Theorem geosergap 11649

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 9437	. . . . . 6 |- ( ph -> M e. ZZ )
3		geoserg.4	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` M ) )
4		eluzelz 9601	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
6		fzofig 10503	. . . . . 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 7965	. . . . . 6 |- 1 e. CC
9		geoserg.1	. . . . . 6 |- ( ph -> A e. CC )
10		subcl 8218	. . . . . 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 10217	. . . . . . 7 |- ( k e. ( M..^ N ) -> k e. ( ZZ>= ` M ) )
14		eluznn0 9664	. . . . . . 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 10744	. . . . 5 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( A ^ k ) e. CC )
17	7, 11, 16	fsummulc1 11592	. . . 4 |- ( ph -> ( sum_ k e. ( M..^ N ) ( A ^ k ) x. ( 1 - A ) ) = sum_ k e. ( M..^ N ) ( ( A ^ k ) x. ( 1 - A ) ) )
18		1cnd 8035	. . . . . . 7 |- ( ( ph /\ k e. ( M..^ N ) ) -> 1 e. CC )
19	16, 18, 12	subdid 8433	. . . . . 6 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. ( 1 - A ) ) = ( ( ( A ^ k ) x. 1 ) - ( ( A ^ k ) x. A ) ) )
20	16	mulridd 8036	. . . . . . 7 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. 1 ) = ( A ^ k ) )
21	12, 15	expp1d 10745	. . . . . . . 8 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
22	21	eqcomd 2199	. . . . . . 7 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. A ) = ( A ^ ( k + 1 ) ) )
23	20, 22	oveq12d 5936	. . . . . 6 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( ( A ^ k ) x. 1 ) - ( ( A ^ k ) x. A ) ) = ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) )
24	19, 23	eqtrd 2226	. . . . 5 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. ( 1 - A ) ) = ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) )
25	24	sumeq2dv 11511	. . . 4 |- ( ph -> sum_ k e. ( M..^ N ) ( ( A ^ k ) x. ( 1 - A ) ) = sum_ k e. ( M..^ N ) ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) )
26		oveq2 5926	. . . . 5 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
27		oveq2 5926	. . . . 5 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
28		oveq2 5926	. . . . 5 |- ( j = M -> ( A ^ j ) = ( A ^ M ) )
29		oveq2 5926	. . . . 5 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
30	9	adantr 276	. . . . . 6 |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )
31		elfzuz 10087	. . . . . . 7 |- ( j e. ( M ... N ) -> j e. ( ZZ>= ` M ) )
32		eluznn0 9664	. . . . . . 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 10744	. . . . 5 |- ( ( ph /\ j e. ( M ... N ) ) -> ( A ^ j ) e. CC )
35	26, 27, 28, 29, 3, 34	telfsumo 11609	. . . 4 |- ( ph -> sum_ k e. ( M..^ N ) ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) = ( ( A ^ M ) - ( A ^ N ) ) )
36	17, 25, 35	3eqtrrd 2231	. . 3 |- ( ph -> ( ( A ^ M ) - ( A ^ N ) ) = ( sum_ k e. ( M..^ N ) ( A ^ k ) x. ( 1 - A ) ) )
37	9, 1	expcld 10744	. . . . 5 |- ( ph -> ( A ^ M ) e. CC )
38		eluznn0 9664	. . . . . . 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 10744	. . . . 5 |- ( ph -> ( A ^ N ) e. CC )
41	37, 40	subcld 8330	. . . 4 |- ( ph -> ( ( A ^ M ) - ( A ^ N ) ) e. CC )
42	7, 16	fsumcl 11543	. . . 4 |- ( ph -> sum_ k e. ( M..^ N ) ( A ^ k ) e. CC )
43		geosergap.2	. . . . . . 7 |- ( ph -> A # 1 )
44		1cnd 8035	. . . . . . . 8 |- ( ph -> 1 e. CC )
45		apneg 8630	. . . . . . . 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 8317	. . . . . . 7 |- ( ph -> -u A e. CC )
49	44	negcld 8317	. . . . . . 7 |- ( ph -> -u 1 e. CC )
50		apadd2 8628	. . . . . . 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 1249	. . . . . 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 8336	. . . . 5 |- ( ph -> ( 1 + -u A ) = ( 1 - A ) )
54		1pneg1e0 9093	. . . . . 6 |- ( 1 + -u 1 ) = 0
55	54	a1i 9	. . . . 5 |- ( ph -> ( 1 + -u 1 ) = 0 )
56	52, 53, 55	3brtr3d 4060	. . . 4 |- ( ph -> ( 1 - A ) # 0 )
57	41, 42, 11, 56	divmulap3d 8844	. . 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 2199	1 |- ( ph -> sum_ k e. ( M..^ N ) ( A ^ k ) = ( ( ( A ^ M ) - ( A ^ N ) ) / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   Fincfn 6794   CCcc 7870   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   - cmin 8190   -ucneg 8191   # cap 8600   / cdiv 8691   NN0cn0 9240   ZZcz 9317   ZZ>=cuz 9592   ...cfz 10074  ..^cfzo 10208   ^cexp 10609   sum_csu 11496

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-oadd 6473  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-exp 10610  df-ihash 10847  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422  df-sumdc 11497

This theorem is referenced by:  geoserap  11650  cvgratnnlemsumlt  11671  cvgcmp2nlemabs  15522