Theorem geosergap 12057

Description: The value of the finite geometric series A ^ M + A ^ ( M + 1 ) +... + A ^ ( N - 1 ). (Contributed by Mario Carneiro, 2-May-2016.) (Revised by Jim Kingdon, 24-Oct-2022.)

Hypotheses

Ref	Expression
geoserg.1	|- ( ph -> A e. CC )
geosergap.2	|- ( ph -> A # 1 )
geoserg.3	|- ( ph -> M e. NN0 )
geoserg.4	|- ( ph -> N e. ( ZZ>= ` M ) )

Assertion

Ref	Expression
geosergap	|- ( ph -> sum_ k e. ( M..^ N ) ( A ^ k ) = ( ( ( A ^ M ) - ( A ^ N ) ) / ( 1 - A ) ) )

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

Proof of Theorem geosergap

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		geoserg.3	. . . . . . 7 |- ( ph -> M e. NN0 )
2	1	nn0zd 9590	. . . . . 6 |- ( ph -> M e. ZZ )
3		geoserg.4	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` M ) )
4		eluzelz 9755	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
6		fzofig 10684	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M..^ N ) e. Fin )
7	2, 5, 6	syl2anc 411	. . . . 5 |- ( ph -> ( M..^ N ) e. Fin )
8		ax-1cn 8115	. . . . . 6 |- 1 e. CC
9		geoserg.1	. . . . . 6 |- ( ph -> A e. CC )
10		subcl 8368	. . . . . 6 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 - A ) e. CC )
11	8, 9, 10	sylancr 414	. . . . 5 |- ( ph -> ( 1 - A ) e. CC )
12	9	adantr 276	. . . . . 6 |- ( ( ph /\ k e. ( M..^ N ) ) -> A e. CC )
13		elfzouz 10376	. . . . . . 7 |- ( k e. ( M..^ N ) -> k e. ( ZZ>= ` M ) )
14		eluznn0 9823	. . . . . . 7 |- ( ( M e. NN0 /\ k e. ( ZZ>= ` M ) ) -> k e. NN0 )
15	1, 13, 14	syl2an 289	. . . . . 6 |- ( ( ph /\ k e. ( M..^ N ) ) -> k e. NN0 )
16	12, 15	expcld 10925	. . . . 5 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( A ^ k ) e. CC )
17	7, 11, 16	fsummulc1 12000	. . . 4 |- ( ph -> ( sum_ k e. ( M..^ N ) ( A ^ k ) x. ( 1 - A ) ) = sum_ k e. ( M..^ N ) ( ( A ^ k ) x. ( 1 - A ) ) )
18		1cnd 8185	. . . . . . 7 |- ( ( ph /\ k e. ( M..^ N ) ) -> 1 e. CC )
19	16, 18, 12	subdid 8583	. . . . . 6 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. ( 1 - A ) ) = ( ( ( A ^ k ) x. 1 ) - ( ( A ^ k ) x. A ) ) )
20	16	mulridd 8186	. . . . . . 7 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. 1 ) = ( A ^ k ) )
21	12, 15	expp1d 10926	. . . . . . . 8 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
22	21	eqcomd 2235	. . . . . . 7 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. A ) = ( A ^ ( k + 1 ) ) )
23	20, 22	oveq12d 6031	. . . . . 6 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( ( A ^ k ) x. 1 ) - ( ( A ^ k ) x. A ) ) = ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) )
24	19, 23	eqtrd 2262	. . . . 5 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. ( 1 - A ) ) = ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) )
25	24	sumeq2dv 11919	. . . 4 |- ( ph -> sum_ k e. ( M..^ N ) ( ( A ^ k ) x. ( 1 - A ) ) = sum_ k e. ( M..^ N ) ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) )
26		oveq2 6021	. . . . 5 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
27		oveq2 6021	. . . . 5 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
28		oveq2 6021	. . . . 5 |- ( j = M -> ( A ^ j ) = ( A ^ M ) )
29		oveq2 6021	. . . . 5 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
30	9	adantr 276	. . . . . 6 |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )
