Theorem trilpolemclim 15975

Description: Lemma for trilpo 15982. 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 5952	. . . . . 6 |- ( n = k -> ( 2 ^ n ) = ( 2 ^ k ) )
3	2	oveq2d 5960	. . . . 5 |- ( n = k -> ( 1 / ( 2 ^ n ) ) = ( 1 / ( 2 ^ k ) ) )
4		fveq2 5576	. . . . 5 |- ( n = k -> ( F ` n ) = ( F ` k ) )
5	3, 4	oveq12d 5962	. . . 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 9780	. . . . . . . . 9 |- 2 e. RR+
8	7	a1i 9	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> 2 e. RR+ )
9	6	nnzd 9494	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> k e. ZZ )
10	8, 9	rpexpcld 10842	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( 2 ^ k ) e. RR+ )
11	10	rpreccld 9829	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( 1 / ( 2 ^ k ) ) e. RR+ )
12	11	rpred 9818	. . . . 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 8072	. . . . . . 7 |- 0 e. RR
15	13, 14	eqeltrdi 2296	. . . . . 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 8071	. . . . . . 7 |- 1 e. RR
18	16, 17	eqeltrdi 2296	. . . . . 6 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( F ` k ) e. RR )
19		trilpolemgt1.f	. . . . . . . 8 |- ( ph -> F : NN --> { 0 , 1 } )
20	19	ffvelcdmda 5715	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. { 0 , 1 } )
21		elpri 3656	. . . . . . 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 800	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. RR )
24	12, 23	remulcld 8103	. . . 4 |- ( ( ph /\ k e. NN ) -> ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) e. RR )
25	1, 5, 6, 24	fvmptd3 5673	. . 3 |- ( ( ph /\ k e. NN ) -> ( G ` k ) = ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) )
26	25, 24	eqeltrd 2282	. 2 |- ( ( ph /\ k e. NN ) -> ( G ` k ) e. RR )
27	11	rpge0d 9822	. . . 4 |- ( ( ph /\ k e. NN ) -> 0 <_ ( 1 / ( 2 ^ k ) ) )
28		0le0 9125	. . . . . 6 |- 0 <_ 0
29	28, 13	breqtrrid 4082	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> 0 <_ ( F ` k ) )
30		0le1 8554	. . . . . 6 |- 0 <_ 1
31	30, 16	breqtrrid 4082	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> 0 <_ ( F ` k ) )
32	29, 31, 22	mpjaodan 800	. . . 4 |- ( ( ph /\ k e. NN ) -> 0 <_ ( F ` k ) )
33	12, 23, 27, 32	mulge0d 8694	. . 3 |- ( ( ph /\ k e. NN ) -> 0 <_ ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) )
34	33, 25	breqtrrd 4072	. 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 5960	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) = ( ( 1 / ( 2 ^ k ) ) x. 0 ) )
37	11	rpcnd 9820	. . . . . . 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 8465	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( ( 1 / ( 2 ^ k ) ) x. 0 ) = 0 )
40	35, 36, 39	3eqtrd 2242	. . . 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 4066	. . 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 5960	. . . . 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 8089	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( ( 1 / ( 2 ^ k ) ) x. 1 ) = ( 1 / ( 2 ^ k ) ) )
47	43, 44, 46	3eqtrd 2242	. . . 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 8587	. . . 4 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( 1 / ( 2 ^ k ) ) <_ ( 1 / ( 2 ^ k ) ) )
50	47, 49	eqbrtrd 4066	. . 3 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( G ` k ) <_ ( 1 / ( 2 ^ k ) ) )
51	42, 50, 22	mpjaodan 800	. 2 |- ( ( ph /\ k e. NN ) -> ( G ` k ) <_ ( 1 / ( 2 ^ k ) ) )
52	26, 34, 51	cvgcmp2n 15972	1 |- ( ph -> seq 1 ( + , G ) e. dom ~~> )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 710   = wceq 1373   e. wcel 2176   {cpr 3634   class class class wbr 4044   |-> cmpt 4105   dom cdm 4675   -->wf 5267   ` cfv 5271  (class class class)co 5944   CCcc 7923   RRcr 7924   0cc0 7925   1c1 7926   + caddc 7928   x. cmul 7930   <_ cle 8108   / cdiv 8745   NNcn 9036   2c2 9087   RR+crp 9775   seqcseq 10592   ^cexp 10683   ~~> cli 11589

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-frec 6477  df-1o 6502  df-oadd 6506  df-er 6620  df-en 6828  df-dom 6829  df-fin 6830  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-ico 10016  df-fz 10131  df-fzo 10265  df-seqfrec 10593  df-exp 10684  df-ihash 10921  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-clim 11590  df-sumdc 11665

This theorem is referenced by:  trilpolemcl  15976  trilpolemisumle  15977  trilpolemeq1  15979  trilpolemlt1  15980  nconstwlpolemgt0  16003