Theorem geoisum1c 12161

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 1024	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> A e. CC )
2		simp2 1025	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> R e. CC )
3		1cnd 8255	. . . 4 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> 1 e. CC )
4	3, 2	subcld 8549	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( 1 - R ) e. CC )
5		1red 8254	. . . . . 6 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> 1 e. RR )
6		simp3 1026	. . . . . 6 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( abs ` R ) < 1 )
7	2, 5, 6	absltap 12150	. . . . 5 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> R # 1 )
8		apsym 8845	. . . . . 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 8883	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( 1 - R ) # 0 )
12	1, 2, 4, 11	divassapd 9065	. 2 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> ( ( A x. R ) / ( 1 - R ) ) = ( A x. ( R / ( 1 - R ) ) ) )
13		geoisum1 12160	. . . 4 |- ( ( R e. CC /\ ( abs ` R ) < 1 ) -> sum_ k e. NN ( R ^ k ) = ( R / ( 1 - R ) ) )
14	13	3adant1 1042	. . 3 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> sum_ k e. NN ( R ^ k ) = ( R / ( 1 - R ) ) )
15	14	oveq2d 6044	. 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 9853	. . 3 |- NN = ( ZZ>= ` 1 )
17		1zzd 9567	. . 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 1028	. . . . 5 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> R e. CC )
20	18	nnnn0d 9516	. . . . 5 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> k e. NN0 )
21	19, 20	expcld 10998	. . . 4 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> ( R ^ k ) e. CC )
22		oveq2 6036	. . . . 5 |- ( n = k -> ( R ^ n ) = ( R ^ k ) )
23		eqid 2231	. . . . 5 |- ( n e. NN |-> ( R ^ n ) ) = ( n e. NN |-> ( R ^ n ) )
24	22, 23	fvmptg 5731	. . . 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 9468	. . . . 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 10998	. . 3 |- ( ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) /\ k e. NN ) -> ( R ^ k ) e. CC )
29		seqex 10774	. . . 4 |- seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) e. _V
30		1nn0 9477	. . . . . . 7 |- 1 e. NN0
31	30	a1i 9	. . . . . 6 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> 1 e. NN0 )
32		elnnuz 9854	. . . . . . 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 12153	. . . . 5 |- ( ( A e. CC /\ R e. CC /\ ( abs ` R ) < 1 ) -> seq 1 ( + , ( n e. NN |-> ( R ^ n ) ) ) ~~> ( ( R ^ 1 ) / ( 1 - R ) ) )
35		climcl 11922	. . . . 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 4943	. . . 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 1379	. . 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 12067	. 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 2271	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 1005   = wceq 1398   e. wcel 2202   _Vcvv 2803   class class class wbr 4093   |-> cmpt 4155   dom cdm 4731   ` cfv 5333  (class class class)co 6028   CCcc 8090   1c1 8093   + caddc 8095   x. cmul 8097   < clt 8273   - cmin 8409   # cap 8820   / cdiv 8911   NNcn 9202   NN0cn0 9461   ZZ>=cuz 9816   seqcseq 10772   ^cexp 10863   abscabs 11637   ~~> cli 11918   sum_csu 11993

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-fz 10306  df-fzo 10440  df-seqfrec 10773  df-exp 10864  df-ihash 11101  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-clim 11919  df-sumdc 11994

This theorem is referenced by:  0.999...  12162