31		elfzuz 10246	. . . . . . 7 |- ( j e. ( M ... N ) -> j e. ( ZZ>= ` M ) )
32		eluznn0 9823	. . . . . . 7 |- ( ( M e. NN0 /\ j e. ( ZZ>= ` M ) ) -> j e. NN0 )
33	1, 31, 32	syl2an 289	. . . . . 6 |- ( ( ph /\ j e. ( M ... N ) ) -> j e. NN0 )
34	30, 33	expcld 10925	. . . . 5 |- ( ( ph /\ j e. ( M ... N ) ) -> ( A ^ j ) e. CC )
35	26, 27, 28, 29, 3, 34	telfsumo 12017	. . . 4 |- ( ph -> sum_ k e. ( M..^ N ) ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) = ( ( A ^ M ) - ( A ^ N ) ) )
36	17, 25, 35	3eqtrrd 2267	. . 3 |- ( ph -> ( ( A ^ M ) - ( A ^ N ) ) = ( sum_ k e. ( M..^ N ) ( A ^ k ) x. ( 1 - A ) ) )
37	9, 1	expcld 10925	. . . . 5 |- ( ph -> ( A ^ M ) e. CC )
38		eluznn0 9823	. . . . . . 7 |- ( ( M e. NN0 /\ N e. ( ZZ>= ` M ) ) -> N e. NN0 )
39	1, 3, 38	syl2anc 411	. . . . . 6 |- ( ph -> N e. NN0 )
40	9, 39	expcld 10925	. . . . 5 |- ( ph -> ( A ^ N ) e. CC )
41	37, 40	subcld 8480	. . . 4 |- ( ph -> ( ( A ^ M ) - ( A ^ N ) ) e. CC )
42	7, 16	fsumcl 11951	. . . 4 |- ( ph -> sum_ k e. ( M..^ N ) ( A ^ k ) e. CC )
43		geosergap.2	. . . . . . 7 |- ( ph -> A # 1 )
44		1cnd 8185	. . . . . . . 8 |- ( ph -> 1 e. CC )
45		apneg 8781	. . . . . . . 8 |- ( ( A e. CC /\ 1 e. CC ) -> ( A # 1 <-> -u A # -u 1 ) )
46	9, 44, 45	syl2anc 411	. . . . . . 7 |- ( ph -> ( A # 1 <-> -u A # -u 1 ) )
47	43, 46	mpbid 147	. . . . . 6 |- ( ph -> -u A # -u 1 )
48	9	negcld 8467	. . . . . . 7 |- ( ph -> -u A e. CC )
49	44	negcld 8467	. . . . . . 7 |- ( ph -> -u 1 e. CC )
50		apadd2 8779	. . . . . . 7 |- ( ( -u A e. CC /\ -u 1 e. CC /\ 1 e. CC ) -> ( -u A # -u 1 <-> ( 1 + -u A ) # ( 1 + -u 1 ) ) )
51	48, 49, 44, 50	syl3anc 1271	. . . . . 6 |- ( ph -> ( -u A # -u 1 <-> ( 1 + -u A ) # ( 1 + -u 1 ) ) )
52	47, 51	mpbid 147	. . . . 5 |- ( ph -> ( 1 + -u A ) # ( 1 + -u 1 ) )
53	44, 9	negsubd 8486	. . . . 5 |- ( ph -> ( 1 + -u A ) = ( 1 - A ) )
54		1pneg1e0 9244	. . . . . 6 |- ( 1 + -u 1 ) = 0
55	54	a1i 9	. . . . 5 |- ( ph -> ( 1 + -u 1 ) = 0 )
56	52, 53, 55	3brtr3d 4117	. . . 4 |- ( ph -> ( 1 - A ) # 0 )
57	41, 42, 11, 56	divmulap3d 8995	. . 3 |- ( ph -> ( ( ( ( A ^ M ) - ( A ^ N ) ) / ( 1 - A ) ) = sum_ k e. ( M..^ N ) ( A ^ k ) <-> ( ( A ^ M ) - ( A ^ N ) ) = ( sum_ k e. ( M..^ N ) ( A ^ k ) x. ( 1 - A ) ) ) )
58	36, 57	mpbird 167	. 2 |- ( ph -> ( ( ( A ^ M ) - ( A ^ N ) ) / ( 1 - A ) ) = sum_ k e. ( M..^ N ) ( A ^ k ) )
59	58	eqcomd 2235	1 |- ( ph -> sum_ k e. ( M..^ N ) ( A ^ k ) = ( ( ( A ^ M ) - ( A ^ N ) ) / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   Fincfn 6904   CCcc 8020   0cc0 8022   1c1 8023   + caddc 8025   x. cmul 8027   - cmin 8340   -ucneg 8341   # cap 8751   / cdiv 8842   NN0cn0 9392   ZZcz 9469   ZZ>=cuz 9745   ...cfz 10233  ..^cfzo 10367   ^cexp 10790   sum_csu 11904

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141  ax-caucvg 8142

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-isom 5333  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-frec 6552  df-1o 6577  df-oadd 6581  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fz 10234  df-fzo 10368  df-seqfrec 10700  df-exp 10791  df-ihash 11028  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550  df-clim 11830  df-sumdc 11905

This theorem is referenced by:  geoserap  12058  cvgratnnlemsumlt  12079  cvgcmp2nlemabs  16572