Theorem pwm1geoserap1 12132

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 12131	. 2 |- ( ph -> sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ N ) ) / ( 1 - A ) ) )
5		eqcom 2233	. . 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 8238	. . . . . 6 |- ( ph -> 1 e. CC )
7	1, 3	expcld 10981	. . . . . 6 |- ( ph -> ( A ^ N ) e. CC )
8		apsym 8828	. . . . . . . 8 |- ( ( A e. CC /\ 1 e. CC ) -> ( A # 1 <-> 1 # A ) )
9	1, 6, 8	syl2anc 411	. . . . . . 7 |- ( ph -> ( A # 1 <-> 1 # A ) )
10	2, 9	mpbid 147	. . . . . 6 |- ( ph -> 1 # A )
11	6, 7, 6, 1, 10	div2subapd 9060	. . . . 5 |- ( ph -> ( ( 1 - ( A ^ N ) ) / ( 1 - A ) ) = ( ( ( A ^ N ) - 1 ) / ( A - 1 ) ) )
12	11	eqeq1d 2240	. . . 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 8487	. . . . . 6 |- ( ( A ^ N ) e. CC -> ( ( A ^ N ) - 1 ) e. CC )
14	7, 13	syl 14	. . . . 5 |- ( ph -> ( ( A ^ N ) - 1 ) e. CC )
15		0zd 9535	. . . . . . 7 |- ( ph -> 0 e. ZZ )
16	3	nn0zd 9644	. . . . . . . 8 |- ( ph -> N e. ZZ )
17		peano2zm 9561	. . . . . . . 8 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
18	16, 17	syl 14	. . . . . . 7 |- ( ph -> ( N - 1 ) e. ZZ )
19	15, 18	fzfigd 10739	. . . . . 6 |- ( ph -> ( 0 ... ( N - 1 ) ) e. Fin )
20	1	adantr 276	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... ( N - 1 ) ) ) -> A e. CC )
21		elfznn0 10394	. . . . . . . 8 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. NN0 )
22	21	adantl 277	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. NN0 )
23	20, 22	expcld 10981	. . . . . 6 |- ( ( ph /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( A ^ k ) e. CC )
24	19, 23	fsumcl 12024	. . . . 5 |- ( ph -> sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) e. CC )
25		peano2cnm 8487	. . . . . 6 |- ( A e. CC -> ( A - 1 ) e. CC )
26	1, 25	syl 14	. . . . 5 |- ( ph -> ( A - 1 ) e. CC )
27	1, 6, 2	subap0d 8866	. . . . 5 |- ( ph -> ( A - 1 ) # 0 )
28	14, 24, 26, 27	divmulap2d 9046	. . . 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 188	. . 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	bitrid 192	. 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 147	1 |- ( ph -> ( ( A ^ N ) - 1 ) = ( ( A - 1 ) x. sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   class class class wbr 4093  (class class class)co 6028   CCcc 8073   0cc0 8075   1c1 8076   x. cmul 8080   - cmin 8392   # cap 8803   / cdiv 8894   NN0cn0 9444   ZZcz 9523   ...cfz 10288   ^cexp 10846   sum_csu 11976

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 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

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 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-fz 10289  df-fzo 10423  df-seqfrec 10756  df-exp 10847  df-ihash 11084  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-clim 11902  df-sumdc 11977

This theorem is referenced by: (None)