Theorem trilpolemclim 14720

Description: Lemma for trilpo 14727. Convergence of the series. (Contributed by Jim Kingdon, 24-Aug-2023.)

Hypotheses

Ref	Expression
trilpolemgt1.f	|- ( ph -> F : NN --> { 0 , 1 } )
trilpolemclim.g	|- G = ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) )

Assertion

Ref	Expression
trilpolemclim	|- ( ph -> seq 1 ( + , G ) e. dom ~~> )

Distinct variable group:   n, F

Allowed substitution hints:   ph( n)   G( n)

Proof of Theorem trilpolemclim

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		trilpolemclim.g	. . . 4 |- G = ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) )
2		oveq2 5882	. . . . . 6 |- ( n = k -> ( 2 ^ n ) = ( 2 ^ k ) )
3	2	oveq2d 5890	. . . . 5 |- ( n = k -> ( 1 / ( 2 ^ n ) ) = ( 1 / ( 2 ^ k ) ) )
4		fveq2 5515	. . . . 5 |- ( n = k -> ( F ` n ) = ( F ` k ) )
5	3, 4	oveq12d 5892	. . . 4 |- ( n = k -> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) = ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) )
6		simpr 110	. . . 4 |- ( ( ph /\ k e. NN ) -> k e. NN )
7		2rp 9657	. . . . . . . . 9 |- 2 e. RR+
8	7	a1i 9	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> 2 e. RR+ )
9	6	nnzd 9373	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> k e. ZZ )
10	8, 9	rpexpcld 10677	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( 2 ^ k ) e. RR+ )
11	10	rpreccld 9706	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( 1 / ( 2 ^ k ) ) e. RR+ )
12	11	rpred 9695	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( 1 / ( 2 ^ k ) ) e. RR )
13		simpr 110	. . . . . . 7 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( F ` k ) = 0 )
14		0re 7956	. . . . . . 7 |- 0 e. RR
15	13, 14	eqeltrdi 2268	. . . . . 6 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( F ` k ) e. RR )
16		simpr 110	. . . . . . 7 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( F ` k ) = 1 )
17		1re 7955	. . . . . . 7 |- 1 e. RR
18	16, 17	eqeltrdi 2268	. . . . . 6 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( F ` k ) e. RR )
19		trilpolemgt1.f	. . . . . . . 8 |- ( ph -> F : NN --> { 0 , 1 } )
20	19	ffvelcdmda 5651	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. { 0 , 1 } )
21		elpri 3615	. . . . . . 7 |- ( ( F ` k ) e. { 0 , 1 } -> ( ( F ` k ) = 0 \/ ( F ` k ) = 1 ) )
22	20, 21	syl 14	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( ( F ` k ) = 0 \/ ( F ` k ) = 1 ) )
23	15, 18, 22	mpjaodan 798	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. RR )
24	12, 23	remulcld 7987	. . . 4 |- ( ( ph /\ k e. NN ) -> ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) e. RR )
25	1, 5, 6, 24	fvmptd3 5609	. . 3 |- ( ( ph /\ k e. NN ) -> ( G ` k ) = ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) )
26	25, 24	eqeltrd 2254	. 2 |- ( ( ph /\ k e. NN ) -> ( G ` k ) e. RR )
27	11	rpge0d 9699	. . . 4 |- ( ( ph /\ k e. NN ) -> 0 <_ ( 1 / ( 2 ^ k ) ) )
28		0le0 9007	. . . . . 6 |- 0 <_ 0
29	28, 13	breqtrrid 4041	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> 0 <_ ( F ` k ) )
30		0le1 8437	. . . . . 6 |- 0 <_ 1
31	30, 16	breqtrrid 4041	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> 0 <_ ( F ` k ) )
32	29, 31, 22	mpjaodan 798	. . . 4 |- ( ( ph /\ k e. NN ) -> 0 <_ ( F ` k ) )
33	12, 23, 27, 32	mulge0d 8577	. . 3 |- ( ( ph /\ k e. NN ) -> 0 <_ ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) )
34	33, 25	breqtrrd 4031	. 2 |- ( ( ph /\ k e. NN ) -> 0 <_ ( G ` k ) )
35	25	adantr 276	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( G ` k ) = ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) )
36	13	oveq2d 5890	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) = ( ( 1 / ( 2 ^ k ) ) x. 0 ) )
37	11	rpcnd 9697	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( 1 / ( 2 ^ k ) ) e. CC )
38	37	adantr 276	. . . . . 6 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( 1 / ( 2 ^ k ) ) e. CC )
39	38	mul01d 8349	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( ( 1 / ( 2 ^ k ) ) x. 0 ) = 0 )
40	35, 36, 39	3eqtrd 2214	. . . 4 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( G ` k ) = 0 )
41	27	adantr 276	. . . 4 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> 0 <_ ( 1 / ( 2 ^ k ) ) )
42	40, 41	eqbrtrd 4025	. . 3 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( G ` k ) <_ ( 1 / ( 2 ^ k ) ) )
43	25	adantr 276	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( G ` k ) = ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) )
44	16	oveq2d 5890	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) = ( ( 1 / ( 2 ^ k ) ) x. 1 ) )
45	37	adantr 276	. . . . . 6 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( 1 / ( 2 ^ k ) ) e. CC )
46	45	mulridd 7973	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( ( 1 / ( 2 ^ k ) ) x. 1 ) = ( 1 / ( 2 ^ k ) ) )
47	43, 44, 46	3eqtrd 2214	. . . 4 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( G ` k ) = ( 1 / ( 2 ^ k ) ) )
48	12	adantr 276	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( 1 / ( 2 ^ k ) ) e. RR )
49	48	leidd 8470	. . . 4 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( 1 / ( 2 ^ k ) ) <_ ( 1 / ( 2 ^ k ) ) )
50	47, 49	eqbrtrd 4025	. . 3 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( G ` k ) <_ ( 1 / ( 2 ^ k ) ) )
51	42, 50, 22	mpjaodan 798	. 2 |- ( ( ph /\ k e. NN ) -> ( G ` k ) <_ ( 1 / ( 2 ^ k ) ) )
52	26, 34, 51	cvgcmp2n 14717	1 |- ( ph -> seq 1 ( + , G ) e. dom ~~> )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 708   = wceq 1353   e. wcel 2148   {cpr 3593   class class class wbr 4003   |-> cmpt 4064   dom cdm 4626   -->wf 5212   ` cfv 5216  (class class class)co 5874   CCcc 7808   RRcr 7809   0cc0 7810   1c1 7811   + caddc 7813   x. cmul 7815   <_ cle 7992   / cdiv 8628   NNcn 8918   2c2 8969   RR+crp 9652   seqcseq 10444   ^cexp 10518   ~~> cli 11285

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-3 8978  df-4 8979  df-n0 9176  df-z 9253  df-uz 9528  df-q 9619  df-rp 9653  df-ico 9893  df-fz 10008  df-fzo 10142  df-seqfrec 10445  df-exp 10519  df-ihash 10755  df-cj 10850  df-re 10851  df-im 10852  df-rsqrt 11006  df-abs 11007  df-clim 11286  df-sumdc 11361

This theorem is referenced by:  trilpolemcl  14721  trilpolemisumle  14722  trilpolemeq1  14724  trilpolemlt1  14725  nconstwlpolemgt0  14747