MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efnnfsumcl Unicode version

Theorem efnnfsumcl 20838
Description: Finite sum closure in the log-integers. (Contributed by Mario Carneiro, 7-Apr-2016.)
Hypotheses
Ref Expression
efnnfsumcl.1  |-  ( ph  ->  A  e.  Fin )
efnnfsumcl.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  RR )
efnnfsumcl.3  |-  ( (
ph  /\  k  e.  A )  ->  ( exp `  B )  e.  NN )
Assertion
Ref Expression
efnnfsumcl  |-  ( ph  ->  ( exp `  sum_ k  e.  A  B
)  e.  NN )
Distinct variable groups:    A, k    ph, k
Allowed substitution hint:    B( k)

Proof of Theorem efnnfsumcl
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3388 . . . . 5  |-  { x  e.  RR  |  ( exp `  x )  e.  NN }  C_  RR
2 ax-resscn 9003 . . . . 5  |-  RR  C_  CC
31, 2sstri 3317 . . . 4  |-  { x  e.  RR  |  ( exp `  x )  e.  NN }  C_  CC
43a1i 11 . . 3  |-  ( ph  ->  { x  e.  RR  |  ( exp `  x
)  e.  NN }  C_  CC )
5 fveq2 5687 . . . . . . 7  |-  ( x  =  y  ->  ( exp `  x )  =  ( exp `  y
) )
65eleq1d 2470 . . . . . 6  |-  ( x  =  y  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  y )  e.  NN ) )
76elrab 3052 . . . . 5  |-  ( y  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( y  e.  RR  /\  ( exp `  y
)  e.  NN ) )
8 fveq2 5687 . . . . . . 7  |-  ( x  =  z  ->  ( exp `  x )  =  ( exp `  z
) )
98eleq1d 2470 . . . . . 6  |-  ( x  =  z  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  z )  e.  NN ) )
109elrab 3052 . . . . 5  |-  ( z  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( z  e.  RR  /\  ( exp `  z
)  e.  NN ) )
11 simpll 731 . . . . . . 7  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  y  e.  RR )
12 simprl 733 . . . . . . 7  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  z  e.  RR )
1311, 12readdcld 9071 . . . . . 6  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  (
y  +  z )  e.  RR )
1411recnd 9070 . . . . . . . 8  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  y  e.  CC )
1512recnd 9070 . . . . . . . 8  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  z  e.  CC )
16 efadd 12651 . . . . . . . 8  |-  ( ( y  e.  CC  /\  z  e.  CC )  ->  ( exp `  (
y  +  z ) )  =  ( ( exp `  y )  x.  ( exp `  z
) ) )
1714, 15, 16syl2anc 643 . . . . . . 7  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  ( exp `  ( y  +  z ) )  =  ( ( exp `  y
)  x.  ( exp `  z ) ) )
18 nnmulcl 9979 . . . . . . . 8  |-  ( ( ( exp `  y
)  e.  NN  /\  ( exp `  z )  e.  NN )  -> 
( ( exp `  y
)  x.  ( exp `  z ) )  e.  NN )
1918ad2ant2l 727 . . . . . . 7  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  (
( exp `  y
)  x.  ( exp `  z ) )  e.  NN )
2017, 19eqeltrd 2478 . . . . . 6  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  ( exp `  ( y  +  z ) )  e.  NN )
21 fveq2 5687 . . . . . . . 8  |-  ( x  =  ( y  +  z )  ->  ( exp `  x )  =  ( exp `  (
y  +  z ) ) )
2221eleq1d 2470 . . . . . . 7  |-  ( x  =  ( y  +  z )  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  ( y  +  z ) )  e.  NN ) )
2322elrab 3052 . . . . . 6  |-  ( ( y  +  z )  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( ( y  +  z )  e.  RR  /\  ( exp `  (
y  +  z ) )  e.  NN ) )
2413, 20, 23sylanbrc 646 . . . . 5  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  (
y  +  z )  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } )
257, 10, 24syl2anb 466 . . . 4  |-  ( ( y  e.  { x  e.  RR  |  ( exp `  x )  e.  NN }  /\  z  e.  {
x  e.  RR  | 
( exp `  x
)  e.  NN }
)  ->  ( y  +  z )  e. 
