Theorem trilpolemclim 14068

Description: Lemma for trilpo 14075. 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 5861	. . . . . 6 |- ( n = k -> ( 2 ^ n ) = ( 2 ^ k ) )
3	2	oveq2d 5869	. . . . 5 |- ( n = k -> ( 1 / ( 2 ^ n ) ) = ( 1 / ( 2 ^ k ) ) )
4		fveq2 5496	. . . . 5 |- ( n = k -> ( F ` n ) = ( F ` k ) )
5	3, 4	oveq12d 5871	. . . 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 9615	. . . . . . . . 9 |- 2 e. RR+
8	7	a1i 9	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> 2 e. RR+ )
9	6	nnzd 9333	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> k e. ZZ )
10	8, 9	rpexpcld 10633	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( 2 ^ k ) e. RR+ )
11	10	rpreccld 9664	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( 1 / ( 2 ^ k ) ) e. RR+ )
12	11	rpred 9653	. . . . 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 7920	. . . . . . 7 |- 0 e. RR
15	13, 14	eqeltrdi 2261	. . . . . 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 7919	. . . . . . 7 |- 1 e. RR
18	16, 17	eqeltrdi 2261	. . . . . 6 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( F ` k ) e. RR )
19		trilpolemgt1.f	. . . . . . . 8 |- ( ph -> F : NN --> { 0 , 1 } )
20	19	ffvelrnda 5631	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. { 0 , 1 } )
21		elpri 3606	. . . . . . 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 793	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. RR )
24	12, 23	remulcld 7950	. . . 4 |- ( ( ph /\ k e. NN ) -> ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) e. RR )
25	1, 5, 6, 24	fvmptd3 5589	. . 3 |- ( ( ph /\ k e. NN ) -> ( G ` k ) = ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) )
26	25, 24	eqeltrd 2247	. 2 |- ( ( ph /\ k e. NN ) -> ( G ` k ) e. RR )
27	11	rpge0d 9657	. . . 4 |- ( ( ph /\ k e. NN ) -> 0 <_ ( 1 / ( 2 ^ k ) ) )
28		0le0 8967	. . . . . 6 |- 0 <_ 0
29	28, 13	breqtrrid 4027	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> 0 <_ ( F ` k ) )
30		0le1 8400	. . . . . 6 |- 0 <_ 1
31	30, 16	breqtrrid 4027	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> 0 <_ ( F ` k ) )
32	29, 31, 22	mpjaodan 793	. . . 4 |- ( ( ph /\ k e. NN ) -> 0 <_ ( F ` k ) )
33	12, 23, 27, 32	mulge0d 8540	. . 3 |- ( ( ph /\ k e. NN ) -> 0 <_ ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) )
34	33, 25	breqtrrd 4017	. 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 5869	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) = ( ( 1 / ( 2 ^ k ) ) x. 0 ) )
37	11	rpcnd 9655	. . . . . . 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 8312	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( ( 1 / ( 2 ^ k ) ) x. 0 ) = 0 )
40	35, 36, 39	3eqtrd 2207	. . . 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 4011	. . 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 5869	. . . . 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 7937	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( ( 1 / ( 2 ^ k ) ) x. 1 ) = ( 1 / ( 2 ^ k ) ) )
47	43, 44, 46	3eqtrd 2207	. . . 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 8433	. . . 4 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( 1 / ( 2 ^ k ) ) <_ ( 1 / ( 2 ^ k ) ) )
50	47, 49	eqbrtrd 4011	. . 3 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( G ` k ) <_ ( 1 / ( 2 ^ k ) ) )
51	42, 50, 22	mpjaodan 793	. 2 |- ( ( ph /\ k e. NN ) -> ( G ` k ) <_ ( 1 / ( 2 ^ k ) ) )
52	26, 34, 51	cvgcmp2n 14065	1 |- ( ph -> seq 1 ( + , G ) e. dom ~~> )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 703   = wceq 1348   e. wcel 2141   {cpr 3584   class class class wbr 3989   |-> cmpt 4050   dom cdm 4611   -->wf 5194   ` cfv 5198  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   <_ cle 7955   / cdiv 8589   NNcn 8878   2c2 8929   RR+crp 9610   seqcseq 10401   ^cexp 10475   ~~> cli 11241

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-ico 9851  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317

This theorem is referenced by:  trilpolemcl  14069  trilpolemisumle  14070  trilpolemeq1  14072  trilpolemlt1  14073  nconstwlpolemgt0  14095