Theorem geoisum1c 12287

Description: The infinite sum of A x. ( R ^ 1 ) + A x. ( R ^ 2 )... is ( A x. R ) / ( 1 - R ). (Contributed by NM, 2-Nov-2007.) (Revised by Mario Carneiro, 26-Apr-2014.)

Assertion

Ref	Expression
geoisum1c	|- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> sum_ k e. NN ( A x. ( R ^ k ) ) = ( ( A x. R ) / ( 1 - R ) ) )

Distinct variable groups:   A, k   R, k

Proof of Theorem geoisum1c

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1028	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> A e. CC )
2		simp2 1029	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> R e. CC )
3		1cnd 8342	. . . 4 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> 1 e. CC )
4	3, 2	subcld 8637	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( 1 - R ) e. CC )
5		1red 8341	. . . . . 6 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> 1 e. RR )
6		simp3 1030	. . . . . 6 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( abs ` R ) < 1 )
7	2, 5, 6	absltap 12276	. . . . 5 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> R # 1 )
8		apsym 8934	. . . . . 6 |- ( ( R e. CC /\ 1 e. CC ) -> ( R # 1 <-> 1 # R ) )
9	2, 3, 8	syl2anc 415	. . . . 5 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( R # 1 <-> 1 # R ) )
10	7, 9	mpbid 147	. . . 4 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> 1 # R )
11	3, 2, 10	subap0d 8972	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( 1 - R ) # 0 )
12	1, 2, 4, 11	divassapd 9156	. 2 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( ( A x. R ) / ( 1 - R ) ) = ( A x. ( R / ( 1 - R ) ) ) )
13		geoisum1 12286	. . . 4 |- ( ( R e. CC /\ ( abs ` R ) < 1 ) -> sum_ k e. NN ( R ^ k ) = ( R / ( 1 - R ) ) )
14	13	3adant1 1046	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> sum_ k e. NN ( R ^ k ) = ( R / ( 1 - R ) ) )
15	14	oveq2d 6101	. 2 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( A x. sum_ k e. NN ( R ^ k ) ) = ( A x. ( R / ( 1 - R ) ) ) )
16		nnuz 9958	. . 3 |- NN = ( ZZ>= ` 1 )
17		1zzd 9671	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> 1 e. ZZ )
18		simpr 110	. . . 4 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> k e. NN )
19		simpl2 1032	. . . . 5 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> R e. CC )
20	18	nnnn0d 9620	. . . . 5 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> k e. NN0 )
21	19, 20	expcld 11111	. . . 4 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> ( R ^ k ) e. CC )
22		oveq2 6093	. . . . 5 |- ( n = k -> ( R ^ n ) = ( R ^ k ) )
23		eqid 2238	. . . . 5 |- ( n e. NN |-> ( R ^ n ) ) = ( n e. NN |-> ( R ^ n ) )
24	22, 23	fvmptg 5781	. . . 4 |- ( ( k e. NN /\ ( R ^ k ) e. CC ) -> ( ( n e. NN |-> ( R ^ n ) ) ` k ) = ( R ^ k ) )
25	18, 21, 24	syl2anc 415	. . 3 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> ( ( n e. NN |-> ( R ^ n ) ) ` k ) = ( R ^ k ) )
26		nnnn0 9570	. . . . 5 |- ( k e. NN -> k e. NN0 )
27	26	adantl 277	. . . 4 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> k e. NN0 )
28	19, 27	expcld 11111	. . 3 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> ( R ^ k ) e. CC )
29		seqex 10886	. . . 4 |- seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) e. _V
30		1nn0 9579	. . . . . . 7 |- 1 e. NN0
31	30	a1i 9	. . . . . 6 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> 1 e. NN0 )
32		elnnuz 9959	. . . . . . 7 |- ( k e. NN <-> k e. ( ZZ>= ` 1 ) )
33	32, 25	sylan2br 288	. . . . . 6 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> ( R ^ n ) ) ` k ) = ( R ^ k ) )
34	2, 6, 31, 33	geolim2 12279	. . . . 5 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) ~~> ( ( R ^ 1 ) / ( 1 - R ) ) )
35		climcl 12048	. . . . 5 |- ( seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) ~~> ( ( R ^ 1 ) / ( 1 - R ) ) -> ( ( R ^ 1 ) / ( 1 - R ) ) e. CC )
36	34, 35	syl 14	. . . 4 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( ( R ^ 1 ) / ( 1 - R ) ) e. CC )
37		breldmg 4987	. . . 4 |- ( ( seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) e. _V /\ ( ( R ^ 1 ) / ( 1 - R ) ) e. CC /\ seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) ~~> ( ( R ^ 1 ) / ( 1 - R ) ) ) -> seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) e. dom ~~> )
38	29, 36, 34, 37	mp3an2i 1383	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) e. dom ~~> )
39	16, 17, 25, 28, 38, 1	isummulc2 12193	. 2 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( A x. sum_ k e. NN ( R ^ k ) ) = sum_ k e. NN ( A x. ( R ^ k ) ) )
40	12, 15, 39	3eqtr2rd 2278	1 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> sum_ k e. NN ( A x. ( R ^ k ) ) = ( ( A x. R ) / ( 1 - R ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   class class class wbr 4130   |-> cmpt 4192   dom cdm 4774   ` cfv 5377  (class class class)co 6085   CCcc 8177   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   - cmin 8497   # cap 8909   / cdiv 9002   NNcn 9304   NN0cn0 9563   ZZ>=cuz 9921   seqcseq 10884   ^cexp 10975   abscabs 11763   ~~> cli 12044   sum_csu 12119

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-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-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-ihash 11215  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-sumdc 12120

This theorem is used by:  0.999...  12288