Theorem efnnfsumcl 16158

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 8271	. . . . 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 5695	. . . . . . 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 5695	. . . . . . 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 5695	. . . . . . 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 8355	. . . . . 6 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> ( y + z ) e. RR )
16	13	recnd 8354	. . . . . . . 8 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> y e. CC )
17	14	recnd 8354	. . . . . . . 8 |- ( ( ( y e. RR /\ ( exp ` y ) e. NN ) /\ ( z e. RR /\ ( exp ` z ) e. NN ) ) -> z e. CC )
18		efadd 12458	. . . . . . . 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 9327	. . . . . . . 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 5695	. . . . 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 8326	. . . . 5 |- 0 e. RR
33		1nn 9317	. . . . 5 |- 1 e. NN
34		fveq2 5695	. . . . . . . 8 |- ( x = 0 -> ( exp ` x ) = ( exp ` 0 ) )
35		ef0 12455	. . . . . . . 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 12182	. 2 |- ( ph -> sum_ k e. A B e. { x e. RR | ( exp ` x ) e. NN } )
42		fveq2 5695	. . . . 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 5377  (class class class)co 6085   Fincfn 7022   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   NNcn 9306   sum_csu 12135   expce 12425

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-2 9365  df-3 9366  df-4 9367  df-n0 9568  df-z 9649  df-uz 9931  df-q 10029  df-rp 10065  df-ico 10306  df-fz 10422  df-fzo 10560  df-seqfrec 10898  df-exp 10989  df-fac 11178  df-bc 11200  df-ihash 11229  df-cj 11621  df-re 11622  df-im 11623  df-rsqrt 11778  df-abs 11779  df-clim 12061  df-sumdc 12136  df-ef 12431

This theorem is used by:  efchtqcl  16165