Theorem efnnfsumcl 24829

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.

Step	Hyp	Ref	Expression
1		ssrab2 3687	. . . . 5 |- { x e. RR | ( exp ` x ) e. NN } C_ RR
2		ax-resscn 9993	. . . . 5 |- RR C_ CC
3	1, 2	sstri 3612	. . . 4 |- { x e. RR | ( exp ` x ) e. NN } C_ CC
4	3	a1i 11	. . 3 |- ( ph -> { x e. RR | ( exp ` x ) e. NN } C_ CC )
5		fveq2 6191	. . . . . . 7 |- ( x = y -> ( exp ` x ) = ( exp ` y ) )
6	5	eleq1d 2686	. . . . . 6 |- ( x = y -> ( ( exp ` x ) e. NN <-> ( exp ` y ) e. NN ) )
7	6	elrab 3363	. . . . 5 |- ( y e. { x e. RR | ( exp ` x ) e. NN } <-> ( y e. RR /\ ( exp ` y ) e. NN ) )
8		fveq2 6191	. . . . . . 7 |- ( x = z -> ( exp ` x ) = ( exp ` z ) )
9	8	eleq1d 2686	. . . . . 6 |- ( x = z -> ( ( exp ` x ) e. NN <-> ( exp ` z ) e. NN ) )
10	9	elrab 3363	. . . . 5 |- ( z e. { x e. RR | ( exp ` x ) e. NN } <-> ( z e. RR /\ ( exp ` z ) e. NN ) )
11		simpll 790	. . . . . . 7 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> y e. RR )
12		simprl 794	. . . . . . 7 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> z e. RR )
13	11, 12	readdcld 10069	. . . . . 6 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> ( y + z ) e. RR )
14	11	recnd 10068	. . . . . . . 8 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> y e. CC )
15	12	recnd 10068	. . . . . . . 8 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> z e. CC )
16		efadd 14824	. . . . . . . 8 |- ( ( y e. CC /\ z e. CC ) -> ( exp ` ( y + z ) ) = ( ( exp ` y ) x. ( exp ` z ) ) )
17	14, 15, 16	syl2anc 693	. . . . . . 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 11043	. . . . . . . 8 |- ( ( ( exp ` y ) e. NN /\ ( exp ` z ) e. NN ) -> ( ( exp ` y ) x. ( exp ` z ) ) e. NN )
19	18	ad2ant2l 782	. . . . . . 7 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> ( ( exp ` y ) x. ( exp ` z ) ) e. NN )
20	17, 19	eqeltrd 2701	. . . . . 6 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> ( exp ` ( y + z ) ) e. NN )
21		fveq2 6191	. . . . . . . 8 |- ( x = ( y + z ) -> ( exp ` x ) = ( exp ` ( y + z ) ) )
22	21	eleq1d 2686	. . . . . . 7 |- ( x = ( y + z ) -> ( ( exp ` x ) e. NN <-> ( exp ` ( y + z ) ) e. NN ) )
23	22	elrab 3363	. . . . . 6 |- ( ( y + z ) e. { x e. RR | ( exp ` x ) e. NN } <-> ( ( y + z ) e. RR /\ ( exp ` ( y + z ) ) e. NN ) )
24	13, 20, 23	sylanbrc 698	. . . . 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 } )
25	7, 10, 24	syl2anb 496	. . . 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 } )
26	25	adantl 482	. . 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 6191	. . . . . 6 |- ( x = B -> ( exp ` x ) = ( exp ` B ) )
31	30	eleq1d 2686	. . . . 5 |- ( x = B -> ( ( exp ` x ) e. NN <-> ( exp ` B ) e. NN ) )
32	31	elrab 3363	. . . 4 |- ( B e. { x e. RR | ( exp ` x ) e. NN } <-> ( B e. RR /\ ( exp ` B ) e. NN ) )
33	28, 29, 32	sylanbrc 698	. . 3 |- ( ( ph /\ k e. A ) -> B e. { x e. RR | ( exp ` x ) e. NN } )
34		0re 10040	. . . . 5 |- 0 e. RR
35		1nn 11031	. . . . 5 |- 1 e. NN
36		fveq2 6191	. . . . . . . 8 |- ( x = 0 -> ( exp ` x ) = ( exp ` 0 ) )
37		ef0 14821	. . . . . . . 8 |- ( exp ` 0 ) = 1
38	36, 37	syl6eq 2672	. . . . . . 7 |- ( x = 0 -> ( exp ` x ) = 1 )
39	38	eleq1d 2686	. . . . . 6 |- ( x = 0 -> ( ( exp ` x ) e. NN <-> 1 e. NN ) )
40	39	elrab 3363	. . . . 5 |- ( 0 e. { x e. RR | ( exp ` x ) e. NN } <-> ( 0 e. RR /\ 1 e. NN ) )
41	34, 35, 40	mpbir2an 955	. . . 4 |- 0 e. { x e. RR | ( exp ` x ) e. NN }
42	41	a1i 11	. . 3 |- ( ph -> 0 e. { x e. RR | ( exp ` x ) e. NN } )
43	4, 26, 27, 33, 42	fsumcllem 14463	. 2 |- ( ph -> sum_ k e. A B e. { x e. RR | ( exp ` x ) e. NN } )
44		fveq2 6191	. . . . 5 |- ( x = sum_ k e. A B -> ( exp ` x ) = ( exp ` sum_ k e. A B ) )
45	44	eleq1d 2686	. . . 4 |- ( x = sum_ k e. A B -> ( ( exp ` x ) e. NN <-> ( exp ` sum_ k e. A B ) e. NN ) )
46	45	elrab 3363	. . 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 ) )
47	46	simprbi 480	. 2 |- ( sum_ k e. A B e. { x e. RR | ( exp ` x ) e. NN } -> ( exp ` sum_ k e. A B ) e. NN )
48	43, 47	syl 17	1 |- ( ph -> ( exp ` sum_ k e. A B ) e. NN )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NNcn 11020   sum_csu 14416   expce 14792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798

This theorem is referenced by:  efchtcl  24837  efchpcl  24851