Theorem geosergap 12085

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 9600	. . . . . 6 |- ( ph -> M e. ZZ )
3		geoserg.4	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` M ) )
4		eluzelz 9765	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
6		fzofig 10695	. . . . . 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 8125	. . . . . 6 |- 1 e. CC
9		geoserg.1	. . . . . 6 |- ( ph -> A e. CC )
10		subcl 8378	. . . . . 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 10386	. . . . . . 7 |- ( k e. ( M..^ N ) -> k e. ( ZZ>= ` M ) )
14		eluznn0 9833	. . . . . . 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 10936	. . . . 5 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( A ^ k ) e. CC )
17	7, 11, 16	fsummulc1 12028	. . . 4 |- ( ph -> ( sum_ k e. ( M..^ N ) ( A ^ k ) x. ( 1 - A ) ) = sum_ k e. ( M..^ N ) ( ( A ^ k ) x. ( 1 - A ) ) )
18		1cnd 8195	. . . . . . 7 |- ( ( ph /\ k e. ( M..^ N ) ) -> 1 e. CC )
19	16, 18, 12	subdid 8593	. . . . . 6 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. ( 1 - A ) ) = ( ( ( A ^ k ) x. 1 ) - ( ( A ^ k ) x. A ) ) )
20	16	mulridd 8196	. . . . . . 7 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. 1 ) = ( A ^ k ) )
21	12, 15	expp1d 10937	. . . . . . . 8 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
22	21	eqcomd 2237	. . . . . . 7 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. A ) = ( A ^ ( k + 1 ) ) )
23	20, 22	oveq12d 6036	. . . . . 6 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( ( A ^ k ) x. 1 ) - ( ( A ^ k ) x. A ) ) = ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) )
24	19, 23	eqtrd 2264	. . . . 5 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. ( 1 - A ) ) = ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) )
25	24	sumeq2dv 11946	. . . 4 |- ( ph -> sum_ k e. ( M..^ N ) ( ( A ^ k ) x. ( 1 - A ) ) = sum_ k e. ( M..^ N ) ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) )
26		oveq2 6026	. . . . 5 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
27		oveq2 6026	. . . . 5 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
28		oveq2 6026	. . . . 5 |- ( j = M -> ( A ^ j ) = ( A ^ M ) )
29		oveq2 6026	. . . . 5 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
30	9	adantr 276	. . . . . 6 |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )
31		elfzuz 10256	. . . . . . 7 |- ( j e. ( M ... N ) -> j e. ( ZZ>= ` M ) )
32		eluznn0 9833	. . . . . . 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 10936	. . . . 5 |- ( ( ph /\ j e. ( M ... N ) ) -> ( A ^ j ) e. CC )
35	26, 27, 28, 29, 3, 34	telfsumo 12045	. . . 4 |- ( ph -> sum_ k e. ( M..^ N ) ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) = ( ( A ^ M ) - ( A ^ N ) ) )
36	17, 25, 35	3eqtrrd 2269	. . 3 |- ( ph -> ( ( A ^ M ) - ( A ^ N ) ) = ( sum_ k e. ( M..^ N ) ( A ^ k ) x. ( 1 - A ) ) )
37	9, 1	expcld 10936	. . . . 5 |- ( ph -> ( A ^ M ) e. CC )
38		eluznn0 9833	. . . . . . 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 10936	. . . . 5 |- ( ph -> ( A ^ N ) e. CC )
41	37, 40	subcld 8490	. . . 4 |- ( ph -> ( ( A ^ M ) - ( A ^ N ) ) e. CC )
42	7, 16	fsumcl 11979	. . . 4 |- ( ph -> sum_ k e. ( M..^ N ) ( A ^ k ) e. CC )
43		geosergap.2	. . . . . . 7 |- ( ph -> A # 1 )
44		1cnd 8195	. . . . . . . 8 |- ( ph -> 1 e. CC )
45		apneg 8791	. . . . . . . 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 8477	. . . . . . 7 |- ( ph -> -u A e. CC )
49	44	negcld 8477	. . . . . . 7 |- ( ph -> -u 1 e. CC )
50		apadd2 8789	. . . . . . 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 1273	. . . . . 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 8496	. . . . 5 |- ( ph -> ( 1 + -u A ) = ( 1 - A ) )
54		1pneg1e0 9254	. . . . . 6 |- ( 1 + -u 1 ) = 0
55	54	a1i 9	. . . . 5 |- ( ph -> ( 1 + -u 1 ) = 0 )
56	52, 53, 55	3brtr3d 4119	. . . 4 |- ( ph -> ( 1 - A ) # 0 )
57	41, 42, 11, 56	divmulap3d 9005	. . 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 2237	1 |- ( ph -> sum_ k e. ( M..^ N ) ( A ^ k ) = ( ( ( A ^ M ) - ( A ^ N ) ) / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   Fincfn 6909   CCcc 8030   0cc0 8032   1c1 8033   + caddc 8035   x. cmul 8037   - cmin 8350   -ucneg 8351   # cap 8761   / cdiv 8852   NN0cn0 9402   ZZcz 9479   ZZ>=cuz 9755   ...cfz 10243  ..^cfzo 10377   ^cexp 10801   sum_csu 11931

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-oadd 6586  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fz 10244  df-fzo 10378  df-seqfrec 10711  df-exp 10802  df-ihash 11039  df-cj 11420  df-re 11421  df-im 11422  df-rsqrt 11576  df-abs 11577  df-clim 11857  df-sumdc 11932

This theorem is referenced by:  geoserap  12086  cvgratnnlemsumlt  12107  cvgcmp2nlemabs  16687