{ x  e.  RR  |  ( exp `  x
)  e.  NN }
)
2625adantl 453 . . 3  |-  ( (
ph  /\  ( y  e.  { x  e.  RR  |  ( exp `  x
)  e.  NN }  /\  z  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } ) )  -> 
( y  +  z )  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } )
27 efnnfsumcl.1 . . 3  |-  ( ph  ->  A  e.  Fin )
28 efnnfsumcl.2 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  RR )
29 efnnfsumcl.3 . . . 4  |-  ( (
ph  /\  k  e.  A )  ->  ( exp `  B )  e.  NN )
30 fveq2 5687 . . . . . 6  |-  ( x  =  B  ->  ( exp `  x )  =  ( exp `  B
) )
3130eleq1d 2470 . . . . 5  |-  ( x  =  B  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  B )  e.  NN ) )
3231elrab 3052 . . . 4  |-  ( B  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( B  e.  RR  /\  ( exp `  B
)  e.  NN ) )
3328, 29, 32sylanbrc 646 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  { x  e.  RR  |  ( exp `  x
)  e.  NN }
)
34 0re 9047 . . . . 5  |-  0  e.  RR
35 1nn 9967 . . . . 5  |-  1  e.  NN
36 fveq2 5687 . . . . . . . 8  |-  ( x  =  0  ->  ( exp `  x )  =  ( exp `  0
) )
37 ef0 12648 . . . . . . . 8  |-  ( exp `  0 )  =  1
3836, 37syl6eq 2452 . . . . . . 7  |-  ( x  =  0  ->  ( exp `  x )  =  1 )
3938eleq1d 2470 . . . . . 6  |-  ( x  =  0  ->  (
( exp `  x
)  e.  NN  <->  1  e.  NN ) )
4039elrab 3052 . . . . 5  |-  ( 0  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( 0  e.  RR  /\  1  e.  NN ) )
4134, 35, 40mpbir2an 887 . . . 4  |-  0  e.  { x  e.  RR  |  ( exp `  x
)  e.  NN }
4241a1i 11 . . 3  |-  ( ph  ->  0  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } )
434, 26, 27, 33, 42fsumcllem 12481 . 2  |-  ( ph  -> 
sum_ k  e.  A  B  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } )
44 fveq2 5687 . . . . 5  |-  ( x  =  sum_ k  e.  A  B  ->  ( exp `  x
)  =  ( exp `  sum_ k  e.  A  B ) )
4544eleq1d 2470 . . . 4  |-  ( x  =  sum_ k  e.  A  B  ->  ( ( exp `  x )  e.  NN  <->  ( exp `  sum_ k  e.  A  B )  e.  NN ) )
4645elrab 3052 . . 3  |-  ( sum_ k  e.  A  B  e.  { x  e.  RR  |  ( exp `  x
)  e.  NN }  <->  (
sum_ k  e.  A  B  e.  RR  /\  ( exp `  sum_ k  e.  A  B )  e.  NN ) )
4746simprbi 451 . 2  |-  ( sum_ k  e.  A  B  e.  { x  e.  RR  |  ( exp `  x
)  e.  NN }  ->  ( exp `  sum_ k  e.  A  B
)  e.  NN )
4843, 47syl 16 1  |-  ( ph  ->  ( exp `  sum_ k  e.  A  B
)  e.  NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2670    C_ wss 3280   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951   NNcn 9956   sum_csu 12434   expce 12619
This theorem is referenced by:  efchtcl  20847  efchpcl  20861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  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-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ico 10878  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625
  Copyright terms: Public domain W3C validator