Theorem trilpolemisumle 16822

Description: Lemma for trilpo 16827. An upper bound for the sum of the digits beyond a certain point. (Contributed by Jim Kingdon, 28-Aug-2023.)

Hypotheses

Ref	Expression
trilpolemgt1.f	|- ( ph -> F : NN --> { 0 , 1 } )
trilpolemgt1.a	|- A = sum_ i e. NN ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) )
trilpolemisumle.z	|- Z = ( ZZ>= ` M )
trilpolemisumle.m	|- ( ph -> M e. NN )

Assertion

Ref	Expression
trilpolemisumle	|- ( ph -> sum_ i e. Z ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) <_ sum_ i e. Z ( 1 / ( 2 ^ i ) ) )

Distinct variable groups:   i, F   i, M   i, Z   ph, i

Allowed substitution hint:   A( i)

Proof of Theorem trilpolemisumle

Dummy variables n j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		trilpolemisumle.z	. 2 |- Z = ( ZZ>= ` M )
2		trilpolemisumle.m	. . 3 |- ( ph -> M e. NN )
3	2	nnzd 9699	. 2 |- ( ph -> M e. ZZ )
4	1	eleq2i 2299	. . . . 5 |- ( i e. Z <-> i e. ( ZZ>= ` M ) )
5	4	biimpi 120	. . . 4 |- ( i e. Z -> i e. ( ZZ>= ` M ) )
6		eluznn 9932	. . . 4 |- ( ( M e. NN /\ i e. ( ZZ>= ` M ) ) -> i e. NN )
7	2, 5, 6	syl2an 289	. . 3 |- ( ( ph /\ i e. Z ) -> i e. NN )
8		eqid 2232	. . . 4 |- ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) = ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) )
9		oveq2 6058	. . . . . 6 |- ( n = i -> ( 2 ^ n ) = ( 2 ^ i ) )
10	9	oveq2d 6066	. . . . 5 |- ( n = i -> ( 1 / ( 2 ^ n ) ) = ( 1 / ( 2 ^ i ) ) )
11		fveq2 5670	. . . . 5 |- ( n = i -> ( F ` n ) = ( F ` i ) )
12	10, 11	oveq12d 6068	. . . 4 |- ( n = i -> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) = ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) )
13		simpr 110	. . . 4 |- ( ( ph /\ i e. NN ) -> i e. NN )
14		2rp 9991	. . . . . . . . 9 |- 2 e. RR+
15	14	a1i 9	. . . . . . . 8 |- ( ( ph /\ i e. NN ) -> 2 e. RR+ )
16	13	nnzd 9699	. . . . . . . 8 |- ( ( ph /\ i e. NN ) -> i e. ZZ )
17	15, 16	rpexpcld 11059	. . . . . . 7 |- ( ( ph /\ i e. NN ) -> ( 2 ^ i ) e. RR+ )
18	17	rpreccld 10040	. . . . . 6 |- ( ( ph /\ i e. NN ) -> ( 1 / ( 2 ^ i ) ) e. RR+ )
19	18	rpred 10029	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( 1 / ( 2 ^ i ) ) e. RR )
20		trilpolemgt1.f	. . . . . . 7 |- ( ph -> F : NN --> { 0 , 1 } )
21		0re 8274	. . . . . . . . 9 |- 0 e. RR
22		1re 8273	. . . . . . . . 9 |- 1 e. RR
23		prssi 3852	. . . . . . . . 9 |- ( ( 0 e. RR /\ 1 e. RR ) -> { 0 , 1 } C_ RR )
24	21, 22, 23	mp2an 426	. . . . . . . 8 |- { 0 , 1 } C_ RR
25	24	a1i 9	. . . . . . 7 |- ( ph -> { 0 , 1 } C_ RR )
26	20, 25	fssd 5522	. . . . . 6 |- ( ph -> F : NN --> RR )
27	26	ffvelcdmda 5812	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( F ` i ) e. RR )
28	19, 27	remulcld 8304	. . . 4 |- ( ( ph /\ i e. NN ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) e. RR )
29	8, 12, 13, 28	fvmptd3 5771	. . 3 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ` i ) = ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) )
30	7, 29	syldan 282	. 2 |- ( ( ph /\ i e. Z ) -> ( ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ` i ) = ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) )
31	7, 28	syldan 282	. 2 |- ( ( ph /\ i e. Z ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) e. RR )
32		eqid 2232	. . . 4 |- ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) = ( n e. NN |-> ( 1 / ( 2 ^ n ) ) )
33	32, 10, 13, 18	fvmptd3 5771	. . 3 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` i ) = ( 1 / ( 2 ^ i ) ) )
34	7, 33	syldan 282	. 2 |- ( ( ph /\ i e. Z ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` i ) = ( 1 / ( 2 ^ i ) ) )
35	7, 19	syldan 282	. 2 |- ( ( ph /\ i e. Z ) -> ( 1 / ( 2 ^ i ) ) e. RR )
36		simpr 110	. . . . . . 7 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( F ` i ) = 0 )
