Theorem pwm1geoserap1 11471

Description: The n-th power of a number decreased by 1 expressed by the finite geometric series 1 + A ^ 1 + A ^ 2 +... + A ^ ( N - 1 ). (Contributed by AV, 14-Aug-2021.) (Revised by Jim Kingdon, 24-Oct-2022.)

Hypotheses

Ref	Expression
pwm1geoser.1	|- ( ph -> A e. CC )
pwm1geoser.3	|- ( ph -> N e. NN0 )
pwm1geoserap1.ap	|- ( ph -> A # 1 )

Assertion

Ref	Expression
pwm1geoserap1	|- ( ph -> ( ( A ^ N ) - 1 ) = ( ( A - 1 ) x. sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) ) )

Distinct variable groups:   A, k   k, N   ph, k

Proof of Theorem pwm1geoserap1

Step	Hyp	Ref	Expression
1		pwm1geoser.1	. . 3 |- ( ph -> A e. CC )
2		pwm1geoserap1.ap	. . 3 |- ( ph -> A # 1 )
3		pwm1geoser.3	. . 3 |- ( ph -> N e. NN0 )
4	1, 2, 3	geoserap 11470	. 2 |- ( ph -> sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ N ) ) / ( 1 - A ) ) )
5		eqcom 2172	. . 3 |- ( sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ N ) ) / ( 1 - A ) ) <-> ( ( 1 - ( A ^ N ) ) / ( 1 - A ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) )
6		1cnd 7936	. . . . . 6 |- ( ph -> 1 e. CC )
7	1, 3	expcld 10609	. . . . . 6 |- ( ph -> ( A ^ N ) e. CC )
8		apsym 8525	. . . . . . . 8 |- ( ( A e. CC /\ 1 e. CC ) -> ( A # 1 <-> 1 # A ) )
9	1, 6, 8	syl2anc 409	. . . . . . 7 |- ( ph -> ( A # 1 <-> 1 # A ) )
10	2, 9	mpbid 146	. . . . . 6 |- ( ph -> 1 # A )
11	6, 7, 6, 1, 10	div2subapd 8755	. . . . 5 |- ( ph -> ( ( 1 - ( A ^ N ) ) / ( 1 - A ) ) = ( ( ( A ^ N ) - 1 ) / ( A - 1 ) ) )
12	11	eqeq1d 2179	. . . 4 |- ( ph -> ( ( ( 1 - ( A ^ N ) ) / ( 1 - A ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) <-> ( ( ( A ^ N ) - 1 ) / ( A - 1 ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) ) )
13		peano2cnm 8185	. . . . . 6 |- ( ( A ^ N ) e. CC -> ( ( A ^ N ) - 1 ) e. CC )
14	7, 13	syl 14	. . . . 5 |- ( ph -> ( ( A ^ N ) - 1 ) e. CC )
15		0zd 9224	. . . . . . 7 |- ( ph -> 0 e. ZZ )
16	3	nn0zd 9332	. . . . . . . 8 |- ( ph -> N e. ZZ )
17		peano2zm 9250	. . . . . . . 8 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
18	16, 17	syl 14	. . . . . . 7 |- ( ph -> ( N - 1 ) e. ZZ )
19	15, 18	fzfigd 10387	. . . . . 6 |- ( ph -> ( 0 ... ( N - 1 ) ) e. Fin )
20	1	adantr 274	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... ( N - 1 ) ) ) -> A e. CC )
21		elfznn0 10070	. . . . . . . 8 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. NN0 )
22	21	adantl 275	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. NN0 )
23	20, 22	expcld 10609	. . . . . 6 |- ( ( ph /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( A ^ k ) e. CC )
24	19, 23	fsumcl 11363	. . . . 5 |- ( ph -> sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) e. CC )
25		peano2cnm 8185	. . . . . 6 |- ( A e. CC -> ( A - 1 ) e. CC )
26	1, 25	syl 14	. . . . 5 |- ( ph -> ( A - 1 ) e. CC )
27	1, 6, 2	subap0d 8563	. . . . 5 |- ( ph -> ( A - 1 ) # 0 )
28	14, 24, 26, 27	divmulap2d 8741	. . . 4 |- ( ph -> ( ( ( ( A ^ N ) - 1 ) / ( A - 1 ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) <-> ( ( A ^ N ) - 1 ) = ( ( A - 1 ) x. sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) ) ) )
29	12, 28	bitrd 187	. . 3 |- ( ph -> ( ( ( 1 - ( A ^ N ) ) / ( 1 - A ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) <-> ( ( A ^ N ) - 1 ) = ( ( A - 1 ) x. sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) ) ) )
30	5, 29	syl5bb 191	. 2 |- ( ph -> ( sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ N ) ) / ( 1 - A ) ) <-> ( ( A ^ N ) - 1 ) = ( ( A - 1 ) x. sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) ) ) )
31	4, 30	mpbid 146	1 |- ( ph -> ( ( A ^ N ) - 1 ) = ( ( A - 1 ) x. sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   x. cmul 7779   - cmin 8090   # cap 8500   / cdiv 8589   NN0cn0 9135   ZZcz 9212   ...cfz 9965   ^cexp 10475   sum_csu 11316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317

This theorem is referenced by: (None)