Theorem geoisum1c 12031

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 1021	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> A e. CC )
2		simp2 1022	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> R e. CC )
3		1cnd 8162	. . . 4 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> 1 e. CC )
4	3, 2	subcld 8457	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( 1 - R ) e. CC )
5		1red 8161	. . . . . 6 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> 1 e. RR )
6		simp3 1023	. . . . . 6 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( abs ` R ) < 1 )
7	2, 5, 6	absltap 12020	. . . . 5 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> R # 1 )
8		apsym 8753	. . . . . 6 |- ( ( R e. CC /\ 1 e. CC ) -> ( R # 1 <-> 1 # R ) )
9	2, 3, 8	syl2anc 411	. . . . 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 8791	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( 1 - R ) # 0 )
12	1, 2, 4, 11	divassapd 8973	. 2 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( ( A x. R ) / ( 1 - R ) ) = ( A x. ( R / ( 1 - R ) ) ) )
13		geoisum1 12030	. . . 4 |- ( ( R e. CC /\ ( abs ` R ) < 1 ) -> sum_ k e. NN ( R ^ k ) = ( R / ( 1 - R ) ) )
14	13	3adant1 1039	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> sum_ k e. NN ( R ^ k ) = ( R / ( 1 - R ) ) )
15	14	oveq2d 6017	. 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 9758	. . 3 |- NN = ( ZZ>= ` 1 )
17		1zzd 9473	. . 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 1025	. . . . 5 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> R e. CC )
20	18	nnnn0d 9422	. . . . 5 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> k e. NN0 )
21	19, 20	expcld 10895	. . . 4 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> ( R ^ k ) e. CC )
22		oveq2 6009	. . . . 5 |- ( n = k -> ( R ^ n ) = ( R ^ k ) )
23		eqid 2229	. . . . 5 |- ( n e. NN |-> ( R ^ n ) ) = ( n e. NN |-> ( R ^ n ) )
24	22, 23	fvmptg 5710	. . . 4 |- ( ( k e. NN /\ ( R ^ k ) e. CC ) -> ( ( n e. NN |-> ( R ^ n ) ) ` k ) = ( R ^ k ) )
25	18, 21, 24	syl2anc 411	. . 3 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> ( ( n e. NN |-> ( R ^ n ) ) ` k ) = ( R ^ k ) )
26		nnnn0 9376	. . . . 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 10895	. . 3 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> ( R ^ k ) e. CC )
29		seqex 10671	. . . 4 |- seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) e. _V
30		1nn0 9385	. . . . . . 7 |- 1 e. NN0
31	30	a1i 9	. . . . . 6 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> 1 e. NN0 )
32		elnnuz 9759	. . . . . . 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 12023	. . . . 5 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) ~~> ( ( R ^ 1 ) / ( 1 - R ) ) )
35		climcl 11793	. . . . 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 4929	. . . 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 1376	. . 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 11937	. 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 2269	1 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> sum_ k e. NN ( A x. ( R ^ k ) ) = ( ( A x. R ) / ( 1 - R ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   _Vcvv 2799   class class class wbr 4083   |-> cmpt 4145   dom cdm 4719   ` cfv 5318  (class class class)co 6001   CCcc 7997   1c1 8000   + caddc 8002   x. cmul 8004   < clt 8181   - cmin 8317   # cap 8728   / cdiv 8819   NNcn 9110   NN0cn0 9369   ZZ>=cuz 9722   seqcseq 10669   ^cexp 10760   abscabs 11508   ~~> cli 11789   sum_csu 11864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-oadd 6566  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-seqfrec 10670  df-exp 10761  df-ihash 10998  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-clim 11790  df-sumdc 11865

This theorem is referenced by:  0.999...  12032