37	36	oveq2d 6066	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) = ( ( 1 / ( 2 ^ i ) ) x. 0 ) )
38	18	rpcnd 10031	. . . . . . . 8 |- ( ( ph /\ i e. NN ) -> ( 1 / ( 2 ^ i ) ) e. CC )
39	38	adantr 276	. . . . . . 7 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( 1 / ( 2 ^ i ) ) e. CC )
40	39	mul01d 8666	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( ( 1 / ( 2 ^ i ) ) x. 0 ) = 0 )
41	37, 40	eqtrd 2265	. . . . 5 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) = 0 )
42	18	adantr 276	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( 1 / ( 2 ^ i ) ) e. RR+ )
43	42	rpge0d 10033	. . . . 5 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> 0 <_ ( 1 / ( 2 ^ i ) ) )
44	41, 43	eqbrtrd 4131	. . . 4 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) <_ ( 1 / ( 2 ^ i ) ) )
45		simpr 110	. . . . . . 7 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( F ` i ) = 1 )
46	45	oveq2d 6066	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) = ( ( 1 / ( 2 ^ i ) ) x. 1 ) )
47	38	adantr 276	. . . . . . 7 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( 1 / ( 2 ^ i ) ) e. CC )
48	47	mulridd 8291	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( ( 1 / ( 2 ^ i ) ) x. 1 ) = ( 1 / ( 2 ^ i ) ) )
49	46, 48	eqtrd 2265	. . . . 5 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) = ( 1 / ( 2 ^ i ) ) )
50	19	adantr 276	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( 1 / ( 2 ^ i ) ) e. RR )
51	50	leidd 8788	. . . . 5 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( 1 / ( 2 ^ i ) ) <_ ( 1 / ( 2 ^ i ) ) )
52	49, 51	eqbrtrd 4131	. . . 4 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) <_ ( 1 / ( 2 ^ i ) ) )
53	20	ffvelcdmda 5812	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( F ` i ) e. { 0 , 1 } )
54		elpri 3712	. . . . 5 |- ( ( F ` i ) e. { 0 , 1 } -> ( ( F ` i ) = 0 \/ ( F ` i ) = 1 ) )
55	53, 54	syl 14	. . . 4 |- ( ( ph /\ i e. NN ) -> ( ( F ` i ) = 0 \/ ( F ` i ) = 1 ) )
56	44, 52, 55	mpjaodan 806	. . 3 |- ( ( ph /\ i e. NN ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) <_ ( 1 / ( 2 ^ i ) ) )
57	7, 56	syldan 282	. 2 |- ( ( ph /\ i e. Z ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) <_ ( 1 / ( 2 ^ i ) ) )
58	20, 8	trilpolemclim 16820	. . 3 |- ( ph -> seq 1 ( + , ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ) e. dom ~~> )
59		nnuz 9890	. . . 4 |- NN = ( ZZ>= ` 1 )
60	29, 28	eqeltrd 2309	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ` i ) e. RR )
61	60	recnd 8302	. . . 4 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ` i ) e. CC )
62	59, 2, 61	iserex 12024	. . 3 |- ( ph -> ( seq 1 ( + , ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ) e. dom ~~> <-> seq M ( + , ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ) e. dom ~~> ) )
63	58, 62	mpbid 147	. 2 |- ( ph -> seq M ( + , ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ) e. dom ~~> )
64		seqex 10811	. . . 4 |- seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) e. _V
65		rpreccl 10013	. . . . . . . 8 |- ( 2 e. RR+ -> ( 1 / 2 ) e. RR+ )
66	14, 65	ax-mp 5	. . . . . . 7 |- ( 1 / 2 ) e. RR+
67	66	a1i 9	. . . . . 6 |- ( ph -> ( 1 / 2 ) e. RR+ )
68		1zzd 9604	. . . . . 6 |- ( ph -> 1 e. ZZ )
69	67, 68	rpexpcld 11059	. . . . 5 |- ( ph -> ( ( 1 / 2 ) ^ 1 ) e. RR+ )
70		1mhlfehlf 9456	. . . . . . 7 |- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 )
71	70, 66	eqeltri 2305	. . . . . 6 |- ( 1 - ( 1 / 2 ) ) e. RR+
72	71	a1i 9	. . . . 5 |- ( ph -> ( 1 - ( 1 / 2 ) ) e. RR+ )
73	69, 72	rpdivcld 10047	. . . 4 |- ( ph -> ( ( ( 1 / 2 ) ^ 1 ) / ( 1 - ( 1 / 2 ) ) ) e. RR+ )
74		halfcn 9452	. . . . . 6 |- ( 1 / 2 ) e. CC
75	74	a1i 9	. . . . 5 |- ( ph -> ( 1 / 2 ) e. CC )
76		halfge0 9454	. . . . . . . 8 |- 0 <_ ( 1 / 2 )
