Theorem trilpolemclim 16990

Description: Lemma for trilpo 16997. 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 6083	. . . . . 6 |- ( n = k -> ( 2 ^ n ) = ( 2 ^ k ) )
3	2	oveq2d 6091	. . . . 5 |- ( n = k -> ( 1 / ( 2 ^ n ) ) = ( 1 / ( 2 ^ k ) ) )
4		fveq2 5690	. . . . 5 |- ( n = k -> ( F ` n ) = ( F ` k ) )
5	3, 4	oveq12d 6093	. . . 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 10038	. . . . . . . . 9 |- 2 e. RR+
8	7	a1i 9	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> 2 e. RR+ )
9	6	nnzd 9746	. . . . . . . 8 |- ( ( ph /\ k e. NN ) -> k e. ZZ )
10	8, 9	rpexpcld 11113	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( 2 ^ k ) e. RR+ )
11	10	rpreccld 10087	. . . . . 6 |- ( ( ph /\ k e. NN ) -> ( 1 / ( 2 ^ k ) ) e. RR+ )
12	11	rpred 10076	. . . . 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 8316	. . . . . . 7 |- 0 e. RR
15	13, 14	eqeltrdi 2329	. . . . . 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 8315	. . . . . . 7 |- 1 e. RR
18	16, 17	eqeltrdi 2329	. . . . . 6 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( F ` k ) e. RR )
19		trilpolemgt1.f	. . . . . . . 8 |- ( ph -> F : NN --> { 0 , 1 } )
20	19	ffvelcdmda 5834	. . . . . . 7 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. { 0 , 1 } )
21		elpri 3728	. . . . . . 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 810	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( F ` k ) e. RR )
24	12, 23	remulcld 8346	. . . 4 |- ( ( ph /\ k e. NN ) -> ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) e. RR )
25	1, 5, 6, 24	fvmptd3 5793	. . 3 |- ( ( ph /\ k e. NN ) -> ( G ` k ) = ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) )
26	25, 24	eqeltrd 2315	. 2 |- ( ( ph /\ k e. NN ) -> ( G ` k ) e. RR )
27	11	rpge0d 10080	. . . 4 |- ( ( ph /\ k e. NN ) -> 0 <_ ( 1 / ( 2 ^ k ) ) )
28		0le0 9372	. . . . . 6 |- 0 <_ 0
29	28, 13	breqtrrid 4163	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> 0 <_ ( F ` k ) )
30		0le1 8799	. . . . . 6 |- 0 <_ 1
31	30, 16	breqtrrid 4163	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> 0 <_ ( F ` k ) )
32	29, 31, 22	mpjaodan 810	. . . 4 |- ( ( ph /\ k e. NN ) -> 0 <_ ( F ` k ) )
33	12, 23, 27, 32	mulge0d 8939	. . 3 |- ( ( ph /\ k e. NN ) -> 0 <_ ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) )
34	33, 25	breqtrrd 4153	. 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 6091	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( ( 1 / ( 2 ^ k ) ) x. ( F ` k ) ) = ( ( 1 / ( 2 ^ k ) ) x. 0 ) )
37	11	rpcnd 10078	. . . . . . 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 8710	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 0 ) -> ( ( 1 / ( 2 ^ k ) ) x. 0 ) = 0 )
40	35, 36, 39	3eqtrd 2275	. . . 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 4147	. . 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 6091	. . . . 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 8333	. . . . 5 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( ( 1 / ( 2 ^ k ) ) x. 1 ) = ( 1 / ( 2 ^ k ) ) )
47	43, 44, 46	3eqtrd 2275	. . . 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 8832	. . . 4 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( 1 / ( 2 ^ k ) ) <_ ( 1 / ( 2 ^ k ) ) )
50	47, 49	eqbrtrd 4147	. . 3 |- ( ( ( ph /\ k e. NN ) /\ ( F ` k ) = 1 ) -> ( G ` k ) <_ ( 1 / ( 2 ^ k ) ) )
51	42, 50, 22	mpjaodan 810	. 2 |- ( ( ph /\ k e. NN ) -> ( G ` k ) <_ ( 1 / ( 2 ^ k ) ) )
52	26, 34, 51	cvgcmp2n 16987	1 |- ( ph -> seq 1 ( + , G ) e. dom ~~> )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   {cpr 3706   class class class wbr 4125   |-> cmpt 4187   dom cdm 4769   -->wf 5368   ` cfv 5372  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   <_ cle 8351   / cdiv 8992   NNcn 9283   2c2 9334   RR+crp 10033   seqcseq 10862   ^cexp 10953   ~~> cli 12022

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-ico 10275  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-ihash 11193  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-sumdc 12098

This theorem is referenced by:  trilpolemcl  16991  trilpolemisumle  16992  trilpolemeq1  16994  trilpolemlt1  16995  nconstwlpolemgt0  17019