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Theorem efnnfsumcl 20356
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 3271 . . . . 5  |-  { x  e.  RR  |  ( exp `  x )  e.  NN }  C_  RR
2 ax-resscn 8810 . . . . 5  |-  RR  C_  CC
31, 2sstri 3201 . . . 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 5541 . . . . . . 7  |-  ( x  =  y  ->  ( exp `  x )  =  ( exp `  y
) )
65eleq1d 2362 . . . . . 6  |-  ( x  =  y  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  y )  e.  NN ) )
76elrab 2936 . . . . 5  |-  ( y  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( y  e.  RR  /\  ( exp `  y
)  e.  NN ) )
8 fveq2 5541 . . . . . . 7  |-  ( x  =  z  ->  ( exp `  x )  =  ( exp `  z
) )
98eleq1d 2362 . . . . . 6  |-  ( x  =  z  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  z )  e.  NN ) )
109elrab 2936 . . . . 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 8878 . . . . . 6  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  (
y  +  z )  e.  RR )
1411recnd 8877 . . . . . . . 8  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  y  e.  CC )
1512recnd 8877 . . . . . . . 8  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  z  e.  CC )
16 efadd 12391 . . . . . . . 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 9785 . . . . . . . 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 2370 . . . . . 6  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  ( exp `  ( y  +  z ) )  e.  NN )
21 fveq2 5541 . . . . . . . 8  |-  ( x  =  ( y  +  z )  ->  ( exp `  x )  =  ( exp `  (
y  +  z ) ) )
2221eleq1d 2362 . . . . . . 7  |-  ( x  =  ( y  +  z )  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  ( y  +  z ) )  e.  NN ) )
2322elrab 2936 . . . . . 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 5541 . . . . . 6  |-  ( x  =  B  ->  ( exp `  x )  =  ( exp `  B
) )
3130eleq1d 2362 . . . . 5  |-  ( x  =  B  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  B )  e.  NN ) )
3231elrab 2936 . . . 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 8854 . . . . 5  |-  0  e.  RR
35 1nn 9773 . . . . 5  |-  1  e.  NN
36 fveq2 5541 . . . . . . . 8  |-  ( x  =  0  ->  ( exp `  x )  =  ( exp `  0
) )
37 ef0 12388 . . . . . . . 8  |-  ( exp `  0 )  =  1
3836, 37syl6eq 2344 . . . . . . 7  |-  ( x  =  0  ->  ( exp `  x )  =  1 )
3938eleq1d 2362 . . . . . 6  |-  ( x  =  0  ->  (
( exp `  x
)  e.  NN  <->  1  e.  NN ) )
4039elrab 2936 . . . . 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 12221 . 2  |-  ( ph  -> 
sum_ k  e.  A  B  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } )
44 fveq2 5541 . . . . 5  |-  ( x  =  sum_ k  e.  A  B  ->  ( exp `  x
)  =  ( exp `  sum_ k  e.  A  B ) )
4544eleq1d 2362 . . . 4  |-  ( x  =  sum_ k  e.  A  B  ->  ( ( exp `  x )  e.  NN  <->  ( exp `  sum_ k  e.  A  B )  e.  NN ) )
4645elrab 2936 . . 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 1632    e. wcel 1696   {crab 2560    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Fincfn 6879   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   NNcn 9762   sum_csu 12174   expce 12359
This theorem is referenced by:  efchtcl  20365  efchpcl  20379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365
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