77		halfre 9451	. . . . . . . . 9 |- ( 1 / 2 ) e. RR
78	77	absidi 11811	. . . . . . . 8 |- ( 0 <_ ( 1 / 2 ) -> ( abs ` ( 1 / 2 ) ) = ( 1 / 2 ) )
79	76, 78	ax-mp 5	. . . . . . 7 |- ( abs ` ( 1 / 2 ) ) = ( 1 / 2 )
80		halflt1 9455	. . . . . . 7 |- ( 1 / 2 ) < 1
81	79, 80	eqbrtri 4130	. . . . . 6 |- ( abs ` ( 1 / 2 ) ) < 1
82	81	a1i 9	. . . . 5 |- ( ph -> ( abs ` ( 1 / 2 ) ) < 1 )
83		1nn0 9512	. . . . . 6 |- 1 e. NN0
84	83	a1i 9	. . . . 5 |- ( ph -> 1 e. NN0 )
85		oveq2 6058	. . . . . . . 8 |- ( n = j -> ( 2 ^ n ) = ( 2 ^ j ) )
86	85	oveq2d 6066	. . . . . . 7 |- ( n = j -> ( 1 / ( 2 ^ n ) ) = ( 1 / ( 2 ^ j ) ) )
87		elnnuz 9891	. . . . . . . . 9 |- ( j e. NN <-> j e. ( ZZ>= ` 1 ) )
88	87	biimpri 133	. . . . . . . 8 |- ( j e. ( ZZ>= ` 1 ) -> j e. NN )
89	88	adantl 277	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> j e. NN )
90	14	a1i 9	. . . . . . . . 9 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> 2 e. RR+ )
91	89	nnzd 9699	. . . . . . . . 9 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> j e. ZZ )
92	90, 91	rpexpcld 11059	. . . . . . . 8 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( 2 ^ j ) e. RR+ )
93	92	rpreccld 10040	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( 1 / ( 2 ^ j ) ) e. RR+ )
94	32, 86, 89, 93	fvmptd3 5771	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` j ) = ( 1 / ( 2 ^ j ) ) )
95		2cnd 9310	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> 2 e. CC )
96	90	rpap0d 10035	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> 2 # 0 )
97	95, 96, 91	exprecapd 11043	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
98	94, 97	eqtr4d 2268	. . . . 5 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` j ) = ( ( 1 / 2 ) ^ j ) )
99	75, 82, 84, 98	geolim2 12198	. . . 4 |- ( ph -> seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) ~~> ( ( ( 1 / 2 ) ^ 1 ) / ( 1 - ( 1 / 2 ) ) ) )
100		breldmg 4962	. . . 4 |- ( ( seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) e. _V /\ ( ( ( 1 / 2 ) ^ 1 ) / ( 1 - ( 1 / 2 ) ) ) e. RR+ /\ seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) ~~> ( ( ( 1 / 2 ) ^ 1 ) / ( 1 - ( 1 / 2 ) ) ) ) -> seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) e. dom ~~> )
101	64, 73, 99, 100	mp3an2i 1379	. . 3 |- ( ph -> seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) e. dom ~~> )
102	33, 38	eqeltrd 2309	. . . 4 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` i ) e. CC )
103	59, 2, 102	iserex 12024	. . 3 |- ( ph -> ( seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) e. dom ~~> <-> seq M ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) e. dom ~~> ) )
104	101, 103	mpbid 147	. 2 |- ( ph -> seq M ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) e. dom ~~> )
105	1, 3, 30, 31, 34, 35, 57, 63, 104	isumle 12181	1 |- ( ph -> sum_ i e. Z ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) <_ sum_ i e. Z ( 1 / ( 2 ^ i ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2203   _Vcvv 2813   C_ wss 3211   {cpr 3690   class class class wbr 4109   |-> cmpt 4171   dom cdm 4749   -->wf 5348   ` cfv 5352  (class class class)co 6050   CCcc 8125   RRcr 8126   0cc0 8127   1c1 8128   + caddc 8130   x. cmul 8132   < clt 8308   <_ cle 8309   - cmin 8444   / cdiv 8946   NNcn 9237   2c2 9288   NN0cn0 9496   ZZ>=cuz 9853   RR+crp 9986   seqcseq 10809   ^cexp 10900   abscabs 11682   ~~> cli 11963   sum_csu 12038

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-oadd 6651  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-ico 10227  df-fz 10343  df-fzo 10477  df-seqfrec 10810  df-exp 10901  df-ihash 11139  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-sumdc 12039

This theorem is referenced by:  trilpolemgt1  16823  trilpolemeq1  16824