Theorem geosergap 11652

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 9440	. . . . . 6 |- ( ph -> M e. ZZ )
3		geoserg.4	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` M ) )
4		eluzelz 9604	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
6		fzofig 10506	. . . . . 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 7967	. . . . . 6 |- 1 e. CC
9		geoserg.1	. . . . . 6 |- ( ph -> A e. CC )
10		subcl 8220	. . . . . 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 10220	. . . . . . 7 |- ( k e. ( M..^ N ) -> k e. ( ZZ>= ` M ) )
14		eluznn0 9667	. . . . . . 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 10747	. . . . 5 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( A ^ k ) e. CC )
17	7, 11, 16	fsummulc1 11595	. . . 4 |- ( ph -> ( sum_ k e. ( M..^ N ) ( A ^ k ) x. ( 1 - A ) ) = sum_ k e. ( M..^ N ) ( ( A ^ k ) x. ( 1 - A ) ) )
18		1cnd 8037	. . . . . . 7 |- ( ( ph /\ k e. ( M..^ N ) ) -> 1 e. CC )
19	16, 18, 12	subdid 8435	. . . . . 6 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. ( 1 - A ) ) = ( ( ( A ^ k ) x. 1 ) - ( ( A ^ k ) x. A ) ) )
20	16	mulridd 8038	. . . . . . 7 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. 1 ) = ( A ^ k ) )
21	12, 15	expp1d 10748	. . . . . . . 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 5937	. . . . . 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 11514	. . . 4 |- ( ph -> sum_ k e. ( M..^ N ) ( ( A ^ k ) x. ( 1 - A ) ) = sum_ k e. ( M..^ N ) ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) )
26		oveq2 5927	. . . . 5 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
27		oveq2 5927	. . . . 5 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
28		oveq2 5927	. . . . 5 |- ( j = M -> ( A ^ j ) = ( A ^ M ) )
29		oveq2 5927	. . . . 5 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
30	9	adantr 276	. . . . . 6 |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )
31		elfzuz 10090	. . . . . . 7 |- ( j e. ( M ... N ) -> j e. ( ZZ>= ` M ) )
32		eluznn0 9667	. . . . . . 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 10747	. . . . 5 |- ( ( ph /\ j e. ( M ... N ) ) -> ( A ^ j ) e. CC )
35	26, 27, 28, 29, 3, 34	telfsumo 11612	. . . 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 10747	. . . . 5 |- ( ph -> ( A ^ M ) e. CC )
38		eluznn0 9667	. . . . . . 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 10747	. . . . 5 |- ( ph -> ( A ^ N ) e. CC )
41	37, 40	subcld 8332	. . . 4 |- ( ph -> ( ( A ^ M ) - ( A ^ N ) ) e. CC )
42	7, 16	fsumcl 11546	. . . 4 |- ( ph -> sum_ k e. ( M..^ N ) ( A ^ k ) e. CC )
43		geosergap.2	. . . . . . 7 |- ( ph -> A # 1 )
44		1cnd 8037	. . . . . . . 8 |- ( ph -> 1 e. CC )
45		apneg 8632	. . . . . . . 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 8319	. . . . . . 7 |- ( ph -> -u A e. CC )
49	44	negcld 8319	. . . . . . 7 |- ( ph -> -u 1 e. CC )
50		apadd2 8630	. . . . . . 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 8338	. . . . 5 |- ( ph -> ( 1 + -u A ) = ( 1 - A ) )
54		1pneg1e0 9095	. . . . . 6 |- ( 1 + -u 1 ) = 0
55	54	a1i 9	. . . . 5 |- ( ph -> ( 1 + -u 1 ) = 0 )
56	52, 53, 55	3brtr3d 4061	. . . 4 |- ( ph -> ( 1 - A ) # 0 )
57	41, 42, 11, 56	divmulap3d 8846	. . 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 4030   ` cfv 5255  (class class class)co 5919   Fincfn 6796   CCcc 7872   0cc0 7874   1c1 7875   + caddc 7877   x. cmul 7879   - cmin 8192   -ucneg 8193   # cap 8602   / cdiv 8693   NN0cn0 9243   ZZcz 9320   ZZ>=cuz 9595   ...cfz 10077  ..^cfzo 10211   ^cexp 10612   sum_csu 11499

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 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

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 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-frec 6446  df-1o 6471  df-oadd 6475  df-er 6589  df-en 6797  df-dom 6798  df-fin 6799  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-fz 10078  df-fzo 10212  df-seqfrec 10522  df-exp 10613  df-ihash 10850  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425  df-sumdc 11500

This theorem is referenced by:  geoserap  11653  cvgratnnlemsumlt  11674  cvgcmp2nlemabs  15592