Theorem trilpolemclim 13569

Description: Lemma for trilpo 13576. 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 5826	. . . . . 6 |- ( n = k -> ( 2 ^ n ) = ( 2 ^ k ) )
3	2	oveq2d 5834	. . . . 5 |- ( n = k -> ( 1 / ( 2 ^ n ) ) = ( 1 / ( 2 ^ k ) ) )
4		fveq2 5465	. . . . 5 |- ( n = k -> ( F ` n ) = ( F ` k ) )
5	3, 4	oveq12d 5836	. . . 4 |- ( n = k -> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) = ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) )
6		simpr 109	. . . 4 |- ( ( ph /\ k e. NN ) -> k e. NN )
7		2rp 9547	. . . . . . . . 9 |- 2 e. RR+
8	7	a1i 9	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> 2 e. RR+ )
9	6	nnzd 9268	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> k e. ZZ )
10	8, 9	rpexpcld 10557	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( 2 ^ k ) e. RR+ )
11	10	rpreccld 9596	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( 1 / ( 2 ^ k ) ) e. RR+ )
12	11	rpred 9585	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( 1 / ( 2 ^ k ) ) e. RR )
13		simpr 109	. . . . . . 7 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( F ` k ) = 0 )
14		0re 7861	. . . . . . 7 |- 0 e. RR
15	13, 14	eqeltrdi 2248	. . . . . 6 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( F ` k ) e. RR )
16		simpr 109	. . . . . . 7 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( F ` k ) = 1 )
17		1re 7860	. . . . . . 7 |- 1 e. RR
18	16, 17	eqeltrdi 2248	. . . . . 6 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( F ` k ) e. RR )
19		trilpolemgt1.f	. . . . . . . 8 |- ( ph -> F : NN --> { 0 , 1 } )
20	19	ffvelrnda 5599	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. { 0 , 1 } )
21		elpri 3583	. . . . . . 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 788	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. RR )
24	12, 23	remulcld 7891	. . . 4 |- ( ( ph /\ k e. NN ) -> ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) e. RR )
25	1, 5, 6, 24	fvmptd3 5558	. . 3 |- ( ( ph /\ k e. NN ) -> ( G ` k ) = ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) )
26	25, 24	eqeltrd 2234	. 2 |- ( ( ph /\ k e. NN ) -> ( G ` k ) e. RR )
27	11	rpge0d 9589	. . . 4 |- ( ( ph /\ k e. NN ) -> 0 <_ ( 1 / ( 2 ^ k ) ) )
28		0le0 8905	. . . . . 6 |- 0 <_ 0
29	28, 13	breqtrrid 4002	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> 0 <_ ( F ` k ) )
30		0le1 8339	. . . . . 6 |- 0 <_ 1
31	30, 16	breqtrrid 4002	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> 0 <_ ( F ` k ) )
32	29, 31, 22	mpjaodan 788	. . . 4 |- ( ( ph /\ k e. NN ) -> 0 <_ ( F ` k ) )
33	12, 23, 27, 32	mulge0d 8479	. . 3 |- ( ( ph /\ k e. NN ) -> 0 <_ ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) )
34	33, 25	breqtrrd 3992	. 2 |- ( ( ph /\ k e. NN ) -> 0 <_ ( G ` k ) )
35	25	adantr 274	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( G ` k ) = ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) )
36	13	oveq2d 5834	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) = ( ( 1 / ( 2 ^ k ) ) x. 0 ) )
37	11	rpcnd 9587	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( 1 / ( 2 ^ k ) ) e. CC )
38	37	adantr 274	. . . . . 6 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( 1 / ( 2 ^ k ) ) e. CC )
39	38	mul01d 8251	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( ( 1 / ( 2 ^ k ) ) x. 0 ) = 0 )
