Theorem pwm1geoserap1 12300

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 12299	. 2 |- ( ph -> sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ N ) ) / ( 1 - A ) ) )
5		eqcom 2240	. . 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 8348	. . . . . 6 |- ( ph -> 1 e. CC )
7	1, 3	expcld 11132	. . . . . 6 |- ( ph -> ( A ^ N ) e. CC )
8		apsym 8942	. . . . . . . 8 |- ( ( A e. CC /\ 1 e. CC ) -> ( A # 1 <-> 1 # A ) )
9	1, 6, 8	syl2anc 415	. . . . . . 7 |- ( ph -> ( A # 1 <-> 1 # A ) )
10	2, 9	mpbid 147	. . . . . 6 |- ( ph -> 1 # A )
11	6, 7, 6, 1, 10	div2subapd 9176	. . . . 5 |- ( ph -> ( ( 1 - ( A ^ N ) ) / ( 1 - A ) ) = ( ( ( A ^ N ) - 1 ) / ( A - 1 ) ) )
12	11	eqeq1d 2247	. . . 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 8599	. . . . . 6 |- ( ( A ^ N ) e. CC -> ( ( A ^ N ) - 1 ) e. CC )
14	7, 13	syl 14	. . . . 5 |- ( ph -> ( ( A ^ N ) - 1 ) e. CC )
15		0zd 9667	. . . . . . 7 |- ( ph -> 0 e. ZZ )
16	3	nn0zd 9777	. . . . . . . 8 |- ( ph -> N e. ZZ )
17		peano2zm 9693	. . . . . . . 8 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
18	16, 17	syl 14	. . . . . . 7 |- ( ph -> ( N - 1 ) e. ZZ )
19	15, 18	fzfigd 10889	. . . . . 6 |- ( ph -> ( 0 ... ( N - 1 ) ) e. Fin )
20	1	adantr 276	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... ( N - 1 ) ) ) -> A e. CC )
21		elfznn0 10538	. . . . . . . 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 11132	. . . . . 6 |- ( ( ph /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( A ^ k ) e. CC )
24	19, 23	fsumcl 12192	. . . . 5 |- ( ph -> sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) e. CC )
25		peano2cnm 8599	. . . . . 6 |- ( A e. CC -> ( A - 1 ) e. CC )
26	1, 25	syl 14	. . . . 5 |- ( ph -> ( A - 1 ) e. CC )
27	1, 6, 2	subap0d 8980	. . . . 5 |- ( ph -> ( A - 1 ) # 0 )
28	14, 24, 26, 27	divmulap2d 9162	. . . 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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4131  (class class class)co 6088   CCcc 8183   0cc0 8185   1c1 8186   x. cmul 8190   - cmin 8504   # cap 8917   / cdiv 9010   NN0cn0 9574   ZZcz 9655   ...cfz 10428   ^cexp 10996   sum_csu 12144

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-exp 10997  df-ihash 11237  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-clim 12070  df-sumdc 12145

This theorem is used by: (None)