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Theorem fsumnncl 37660
Description: Closure of a non empty, finite sum of positive integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
fsumnncl.an0  |-  ( ph  ->  A  =/=  (/) )
fsumnncl.afi  |-  ( ph  ->  A  e.  Fin )
fsumnncl.b  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  NN )
Assertion
Ref Expression
fsumnncl  |-  ( ph  -> 
sum_ k  e.  A  B  e.  NN )
Distinct variable groups:    A, k    ph, k
Allowed substitution hint:    B( k)

Proof of Theorem fsumnncl
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 fsumnncl.afi . . . 4  |-  ( ph  ->  A  e.  Fin )
2 fsumnncl.b . . . . 5  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  NN )
32nnnn0d 10932 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  NN0 )
41, 3fsumnn0cl 13814 . . 3  |-  ( ph  -> 
sum_ k  e.  A  B  e.  NN0 )
5 fsumnncl.an0 . . . . 5  |-  ( ph  ->  A  =/=  (/) )
6 n0 3743 . . . . 5  |-  ( A  =/=  (/)  <->  E. j  j  e.  A )
75, 6sylib 200 . . . 4  |-  ( ph  ->  E. j  j  e.  A )
8 0red 9649 . . . . . . . 8  |-  ( (
ph  /\  j  e.  A )  ->  0  e.  RR )
9 nfv 1763 . . . . . . . . . . . 12  |-  F/ k ( ph  /\  j  e.  A )
10 nfcsb1v 3381 . . . . . . . . . . . . 13  |-  F/_ k [_ j  /  k ]_ B
1110nfel1 2608 . . . . . . . . . . . 12  |-  F/ k
[_ j  /  k ]_ B  e.  NN
129, 11nfim 2005 . . . . . . . . . . 11  |-  F/ k ( ( ph  /\  j  e.  A )  ->  [_ j  /  k ]_ B  e.  NN )
13 eleq1 2519 . . . . . . . . . . . . 13  |-  ( k  =  j  ->  (
k  e.  A  <->  j  e.  A ) )
1413anbi2d 711 . . . . . . . . . . . 12  |-  ( k  =  j  ->  (
( ph  /\  k  e.  A )  <->  ( ph  /\  j  e.  A ) ) )
15 csbeq1a 3374 . . . . . . . . . . . . 13  |-  ( k  =  j  ->  B  =  [_ j  /  k ]_ B )
1615eleq1d 2515 . . . . . . . . . . . 12  |-  ( k  =  j  ->  ( B  e.  NN  <->  [_ j  / 
k ]_ B  e.  NN ) )
1714, 16imbi12d 322 . . . . . . . . . . 11  |-  ( k  =  j  ->  (
( ( ph  /\  k  e.  A )  ->  B  e.  NN )  <-> 
( ( ph  /\  j  e.  A )  ->  [_ j  /  k ]_ B  e.  NN ) ) )
1812, 17, 2chvar 2108 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  A )  ->  [_ j  /  k ]_ B  e.  NN )
1918nnred 10631 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  A )  ->  [_ j  /  k ]_ B  e.  RR )
208, 19readdcld 9675 . . . . . . . 8  |-  ( (
ph  /\  j  e.  A )  ->  (
0  +  [_ j  /  k ]_ B
)  e.  RR )
21 diffi 7808 . . . . . . . . . . . . 13  |-  ( A  e.  Fin  ->  ( A  \  { j } )  e.  Fin )
221, 21syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  \  {
j } )  e. 
Fin )
23 eldifi 3557 . . . . . . . . . . . . . 14  |-  ( k  e.  ( A  \  { j } )  ->  k  e.  A
)
2423adantl 468 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( A  \  { j } ) )  -> 
k  e.  A )
2524, 3syldan 473 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( A  \  { j } ) )  ->  B  e.  NN0 )
2622, 25fsumnn0cl 13814 . . . . . . . . . . 11  |-  ( ph  -> 
sum_ k  e.  ( A  \  { j } ) B  e. 
