Theorem pwm1geoserap1 11904

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 11903	. 2 |- ( ph -> sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ N ) ) / ( 1 - A ) ) )
5		eqcom 2208	. . 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 8118	. . . . . 6 |- ( ph -> 1 e. CC )
7	1, 3	expcld 10850	. . . . . 6 |- ( ph -> ( A ^ N ) e. CC )
8		apsym 8709	. . . . . . . 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 8941	. . . . 5 |- ( ph -> ( ( 1 - ( A ^ N ) ) / ( 1 - A ) ) = ( ( ( A ^ N ) - 1 ) / ( A - 1 ) ) )
12	11	eqeq1d 2215	. . . 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 8368	. . . . . 6 |- ( ( A ^ N ) e. CC -> ( ( A ^ N ) - 1 ) e. CC )
14	7, 13	syl 14	. . . . 5 |- ( ph -> ( ( A ^ N ) - 1 ) e. CC )
15		0zd 9414	. . . . . . 7 |- ( ph -> 0 e. ZZ )
16	3	nn0zd 9523	. . . . . . . 8 |- ( ph -> N e. ZZ )
17		peano2zm 9440	. . . . . . . 8 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
18	16, 17	syl 14	. . . . . . 7 |- ( ph -> ( N - 1 ) e. ZZ )
19	15, 18	fzfigd 10608	. . . . . 6 |- ( ph -> ( 0 ... ( N - 1 ) ) e. Fin )
20	1	adantr 276	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... ( N - 1 ) ) ) -> A e. CC )
21		elfznn0 10266	. . . . . . . 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 10850	. . . . . 6 |- ( ( ph /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( A ^ k ) e. CC )
24	19, 23	fsumcl 11796	. . . . 5 |- ( ph -> sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) e. CC )
25		peano2cnm 8368	. . . . . 6 |- ( A e. CC -> ( A - 1 ) e. CC )
26	1, 25	syl 14	. . . . 5 |- ( ph -> ( A - 1 ) e. CC )
27	1, 6, 2	subap0d 8747	. . . . 5 |- ( ph -> ( A - 1 ) # 0 )
28	14, 24, 26, 27	divmulap2d 8927	. . . 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 1373   e. wcel 2177   class class class wbr 4054  (class class class)co 5962   CCcc 7953   0cc0 7955   1c1 7956   x. cmul 7960   - cmin 8273   # cap 8684   / cdiv 8775   NN0cn0 9325   ZZcz 9402   ...cfz 10160   ^cexp 10715   sum_csu 11749

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-mulrcl 8054  ax-addcom 8055  ax-mulcom 8056  ax-addass 8057  ax-mulass 8058  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-1rid 8062  ax-0id 8063  ax-rnegex 8064  ax-precex 8065  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-apti 8070  ax-pre-ltadd 8071  ax-pre-mulgt0 8072  ax-pre-mulext 8073  ax-arch 8074  ax-caucvg 8075

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-po 4356  df-iso 4357  df-iord 4426  df-on 4428  df-ilim 4429  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-isom 5294  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-irdg 6474  df-frec 6495  df-1o 6520  df-oadd 6524  df-er 6638  df-en 6846  df-dom 6847  df-fin 6848  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-reap 8678  df-ap 8685  df-div 8776  df-inn 9067  df-2 9125  df-3 9126  df-4 9127  df-n0 9326  df-z 9403  df-uz 9679  df-q 9771  df-rp 9806  df-fz 10161  df-fzo 10295  df-seqfrec 10625  df-exp 10716  df-ihash 10953  df-cj 11238  df-re 11239  df-im 11240  df-rsqrt 11394  df-abs 11395  df-clim 11675  df-sumdc 11750

This theorem is referenced by: (None)