Theorem geosergap 12217

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 9716	. . . . . 6 |- ( ph -> M e. ZZ )
3		geoserg.4	. . . . . . 7 |- ( ph -> N e. ( ZZ>= ` M ) )
4		eluzelz 9881	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> N e. ZZ )
6		fzofig 10818	. . . . . 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 8236	. . . . . 6 |- 1 e. CC
9		geoserg.1	. . . . . 6 |- ( ph -> A e. CC )
10		subcl 8488	. . . . . 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 10507	. . . . . . 7 |- ( k e. ( M..^ N ) -> k e. ( ZZ>= ` M ) )
14		eluznn0 9949	. . . . . . 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 11060	. . . . 5 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( A ^ k ) e. CC )
17	7, 11, 16	fsummulc1 12160	. . . 4 |- ( ph -> ( sum_ k e. ( M..^ N ) ( A ^ k ) x. ( 1 - A ) ) = sum_ k e. ( M..^ N ) ( ( A ^ k ) x. ( 1 - A ) ) )
18		1cnd 8306	. . . . . . 7 |- ( ( ph /\ k e. ( M..^ N ) ) -> 1 e. CC )
19	16, 18, 12	subdid 8704	. . . . . 6 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. ( 1 - A ) ) = ( ( ( A ^ k ) x. 1 ) - ( ( A ^ k ) x. A ) ) )
20	16	mulridd 8307	. . . . . . 7 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. 1 ) = ( A ^ k ) )
21	12, 15	expp1d 11061	. . . . . . . 8 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) )
22	21	eqcomd 2240	. . . . . . 7 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. A ) = ( A ^ ( k + 1 ) ) )
23	20, 22	oveq12d 6076	. . . . . 6 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( ( A ^ k ) x. 1 ) - ( ( A ^ k ) x. A ) ) = ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) )
24	19, 23	eqtrd 2267	. . . . 5 |- ( ( ph /\ k e. ( M..^ N ) ) -> ( ( A ^ k ) x. ( 1 - A ) ) = ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) )
25	24	sumeq2dv 12078	. . . 4 |- ( ph -> sum_ k e. ( M..^ N ) ( ( A ^ k ) x. ( 1 - A ) ) = sum_ k e. ( M..^ N ) ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) )
26		oveq2 6066	. . . . 5 |- ( j = k -> ( A ^ j ) = ( A ^ k ) )
27		oveq2 6066	. . . . 5 |- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) )
28		oveq2 6066	. . . . 5 |- ( j = M -> ( A ^ j ) = ( A ^ M ) )
29		oveq2 6066	. . . . 5 |- ( j = N -> ( A ^ j ) = ( A ^ N ) )
30	9	adantr 276	. . . . . 6 |- ( ( ph /\ j e. ( M ... N ) ) -> A e. CC )
31		elfzuz 10374	. . . . . . 7 |- ( j e. ( M ... N ) -> j e. ( ZZ>= ` M ) )
32		eluznn0 9949	. . . . . . 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 11060	. . . . 5 |- ( ( ph /\ j e. ( M ... N ) ) -> ( A ^ j ) e. CC )
35	26, 27, 28, 29, 3, 34	telfsumo 12177	. . . 4 |- ( ph -> sum_ k e. ( M..^ N ) ( ( A ^ k ) - ( A ^ ( k + 1 ) ) ) = ( ( A ^ M ) - ( A ^ N ) ) )
36	17, 25, 35	3eqtrrd 2272	. . 3 |- ( ph -> ( ( A ^ M ) - ( A ^ N ) ) = ( sum_ k e. ( M..^ N ) ( A ^ k ) x. ( 1 - A ) ) )
37	9, 1	expcld 11060	. . . . 5 |- ( ph -> ( A ^ M ) e. CC )
38		eluznn0 9949	. . . . . . 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 11060	. . . . 5 |- ( ph -> ( A ^ N ) e. CC )
41	37, 40	subcld 8600	. . . 4 |- ( ph -> ( ( A ^ M ) - ( A ^ N ) ) e. CC )
42	7, 16	fsumcl 12111	. . . 4 |- ( ph -> sum_ k e. ( M..^ N ) ( A ^ k ) e. CC )
43		geosergap.2	. . . . . . 7 |- ( ph -> A # 1 )
44		1cnd 8306	. . . . . . . 8 |- ( ph -> 1 e. CC )
45		apneg 8902	. . . . . . . 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 8587	. . . . . . 7 |- ( ph -> -u A e. CC )
49	44	negcld 8587	. . . . . . 7 |- ( ph -> -u 1 e. CC )
50		apadd2 8900	. . . . . . 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 1274	. . . . . 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 8606	. . . . 5 |- ( ph -> ( 1 + -u A ) = ( 1 - A ) )
54		1pneg1e0 9365	. . . . . 6 |- ( 1 + -u 1 ) = 0
55	54	a1i 9	. . . . 5 |- ( ph -> ( 1 + -u 1 ) = 0 )
56	52, 53, 55	3brtr3d 4145	. . . 4 |- ( ph -> ( 1 - A ) # 0 )
57	41, 42, 11, 56	divmulap3d 9116	. . 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 2240	1 |- ( ph -> sum_ k e. ( M..^ N ) ( A ^ k ) = ( ( ( A ^ M ) - ( A ^ N ) ) / ( 1 - A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   Fincfn 6988   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   - cmin 8460   -ucneg 8461   # cap 8872   / cdiv 8963   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361  ..^cfzo 10498   ^cexp 10924   sum_csu 12063

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-sumdc 12064

This theorem is referenced by:  geoserap  12218  cvgratnnlemsumlt  12239  cvgcmp2nlemabs  16942