NN0 )
2726nn0red 10933 . . . . . . . . . 10  |-  ( ph  -> 
sum_ k  e.  ( A  \  { j } ) B  e.  RR )
2827adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  A )  ->  sum_ k  e.  ( A  \  {
j } ) B  e.  RR )
2928, 19readdcld 9675 . . . . . . . 8  |-  ( (
ph  /\  j  e.  A )  ->  ( sum_ k  e.  ( A 
\  { j } ) B  +  [_ j  /  k ]_ B
)  e.  RR )
3018nnrpd 11346 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  A )  ->  [_ j  /  k ]_ B  e.  RR+ )
318, 30ltaddrpd 11378 . . . . . . . 8  |-  ( (
ph  /\  j  e.  A )  ->  0  <  ( 0  +  [_ j  /  k ]_ B
) )
3226nn0ge0d 10935 . . . . . . . . . 10  |-  ( ph  ->  0  <_  sum_ k  e.  ( A  \  {
j } ) B )
3332adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  A )  ->  0  <_ 
sum_ k  e.  ( A  \  { j } ) B )
348, 28, 19, 33leadd1dd 10234 . . . . . . . 8  |-  ( (
ph  /\  j  e.  A )  ->  (
0  +  [_ j  /  k ]_ B
)  <_  ( sum_ k  e.  ( A  \  { j } ) B  +  [_ j  /  k ]_ B
) )
358, 20, 29, 31, 34ltletrd 9800 . . . . . . 7  |-  ( (
ph  /\  j  e.  A )  ->  0  <  ( sum_ k  e.  ( A  \  { j } ) B  +  [_ j  /  k ]_ B ) )
36 difsnid 4121 . . . . . . . . . . 11  |-  ( j  e.  A  ->  (
( A  \  {
j } )  u. 
{ j } )  =  A )
3736adantl 468 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  A )  ->  (
( A  \  {
j } )  u. 
{ j } )  =  A )
3837eqcomd 2459 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  A )  ->  A  =  ( ( A 
\  { j } )  u.  { j } ) )
3938sumeq1d 13779 . . . . . . . 8  |-  ( (
ph  /\  j  e.  A )  ->  sum_ k  e.  A  B  =  sum_ k  e.  ( ( A  \  { j } )  u.  {
j } ) B )
4022adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  A )  ->  ( A  \  { j } )  e.  Fin )
41 simpr 463 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  A )  ->  j  e.  A )
42 neldifsnd 4103 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  A )  ->  -.  j  e.  ( A  \  { j } ) )
43 simpl 459 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( A  \  { j } ) )  ->  ph )
4443, 24, 2syl2anc 667 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( A  \  { j } ) )  ->  B  e.  NN )
4544nncnd 10632 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( A  \  { j } ) )  ->  B  e.  CC )
4645adantlr 722 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  A )  /\  k  e.  ( A  \  {
j } ) )  ->  B  e.  CC )
47 nnsscn 10621 . . . . . . . . . . 11  |-  NN  C_  CC
4847a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  j  e.  A )  ->  NN  C_  CC )
4948, 18sseldd 3435 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  A )  ->  [_ j  /  k ]_ B  e.  CC )
509, 10, 40, 41, 42, 46, 15, 49fsumsplitsn 37659 . . . . . . . 8  |-  ( (
ph  /\  j  e.  A )  ->  sum_ k  e.  ( ( A  \  { j } )  u.  { j } ) B  =  (
sum_ k  e.  ( A  \  { j } ) B  +  [_ j  /  k ]_ B ) )
5139, 50eqtr2d 2488 . . . . . . 7  |-  ( (
ph  /\  j  e.  A )  ->  ( sum_ k  e.  ( A 
\  { j } ) B  +  [_ j  /  k ]_ B
)  =  sum_ k  e.  A  B )
5235, 51breqtrd 4430 . . . . . 6  |-  ( (
ph  /\  j  e.  A )  ->  0  <  sum_ k  e.  A  B )
5352ex 436 . . . . 5  |-  ( ph  ->  ( j  e.  A  ->  0  <  sum_ k  e.  A  B )
)
5453exlimdv 1781 . . . 4  |-  ( ph  ->  ( E. j  j  e.  A  ->  0  <  sum_ k  e.  A  B ) )
557, 54mpd 15 . . 3  |-  ( ph  ->  0  <  sum_ k  e.  A  B )
564, 55jca 535 . 2  |-  ( ph  ->  ( sum_ k  e.  A  B  e.  NN0  /\  0  <  sum_ k  e.  A  B ) )
57 elnnnn0b 10921 . 2  |-  ( sum_ k  e.  A  B  e.  NN  <->  ( sum_ k  e.  A  B  e.  NN0 
/\  0  <  sum_ k  e.  A  B
) )
5856, 57sylibr 216 1  |-  ( ph  -> 
sum_ k  e.  A  B  e.  NN )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1446   E.wex 1665    e. wcel 1889    =/= wne 2624   [_csb 3365    \ cdif 3403    u. cun 3404    C_ wss 3406   (/)c0 3733   {csn 3970   class class class wbr 4405  (class class class)co 6295   Fincfn 7574   CCcc 9542   RRcr 9543   0cc0 9544    + caddc 9547    < clt 9680    <_ cle 9681   NNcn 10616   NN0cn0 10876   sum_csu 13764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-sup 7961  df-oi 8030  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-fz 11792  df-fzo 11923  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-sum 13765
This theorem is referenced by: (None)
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