +oo )
42 fveq2 6191 . . . . . . . . 9 |- ( y = A -> ( F ` y ) = ( F ` A ) )
4342adantl 482 . . . . . . . 8 |- ( ( ( ph /\ -. ( F ` A ) = +oo ) /\ y = A ) -> ( F ` y ) = ( F ` A ) )
4411adantr 481 . . . . . . . 8 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> A e. V )
45 rge0ssre 12280 . . . . . . . . . 10 |- ( 0 [,) +oo ) C_ RR
46 ax-resscn 9993 . . . . . . . . . 10 |- RR C_ CC
4745, 46sstri 3612 . . . . . . . . 9 |- ( 0 [,) +oo ) C_ CC
4844, 12syl 17 . . . . . . . . . 10 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> A e. { A } )
4935, 48ffvelrnd 6360 . . . . . . . . 9 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> ( F ` A ) e. ( 0 [,) +oo ) )
5047, 49sseldi 3601 . . . . . . . 8 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> ( F ` A ) e. CC )
5140, 41, 43, 44, 50sumsnd 39185 . . . . . . 7 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> sum_ y e. { A } ( F ` y ) = ( F ` A ) )
5251eqcomd 2628 . . . . . 6 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> ( F ` A ) = sum_ y e. { A } ( F ` y ) )
5339, 52preq12d 4276 . . . . 5 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> { 0 , ( F ` A ) } = { sum_ y e. (/) ( F ` y ) , sum_ y e. { A } ( F ` y ) } )
5453supeq1d 8352 . . . 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 11974 . . . . . . . 8 |- < Or RR*
5655a1i 11 . . . . . . 7 |- ( ph -> < Or RR* )
57 0xr 10086 . . . . . . . 8 |- 0 e. RR*
5857a1i 11 . . . . . . 7 |- ( ph -> 0 e. RR* )
59 iccssxr 12256 . . . . . . . 8 |- ( 0 [,] +oo ) C_ RR*
603, 13ffvelrnd 6360 . . . . . . . 8 |- ( ph -> ( F ` A ) e. ( 0 [,] +oo ) )
6159, 60sseldi 3601 . . . . . . 7 |- ( ph -> ( F ` A ) e. RR* )
62 suppr 8377 . . . . . . 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 1326 . . . . . 6 |- ( ph -> sup ( { 0 , ( F ` A ) } , RR* , < ) = if ( ( F ` A ) < 0 , 0 , ( F ` A ) ) )
64 pnfxr 10092 . . . . . . . . . . 11 |- +oo e. RR*
6564a1i 11 . . . . . . . . . 10 |- ( ph -> +oo e. RR* )
6658, 65, 603jca 1242 . . . . . . . . 9 |- ( ph -> ( 0 e. RR* /\ +oo e. RR* /\ ( F ` A ) e. ( 0 [,] +oo ) ) )
67 iccgelb 12230 . . . . . . . . 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 10104 . . . . . . . 8 |- ( ph -> ( 0 <_ ( F ` A ) <-> -. ( F ` A ) < 0 ) )
7068, 69mpbid 222 . . . . . . 7 |- ( ph -> -. ( F ` A ) < 0 )
7170iffalsed 4097 . . . . . 6 |- ( ph -> if ( ( F ` A ) < 0 , 0 , ( F ` A ) ) = ( F ` A ) )
7263, 71eqtr2d 2657 . . . . 5 |- ( ph -> ( F ` A ) = sup ( { 0 , ( F ` A ) } , RR* , < ) )
7372adantr 481 . . . 4 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> ( F ` A ) = sup ( { 0 , ( F ` A ) } , RR* , < ) )
74 pwsn 4428 . . . . . . . . . . . 12 |- ~P { A } = { (/) , { A } }
7574ineq1i 3810 . . . . . . . . . . 11 |- ( ~P { A } i^i Fin ) = ( { (/) , { A } } i^i Fin )
76 0fin 8188 . . . . . . . . . . . . 13 |- (/) e. Fin
77 snfi 8038 . . . . . . . . . . . . 13 |- { A } e. Fin
78 prssi 4353 . . . . . . . . . . . . 13 |- ( ( (/) e. Fin /\ { A } e. Fin ) -> { (/) , { A } } C_ Fin )
7976, 77, 78mp2an 708 . . . . . . . . . . . 12 |- { (/) , { A } } C_ Fin
80 df-ss 3588 . . . . . . . . . . . . 13 |- ( { (/) , { A } } C_ Fin <-> ( { (/) , { A } } i^i Fin ) = { (/) , { A } } )
