Theorem pwm1geoser 14600

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.)

Hypotheses

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

Assertion

Ref	Expression
pwm1geoser	|- ( 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 pwm1geoser

Step	Hyp	Ref	Expression
1		1m1e0 11089	. . . 4 |- ( 1 - 1 ) = 0
2		pwm1geoser.3	. . . . . . 7 |- ( ph -> N e. NN0 )
3	2	nn0zd 11480	. . . . . 6 |- ( ph -> N e. ZZ )
4		1exp 12889	. . . . . 6 |- ( N e. ZZ -> ( 1 ^ N ) = 1 )
5	3, 4	syl 17	. . . . 5 |- ( ph -> ( 1 ^ N ) = 1 )
6	5	oveq1d 6665	. . . 4 |- ( ph -> ( ( 1 ^ N ) - 1 ) = ( 1 - 1 ) )
7		fzfid 12772	. . . . . 6 |- ( ph -> ( 0 ... ( N - 1 ) ) e. Fin )
8		1cnd 10056	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... ( N - 1 ) ) ) -> 1 e. CC )
9		elfznn0 12433	. . . . . . . 8 |- ( k e. ( 0 ... ( N - 1 ) ) -> k e. NN0 )
10	9	adantl 482	. . . . . . 7 |- ( ( ph /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. NN0 )
11	8, 10	expcld 13008	. . . . . 6 |- ( ( ph /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( 1 ^ k ) e. CC )
12	7, 11	fsumcl 14464	. . . . 5 |- ( ph -> sum_ k e. ( 0 ... ( N - 1 ) ) ( 1 ^ k ) e. CC )
13	12	mul02d 10234	. . . 4 |- ( ph -> ( 0 x. sum_ k e. ( 0 ... ( N - 1 ) ) ( 1 ^ k ) ) = 0 )
14	1, 6, 13	3eqtr4a 2682	. . 3 |- ( ph -> ( ( 1 ^ N ) - 1 ) = ( 0 x. sum_ k e. ( 0 ... ( N - 1 ) ) ( 1 ^ k ) ) )
15		oveq1 6657	. . . . 5 |- ( A = 1 -> ( A ^ N ) = ( 1 ^ N ) )
16	15	oveq1d 6665	. . . 4 |- ( A = 1 -> ( ( A ^ N ) - 1 ) = ( ( 1 ^ N ) - 1 ) )
17		oveq1 6657	. . . . . 6 |- ( A = 1 -> ( A - 1 ) = ( 1 - 1 ) )
18	17, 1	syl6eq 2672	. . . . 5 |- ( A = 1 -> ( A - 1 ) = 0 )
19		oveq1 6657	. . . . . . . 8 |- ( A = 1 -> ( A ^ k ) = ( 1 ^ k ) )
20	19	adantr 481	. . . . . . 7 |- ( ( A = 1 /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( A ^ k ) = ( 1 ^ k ) )
21	20	ralrimiva 2966	. . . . . 6 |- ( A = 1 -> A. k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) = ( 1 ^ k ) )
22	21	sumeq2d 14432	. . . . 5 |- ( A = 1 -> sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( 1 ^ k ) )
23	18, 22	oveq12d 6668	. . . 4 |- ( A = 1 -> ( ( A - 1 ) x. sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) ) = ( 0 x. sum_ k e. ( 0 ... ( N - 1 ) ) ( 1 ^ k ) ) )
24	16, 23	eqeq12d 2637	. . 3 |- ( A = 1 -> ( ( ( A ^ N ) - 1 ) = ( ( A - 1 ) x. sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) ) <-> ( ( 1 ^ N ) - 1 ) = ( 0 x. sum_ k e. ( 0 ... ( N - 1 ) ) ( 1 ^ k ) ) ) )
25	14, 24	syl5ibr 236	. 2 |- ( A = 1 -> ( ph -> ( ( A ^ N ) - 1 ) = ( ( A - 1 ) x. sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) ) ) )
26		pwm1geoser.1	. . . . . 6 |- ( ph -> A e. CC )
27	26	adantl 482	. . . . 5 |- ( ( -. A = 1 /\ ph ) -> A e. CC )
28		neqne 2802	. . . . . 6 |- ( -. A = 1 -> A =/= 1 )
29	28	adantr 481	. . . . 5 |- ( ( -. A = 1 /\ ph ) -> A =/= 1 )
30	2	adantl 482	. . . . 5 |- ( ( -. A = 1 /\ ph ) -> N e. NN0 )
31	27, 29, 30	geoser 14599	. . . 4 |- ( ( -. A = 1 /\ ph ) -> sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) = ( ( 1 - ( A ^ N ) ) / ( 1 - A ) ) )
32		eqcom 2629	. . . . 5 |- ( 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 ) )
33		1cnd 10056	. . . . . . . 8 |- ( ( -. A = 1 /\ ph ) -> 1 e. CC )
34	26, 2	expcld 13008	. . . . . . . . 9 |- ( ph -> ( A ^ N ) e. CC )
35	34	adantl 482	. . . . . . . 8 |- ( ( -. A = 1 /\ ph ) -> ( A ^ N ) e. CC )
36		nesym 2850	. . . . . . . . . 10 |- ( 1 =/= A <-> -. A = 1 )
