Theorem efnnfsumcl 16251

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 3333	. . . . 5 |- { x e. RR | ( exp ` x ) e. NN } C_ RR
2		ax-resscn 8278	. . . . 5 |- RR C_ CC
3	1, 2	sstri 3257	. . . 4 |- { x e. RR | ( exp ` x ) e. NN } C_ CC
4	3	a1i 9	. . 3 |- ( ph -> { x e. RR | ( exp ` x ) e. NN } C_ CC )
5		fveq2 5698	. . . . . . 7 |- ( x = y -> ( exp ` x ) = ( exp ` y ) )
6	5	eleq1d 2307	. . . . . 6 |- ( x = y -> ( ( exp ` x ) e. NN <-> ( exp ` y ) e. NN ) )
7	6	elrab 2982	. . . . 5 |- ( y e. { x e. RR | ( exp ` x ) e. NN } <-> ( y e. RR /\ ( exp ` y ) e. NN ) )
8		fveq2 5698	. . . . . . 7 |- ( x = z -> ( exp ` x ) = ( exp ` z ) )
9	8	eleq1d 2307	. . . . . 6 |- ( x = z -> ( ( exp ` x ) e. NN <-> ( exp ` z ) e. NN ) )
10	9	elrab 2982	. . . . 5 |- ( z e. { x e. RR | ( exp ` x ) e. NN } <-> ( z e. RR /\ ( exp ` z ) e. NN ) )
11		fveq2 5698	. . . . . . 7 |- ( x = ( y + z ) -> ( exp ` x ) = ( exp ` ( y + z ) ) )
12	11	eleq1d 2307	. . . . . 6 |- ( x = ( y + z ) -> ( ( exp ` x ) e. NN <-> ( exp ` ( y + z ) ) e. NN ) )
13		simpll 531	. . . . . . 7 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> y e. RR )
14		simprl 535	. . . . . . 7 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> z e. RR )
15	13, 14	readdcld 8362	. . . . . 6 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> ( y + z ) e. RR )
16	13	recnd 8361	. . . . . . . 8 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> y e. CC )
17	14	recnd 8361	. . . . . . . 8 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> z e. CC )
18		efadd 12468	. . . . . . . 8 |- ( ( y e. CC /\ z e. CC ) -> ( exp ` ( y + z ) ) = ( ( exp ` y ) x. ( exp ` z ) ) )
19	16, 17, 18	syl2anc 415	. . . . . . 7 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> ( exp ` ( y + z ) ) = ( ( exp ` y ) x. ( exp ` z ) ) )
20		nnmulcl 9335	. . . . . . . 8 |- ( ( ( exp ` y ) e. NN /\ ( exp ` z ) e. NN ) -> ( ( exp ` y ) x. ( exp ` z ) ) e. NN )
21	20	ad2ant2l 512	. . . . . . 7 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> ( ( exp ` y ) x. ( exp ` z ) ) e. NN )
22	19, 21	eqeltrd 2315	. . . . . 6 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> ( exp ` ( y + z ) ) e. NN )
23	12, 15, 22	elrabd 2984	. . . . 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 } )
24	7, 10, 23	syl2anb 291	. . . 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 } )
25	24	adantl 277	. . 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 } )
26		efnnfsumcl.1	. . 3 |- ( ph -> A e. Fin )
27		fveq2 5698	. . . . 5 |- ( x = B -> ( exp ` x ) = ( exp ` B ) )
28	27	eleq1d 2307	. . . 4 |- ( x = B -> ( ( exp ` x ) e. NN <-> ( exp ` B ) e. NN ) )
29		efnnfsumcl.2	. . . 4 |- ( ( ph /\ k e. A ) -> B e. RR )
30		efnnfsumcl.3	. . . 4 |- ( ( ph /\ k e. A ) -> ( exp ` B ) e. NN )
31	28, 29, 30	elrabd 2984	. . 3 |- ( ( ph /\ k e. A ) -> B e. { x e. RR | ( exp ` x ) e. NN } )
32		0re 8333	. . . . 5 |- 0 e. RR
33		1nn 9325	. . . . 5 |- 1 e. NN
34		fveq2 5698	. . . . . . . 8 |- ( x = 0 -> ( exp ` x ) = ( exp ` 0 ) )
35		ef0 12465	. . . . . . . 8 |- ( exp ` 0 ) = 1
36	34, 35	eqtrdi 2287	. . . . . . 7 |- ( x = 0 -> ( exp ` x ) = 1 )
37	36	eleq1d 2307	. . . . . 6 |- ( x = 0 -> ( ( exp ` x ) e. NN <-> 1 e. NN ) )
38	37	elrab 2982	. . . . 5 |- ( 0 e. { x e. RR | ( exp ` x ) e. NN } <-> ( 0 e. RR /\ 1 e. NN ) )
39	32, 33, 38	mpbir2an 955	. . . 4 |- 0 e. { x e. RR | ( exp ` x ) e. NN }
40	39	a1i 9	. . 3 |- ( ph -> 0 e. { x e. RR | ( exp ` x ) e. NN } )
41	4, 25, 26, 31, 40	fsumcllem 12192	. 2 |- ( ph -> sum_ k e. A B e. { x e. RR | ( exp ` x ) e. NN } )
42		fveq2 5698	. . . . 5 |- ( x = sum_ k e. A B -> ( exp ` x ) = ( exp ` sum_ k e. A B ) )
43	42	eleq1d 2307	. . . 4 |- ( x = sum_ k e. A B -> ( ( exp ` x ) e. NN <-> ( exp ` sum_ k e. A B ) e. NN ) )
44	43	elrab 2982	. . 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 ) )
45	44	simprbi 275	. 2 |- ( sum_ k e. A B e. { x e. RR | ( exp ` x ) e. NN } -> ( exp ` sum_ k e. A B ) e. NN )
46	41, 45	syl 14	1 |- ( ph -> ( exp ` sum_ k e. A B ) e. NN )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {crab 2532   C_ wss 3220   ` cfv 5380  (class class class)co 6088   Fincfn 7027   CCcc 8184   RRcr 8185   0cc0 8186   1c1 8187   + caddc 8189   x. cmul 8191   NNcn 9314   sum_csu 12145   expce 12435

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-en 7028  df-dom 7029  df-fin 7030  df-sup 7331  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-n0 9576  df-z 9657  df-uz 9939  df-q 10037  df-rp 10073  df-ico 10314  df-fz 10430  df-fzo 10568  df-seqfrec 10907  df-exp 10998  df-fac 11187  df-bc 11209  df-ihash 11238  df-cj 11630  df-re 11631  df-im 11632  df-rsqrt 11787  df-abs 11788  df-clim 12071  df-sumdc 12146  df-ef 12441

This theorem is used by:  efchtqcl  16258