40	35, 36, 39	3eqtrd 2194	. . . 4 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( G ` k ) = 0 )
41	27	adantr 274	. . . 4 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> 0 <_ ( 1 / ( 2 ^ k ) ) )
42	40, 41	eqbrtrd 3986	. . 3 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( G ` k ) <_ ( 1 / ( 2 ^ k ) ) )
43	25	adantr 274	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( G ` k ) = ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) )
44	16	oveq2d 5834	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) = ( ( 1 / ( 2 ^ k ) ) x. 1 ) )
45	37	adantr 274	. . . . . 6 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( 1 / ( 2 ^ k ) ) e. CC )
46	45	mulid1d 7878	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( ( 1 / ( 2 ^ k ) ) x. 1 ) = ( 1 / ( 2 ^ k ) ) )
47	43, 44, 46	3eqtrd 2194	. . . 4 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( G ` k ) = ( 1 / ( 2 ^ k ) ) )
48	12	adantr 274	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( 1 / ( 2 ^ k ) ) e. RR )
49	48	leidd 8372	. . . 4 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( 1 / ( 2 ^ k ) ) <_ ( 1 / ( 2 ^ k ) ) )
50	47, 49	eqbrtrd 3986	. . 3 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( G ` k ) <_ ( 1 / ( 2 ^ k ) ) )
51	42, 50, 22	mpjaodan 788	. 2 |- ( ( ph /\ k e. NN ) -> ( G ` k ) <_ ( 1 / ( 2 ^ k ) ) )
52	26, 34, 51	cvgcmp2n 13566	1 |- ( ph -> seq 1 ( + , G ) e. dom ~~> )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698   = wceq 1335   e. wcel 2128   {cpr 3561   class class class wbr 3965   |-> cmpt 4025   dom cdm 4583   -->wf 5163   ` cfv 5167  (class class class)co 5818   CCcc 7713   RRcr 7714   0cc0 7715   1c1 7716   + caddc 7718   x. cmul 7720   <_ cle 7896   / cdiv 8528   NNcn 8816   2c2 8867   RR+crp 9542   seqcseq 10326   ^cexp 10400   ~~> cli 11157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-setind 4494  ax-iinf 4545  ax-cnex 7806  ax-resscn 7807  ax-1cn 7808  ax-1re 7809  ax-icn 7810  ax-addcl 7811  ax-addrcl 7812  ax-mulcl 7813  ax-mulrcl 7814  ax-addcom 7815  ax-mulcom 7816  ax-addass 7817  ax-mulass 7818  ax-distr 7819  ax-i2m1 7820  ax-0lt1 7821  ax-1rid 7822  ax-0id 7823  ax-rnegex 7824  ax-precex 7825  ax-cnre 7826  ax-pre-ltirr 7827  ax-pre-ltwlin 7828  ax-pre-lttrn 7829  ax-pre-apti 7830  ax-pre-ltadd 7831  ax-pre-mulgt0 7832  ax-pre-mulext 7833  ax-arch 7834  ax-caucvg 7835

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4252  df-po 4255  df-iso 4256  df-iord 4325  df-on 4327  df-ilim 4328  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-isom 5176  df-riota 5774  df-ov 5821  df-oprab 5822  df-mpo 5823  df-1st 6082  df-2nd 6083  df-recs 6246  df-irdg 6311  df-frec 6332  df-1o 6357  df-oadd 6361  df-er 6473  df-en 6679  df-dom 6680  df-fin 6681  df-pnf 7897  df-mnf 7898  df-xr 7899  df-ltxr 7900  df-le 7901  df-sub 8031  df-neg 8032  df-reap 8433  df-ap 8440  df-div 8529  df-inn 8817  df-2 8875  df-3 8876  df-4 8877  df-n0 9074  df-z 9151  df-uz 9423  df-q 9511  df-rp 9543  df-ico 9780  df-fz 9895  df-fzo 10024  df-seqfrec 10327  df-exp 10401  df-ihash 10632  df-cj 10724  df-re 10725  df-im 10726  df-rsqrt 10880  df-abs 10881  df-clim 11158  df-sumdc 11233

This theorem is referenced by:  trilpolemcl  13570  trilpolemisumle  13571  trilpolemeq1  13573  trilpolemlt1  13574  nconstwlpolemgt0  13596