37	36	biimpri 218	. . . . . . . . 9 |- ( -. A = 1 -> 1 =/= A )
38	37	adantr 481	. . . . . . . 8 |- ( ( -. A = 1 /\ ph ) -> 1 =/= A )
39	33, 35, 33, 27, 38	div2subd 10851	. . . . . . 7 |- ( ( -. A = 1 /\ ph ) -> ( ( 1 - ( A ^ N ) ) / ( 1 - A ) ) = ( ( ( A ^ N ) - 1 ) / ( A - 1 ) ) )
40	39	eqeq1d 2624	. . . . . 6 |- ( ( -. A = 1 /\ 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 ) ) )
41		peano2cnm 10347	. . . . . . . . 9 |- ( ( A ^ N ) e. CC -> ( ( A ^ N ) - 1 ) e. CC )
42	34, 41	syl 17	. . . . . . . 8 |- ( ph -> ( ( A ^ N ) - 1 ) e. CC )
43	42	adantl 482	. . . . . . 7 |- ( ( -. A = 1 /\ ph ) -> ( ( A ^ N ) - 1 ) e. CC )
44		fzfid 12772	. . . . . . . 8 |- ( ( -. A = 1 /\ ph ) -> ( 0 ... ( N - 1 ) ) e. Fin )
45	27	adantr 481	. . . . . . . . 9 |- ( ( ( -. A = 1 /\ ph ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> A e. CC )
46	9	adantl 482	. . . . . . . . 9 |- ( ( ( -. A = 1 /\ ph ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. NN0 )
47	45, 46	expcld 13008	. . . . . . . 8 |- ( ( ( -. A = 1 /\ ph ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( A ^ k ) e. CC )
48	44, 47	fsumcl 14464	. . . . . . 7 |- ( ( -. A = 1 /\ ph ) -> sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) e. CC )
49		peano2cnm 10347	. . . . . . . . . . . 12 |- ( A e. CC -> ( A - 1 ) e. CC )
50	49	adantr 481	. . . . . . . . . . 11 |- ( ( A e. CC /\ -. A = 1 ) -> ( A - 1 ) e. CC )
51		simpl 473	. . . . . . . . . . . 12 |- ( ( A e. CC /\ -. A = 1 ) -> A e. CC )
52		1cnd 10056	. . . . . . . . . . . 12 |- ( ( A e. CC /\ -. A = 1 ) -> 1 e. CC )
53	28	adantl 482	. . . . . . . . . . . 12 |- ( ( A e. CC /\ -. A = 1 ) -> A =/= 1 )
54	51, 52, 53	subne0d 10401	. . . . . . . . . . 11 |- ( ( A e. CC /\ -. A = 1 ) -> ( A - 1 ) =/= 0 )
55	50, 54	jca 554	. . . . . . . . . 10 |- ( ( A e. CC /\ -. A = 1 ) -> ( ( A - 1 ) e. CC /\ ( A - 1 ) =/= 0 ) )
56	55	ex 450	. . . . . . . . 9 |- ( A e. CC -> ( -. A = 1 -> ( ( A - 1 ) e. CC /\ ( A - 1 ) =/= 0 ) ) )
57	26, 56	syl 17	. . . . . . . 8 |- ( ph -> ( -. A = 1 -> ( ( A - 1 ) e. CC /\ ( A - 1 ) =/= 0 ) ) )
58	57	impcom 446	. . . . . . 7 |- ( ( -. A = 1 /\ ph ) -> ( ( A - 1 ) e. CC /\ ( A - 1 ) =/= 0 ) )
59		divmul2 10689	. . . . . . 7 |- ( ( ( ( A ^ N ) - 1 ) e. CC /\ sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) e. CC /\ ( ( A - 1 ) e. CC /\ ( A - 1 ) =/= 0 ) ) -> ( ( ( ( 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 ) ) ) )
60	43, 48, 58, 59	syl3anc 1326	. . . . . 6 |- ( ( -. A = 1 /\ 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 ) ) ) )
61	40, 60	bitrd 268	. . . . 5 |- ( ( -. A = 1 /\ 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 ) ) ) )
62	32, 61	syl5bb 272	. . . 4 |- ( ( -. A = 1 /\ 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 ) ) ) )
63	31, 62	mpbid 222	. . 3 |- ( ( -. A = 1 /\ ph ) -> ( ( A ^ N ) - 1 ) = ( ( A - 1 ) x. sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) ) )
64	63	ex 450	. 2 |- ( -. A = 1 -> ( ph -> ( ( A ^ N ) - 1 ) = ( ( A - 1 ) x. sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) ) ) )
65	25, 64	pm2.61i 176	1 |- ( ph -> ( ( A ^ N ) - 1 ) = ( ( A - 1 ) x. sum_ k e. ( 0 ... ( N - 1 ) ) ( A ^ k ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   - cmin 10266   / cdiv 10684   NN0cn0 11292   ZZcz 11377   ...cfz 12326   ^cexp 12860   sum_csu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417

This theorem is referenced by:  lighneallem3  41524