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Theorem efnnfsumcl 20340
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 3258 . . . . 5  |-  { x  e.  RR  |  ( exp `  x )  e.  NN }  C_  RR
2 ax-resscn 8794 . . . . 5  |-  RR  C_  CC
31, 2sstri 3188 . . . 4  |-  { x  e.  RR  |  ( exp `  x )  e.  NN }  C_  CC
43a1i 10 . . 3  |-  ( ph  ->  { x  e.  RR  |  ( exp `  x
)  e.  NN }  C_  CC )
5 fveq2 5525 . . . . . . 7  |-  ( x  =  y  ->  ( exp `  x )  =  ( exp `  y
) )
65eleq1d 2349 . . . . . 6  |-  ( x  =  y  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  y )  e.  NN ) )
76elrab 2923 . . . . 5  |-  ( y  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( y  e.  RR  /\  ( exp `  y
)  e.  NN ) )
8 fveq2 5525 . . . . . . 7  |-  ( x  =  z  ->  ( exp `  x )  =  ( exp `  z
) )
98eleq1d 2349 . . . . . 6  |-  ( x  =  z  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  z )  e.  NN ) )
109elrab 2923 . . . . 5  |-  ( z  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( z  e.  RR  /\  ( exp `  z
)  e.  NN ) )
11 simpll 730 . . . . . . 7  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  y  e.  RR )
12 simprl 732 . . . . . . 7  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  z  e.  RR )
1311, 12readdcld 8862 . . . . . 6  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  (
y  +  z )  e.  RR )
1411recnd 8861 . . . . . . . 8  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  y  e.  CC )
1512recnd 8861 . . . . . . . 8  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  z  e.  CC )
16 efadd 12375 . . . . . . . 8  |-  ( ( y  e.  CC  /\  z  e.  CC )  ->  ( exp `  (
y  +  z ) )  =  ( ( exp `  y )  x.  ( exp `  z
) ) )
1714, 15, 16syl2anc 642 . . . . . . 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 9769 . . . . . . . 8  |-  ( ( ( exp `  y
)  e.  NN  /\  ( exp `  z )  e.  NN )  -> 
( ( exp `  y
)  x.  ( exp `  z ) )  e.  NN )
1918ad2ant2l 726 . . . . . . 7  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  (
( exp `  y
)  x.  ( exp `  z ) )  e.  NN )
2017, 19eqeltrd 2357 . . . . . 6  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  ( exp `  ( y  +  z ) )  e.  NN )
21 fveq2 5525 . . . . . . . 8  |-  ( x  =  ( y  +  z )  ->  ( exp `  x )  =  ( exp `  (
y  +  z ) ) )
2221eleq1d 2349 . . . . . . 7  |-  ( x  =  ( y  +  z )  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  ( y  +  z ) )  e.  NN ) )
2322elrab 2923 . . . . . 6  |-  ( ( y  +  z )  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( ( y  +  z )  e.  RR  /\  ( exp `  (
y  +  z ) )  e.  NN ) )
2413, 20, 23sylanbrc 645 . . . . 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 465 . . . 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 452 . . 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 5525 . . . . . 6  |-  ( x  =  B  ->  ( exp `  x )  =  ( exp `  B
) )
3130eleq1d 2349 . . . . 5  |-  ( x  =  B  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  B )  e.  NN ) )
3231elrab 2923 . . . 4  |-  ( B  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( B  e.  RR  /\  ( exp `  B
)  e.  NN ) )
3328, 29, 32sylanbrc 645 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  { x  e.  RR  |  ( exp `  x
)  e.  NN }
)
34 0re 8838 . . . . 5  |-  0  e.  RR
35 1nn 9757 . . . . 5  |-  1  e.  NN
36 fveq2 5525 . . . . . . . 8  |-  ( x  =  0  ->  ( exp `  x )  =  ( exp `  0
) )
37 ef0 12372 . . . . . . . 8  |-  ( exp `  0 )  =  1
3836, 37syl6eq 2331 . . . . . . 7  |-  ( x  =  0  ->  ( exp `  x )  =  1 )
3938eleq1d 2349 . . . . . 6  |-  ( x  =  0  ->  (
( exp `  x
)  e.  NN  <->  1  e.  NN ) )
4039elrab 2923 . . . . 5  |-  ( 0  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( 0  e.  RR  /\  1  e.  NN ) )
4134, 35, 40mpbir2an 886 . . . 4  |-  0  e.  { x  e.  RR  |  ( exp `  x
)  e.  NN }
4241a1i 10 . . 3  |-  ( ph  ->  0  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } )
434, 26, 27, 33, 42fsumcllem 12205 . 2  |-  ( ph  -> 
sum_ k  e.  A  B  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } )
44 fveq2 5525 . . . . 5  |-  ( x  =  sum_ k  e.  A  B  ->  ( exp `  x
)  =  ( exp `  sum_ k  e.  A  B ) )
4544eleq1d 2349 . . . 4  |-  ( x  =  sum_ k  e.  A  B  ->  ( ( exp `  x )  e.  NN  <->  ( exp `  sum_ k  e.  A  B )  e.  NN ) )
4645elrab 2923 . . 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 450 . 2  |-  ( sum_ k  e.  A  B  e.  { x  e.  RR  |  ( exp `  x
)  e.  NN }  ->  ( exp `  sum_ k  e.  A  B
)  e.  NN )
4843, 47syl 15 1  |-  ( ph  ->  ( exp `  sum_ k  e.  A  B
)  e.  NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   NNcn 9746   sum_csu 12158   expce 12343
This theorem is referenced by:  efchtcl  20349  efchpcl  20363
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349
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