8180biimpi 206 . . . . . . . . . . . 12 |- ( { (/) , { A } } C_ Fin -> ( { (/) , { A } } i^i Fin ) = { (/) , { A } } )
8279, 81ax-mp 5 . . . . . . . . . . 11 |- ( { (/) , { A } } i^i Fin ) = { (/) , { A } }
8375, 82eqtri 2644 . . . . . . . . . 10 |- ( ~P { A } i^i Fin ) = { (/) , { A } }
84 mpteq1 4737 . . . . . . . . . 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 4790 . . . . . . . . . . . . 13 |- (/) e. _V
8786a1i 11 . . . . . . . . . . . 12 |- ( T. -> (/) e. _V )
881a1i 11 . . . . . . . . . . . 12 |- ( T. -> { A } e. _V )
89 sumex 14418 . . . . . . . . . . . . 13 |- sum_ y e. (/) ( F ` y ) e. _V
9089a1i 11 . . . . . . . . . . . 12 |- ( T. -> sum_ y e. (/) ( F ` y ) e. _V )
91 sumex 14418 . . . . . . . . . . . . 13 |- sum_ y e. { A } ( F ` y ) e. _V
9291a1i 11 . . . . . . . . . . . 12 |- ( T. -> sum_ y e. { A } ( F ` y ) e. _V )
93 sumeq1 14419 . . . . . . . . . . . . 13 |- ( x = (/) -> sum_ y e. x ( F ` y ) = sum_ y e. (/) ( F ` y ) )
9493adantl 482 . . . . . . . . . . . 12 |- ( ( T. /\ x = (/) ) -> sum_ y e. x ( F ` y ) = sum_ y e. (/) ( F ` y ) )
95 sumeq1 14419 . . . . . . . . . . . . 13 |- ( x = { A } -> sum_ y e. x ( F ` y ) = sum_ y e. { A } ( F ` y ) )
9695adantl 482 . . . . . . . . . . . 12 |- ( ( T. /\ x = { A } ) -> sum_ y e. x ( F ` y ) = sum_ y e. { A } ( F ` y ) )
9787, 88, 90, 92, 94, 96fmptpr 6438 . . . . . . . . . . 11 |- ( T. -> { <. (/) , sum_ y e. (/) ( F ` y ) >. , <. { A } , sum_ y e. { A } ( F ` y ) >. } = ( x e. { (/) , { A } } |-> sum_ y e. x ( F ` y ) ) )
9897trud 1493 . . . . . . . . . 10 |- { <. (/) , sum_ y e. (/) ( F ` y ) >. , <. { A } , sum_ y e. { A } ( F ` y ) >. } = ( x e. { (/) , { A } } |-> sum_ y e. x ( F ` y ) )
9998eqcomi 2631 . . . . . . . . 9 |- ( x e. { (/) , { A } } |-> sum_ y e. x ( F ` y ) ) = { <. (/) , sum_ y e. (/) ( F ` y ) >. , <. { A } , sum_ y e. { A } ( F ` y ) >. }
10085, 99eqtri 2644 . . . . . . . 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 5352 . . . . . . 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 5615 . . . . . . . 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 708 . . . . . . 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 2644 . . . . . 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 8353 . . . . 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 2666 . . 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 2659 . 2 |- ( ( ph /\ -. ( F ` A ) = +oo ) -> (Σ^ ` F ) = ( F ` A ) )
10924, 108pm2.61dan 832 1 |- ( ph -> (Σ^ ` F ) = ( F ` A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   T. wtru 1484   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {csn 4177   {cpr 4179   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   Or wor 5034   dom cdm 5114   ran crn 5115   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   [,)cico 12177   [,]cicc 12178   sum_csu 14416  Σ^csumge0 40579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-sumge0 40580
This theorem is referenced by:  sge0snmpt  40600  sge0sup  40608  sge0snmptf  40654  caratheodorylem1  40740
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