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Theorem efnnfsumcl 22438
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 3435 . . . . 5  |-  { x  e.  RR  |  ( exp `  x )  e.  NN }  C_  RR
2 ax-resscn 9337 . . . . 5  |-  RR  C_  CC
31, 2sstri 3363 . . . 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 5689 . . . . . . 7  |-  ( x  =  y  ->  ( exp `  x )  =  ( exp `  y
) )
65eleq1d 2507 . . . . . 6  |-  ( x  =  y  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  y )  e.  NN ) )
76elrab 3115 . . . . 5  |-  ( y  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( y  e.  RR  /\  ( exp `  y
)  e.  NN ) )
8 fveq2 5689 . . . . . . 7  |-  ( x  =  z  ->  ( exp `  x )  =  ( exp `  z
) )
98eleq1d 2507 . . . . . 6  |-  ( x  =  z  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  z )  e.  NN ) )
109elrab 3115 . . . . 5  |-  ( z  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( z  e.  RR  /\  ( exp `  z
)  e.  NN ) )
11 simpll 753 . . . . . . 7  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  y  e.  RR )
12 simprl 755 . . . . . . 7  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  z  e.  RR )
1311, 12readdcld 9411 . . . . . 6  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  (
y  +  z )  e.  RR )
1411recnd 9410 . . . . . . . 8  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  y  e.  CC )
1512recnd 9410 . . . . . . . 8  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  z  e.  CC )
16 efadd 13377 . . . . . . . 8  |-  ( ( y  e.  CC  /\  z  e.  CC )  ->  ( exp `  (
y  +  z ) )  =  ( ( exp `  y )  x.  ( exp `  z
) ) )
1714, 15, 16syl2anc 661 . . . . . . 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 10343 . . . . . . . 8  |-  ( ( ( exp `  y
)  e.  NN  /\  ( exp `  z )  e.  NN )  -> 
( ( exp `  y
)  x.  ( exp `  z ) )  e.  NN )
1918ad2ant2l 745 . . . . . . 7  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  (
( exp `  y
)  x.  ( exp `  z ) )  e.  NN )
2017, 19eqeltrd 2515 . . . . . 6  |-  ( ( ( y  e.  RR  /\  ( exp `  y
)  e.  NN )  /\  ( z  e.  RR  /\  ( exp `  z )  e.  NN ) )  ->  ( exp `  ( y  +  z ) )  e.  NN )
21 fveq2 5689 . . . . . . . 8  |-  ( x  =  ( y  +  z )  ->  ( exp `  x )  =  ( exp `  (
y  +  z ) ) )
2221eleq1d 2507 . . . . . . 7  |-  ( x  =  ( y  +  z )  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  ( y  +  z ) )  e.  NN ) )
2322elrab 3115 . . . . . 6  |-  ( ( y  +  z )  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( ( y  +  z )  e.  RR  /\  ( exp `  (
y  +  z ) )  e.  NN ) )
2413, 20, 23sylanbrc 664 . . . . 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 479 . . . 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 466 . . 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 5689 . . . . . 6  |-  ( x  =  B  ->  ( exp `  x )  =  ( exp `  B
) )
3130eleq1d 2507 . . . . 5  |-  ( x  =  B  ->  (
( exp `  x
)  e.  NN  <->  ( exp `  B )  e.  NN ) )
3231elrab 3115 . . . 4  |-  ( B  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( B  e.  RR  /\  ( exp `  B
)  e.  NN ) )
3328, 29, 32sylanbrc 664 . . 3  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  { x  e.  RR  |  ( exp `  x
)  e.  NN }
)
34 0re 9384 . . . . 5  |-  0  e.  RR
35 1nn 10331 . . . . 5  |-  1  e.  NN
36 fveq2 5689 . . . . . . . 8  |-  ( x  =  0  ->  ( exp `  x )  =  ( exp `  0
) )
37 ef0 13374 . . . . . . . 8  |-  ( exp `  0 )  =  1
3836, 37syl6eq 2489 . . . . . . 7  |-  ( x  =  0  ->  ( exp `  x )  =  1 )
3938eleq1d 2507 . . . . . 6  |-  ( x  =  0  ->  (
( exp `  x
)  e.  NN  <->  1  e.  NN ) )
4039elrab 3115 . . . . 5  |-  ( 0  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } 
<->  ( 0  e.  RR  /\  1  e.  NN ) )
4134, 35, 40mpbir2an 911 . . . 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 13207 . 2  |-  ( ph  -> 
sum_ k  e.  A  B  e.  { x  e.  RR  |  ( exp `  x )  e.  NN } )
44 fveq2 5689 . . . . 5  |-  ( x  =  sum_ k  e.  A  B  ->  ( exp `  x
)  =  ( exp `  sum_ k  e.  A  B ) )
4544eleq1d 2507 . . . 4  |-  ( x  =  sum_ k  e.  A  B  ->  ( ( exp `  x )  e.  NN  <->  ( exp `  sum_ k  e.  A  B )  e.  NN ) )
4645elrab 3115 . . 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 464 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2717    C_ wss 3326   ` cfv 5416  (class class class)co 6089   Fincfn 7308   CCcc 9278   RRcr 9279   0cc0 9280   1c1 9281    + caddc 9283    x. cmul 9285   NNcn 10320   sum_csu 13161   expce 13345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358  ax-addf 9359  ax-mulf 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-sup 7689  df-oi 7722  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-n0 10578  df-z 10645  df-uz 10860  df-rp 10990  df-ico 11304  df-fz 11436  df-fzo 11547  df-fl 11640  df-seq 11805  df-exp 11864  df-fac 12050  df-bc 12077  df-hash 12102  df-shft 12554  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-limsup 12947  df-clim 12964  df-rlim 12965  df-sum 13162  df-ef 13351
This theorem is referenced by:  efchtcl  22447  efchpcl  22461
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