Theorem trilpolemisumle 14070

Description: Lemma for trilpo 14075. 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 9333	. 2 |- ( ph -> M e. ZZ )
4	1	eleq2i 2237	. . . . 5 |- ( i e. Z <-> i e. ( ZZ>= ` M ) )
5	4	biimpi 119	. . . 4 |- ( i e. Z -> i e. ( ZZ>= ` M ) )
6		eluznn 9559	. . . 4 |- ( ( M e. NN /\ i e. ( ZZ>= ` M ) ) -> i e. NN )
7	2, 5, 6	syl2an 287	. . 3 |- ( ( ph /\ i e. Z ) -> i e. NN )
8		eqid 2170	. . . 4 |- ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) = ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) )
9		oveq2 5861	. . . . . 6 |- ( n = i -> ( 2 ^ n ) = ( 2 ^ i ) )
10	9	oveq2d 5869	. . . . 5 |- ( n = i -> ( 1 / ( 2 ^ n ) ) = ( 1 / ( 2 ^ i ) ) )
11		fveq2 5496	. . . . 5 |- ( n = i -> ( F ` n ) = ( F ` i ) )
12	10, 11	oveq12d 5871	. . . 4 |- ( n = i -> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) = ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) )
13		simpr 109	. . . 4 |- ( ( ph /\ i e. NN ) -> i e. NN )
14		2rp 9615	. . . . . . . . 9 |- 2 e. RR+
15	14	a1i 9	. . . . . . . 8 |- ( ( ph /\ i e. NN ) -> 2 e. RR+ )
16	13	nnzd 9333	. . . . . . . 8 |- ( ( ph /\ i e. NN ) -> i e. ZZ )
17	15, 16	rpexpcld 10633	. . . . . . 7 |- ( ( ph /\ i e. NN ) -> ( 2 ^ i ) e. RR+ )
18	17	rpreccld 9664	. . . . . 6 |- ( ( ph /\ i e. NN ) -> ( 1 / ( 2 ^ i ) ) e. RR+ )
19	18	rpred 9653	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( 1 / ( 2 ^ i ) ) e. RR )
20		trilpolemgt1.f	. . . . . . 7 |- ( ph -> F : NN --> { 0 , 1 } )
21		0re 7920	. . . . . . . . 9 |- 0 e. RR
22		1re 7919	. . . . . . . . 9 |- 1 e. RR
23		prssi 3738	. . . . . . . . 9 |- ( ( 0 e. RR /\ 1 e. RR ) -> { 0 , 1 } C_ RR )
24	21, 22, 23	mp2an 424	. . . . . . . 8 |- { 0 , 1 } C_ RR
25	24	a1i 9	. . . . . . 7 |- ( ph -> { 0 , 1 } C_ RR )
26	20, 25	fssd 5360	. . . . . 6 |- ( ph -> F : NN --> RR )
27	26	ffvelrnda 5631	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( F ` i ) e. RR )
28	19, 27	remulcld 7950	. . . 4 |- ( ( ph /\ i e. NN ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) e. RR )
29	8, 12, 13, 28	fvmptd3 5589	. . 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 280	. 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 280	. 2 |- ( ( ph /\ i e. Z ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) e. RR )
32		eqid 2170	. . . 4 |- ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) = ( n e. NN |-> ( 1 / ( 2 ^ n ) ) )
33	32, 10, 13, 18	fvmptd3 5589	. . 3 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` i ) = ( 1 / ( 2 ^ i ) ) )
34	7, 33	syldan 280	. 2 |- ( ( ph /\ i e. Z ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` i ) = ( 1 / ( 2 ^ i ) ) )
35	7, 19	syldan 280	. 2 |- ( ( ph /\ i e. Z ) -> ( 1 / ( 2 ^ i ) ) e. RR )
36		simpr 109	. . . . . . 7 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( F ` i ) = 0 )
37	36	oveq2d 5869	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) = ( ( 1 / ( 2 ^ i ) ) x. 0 ) )
38	18	rpcnd 9655	. . . . . . . 8 |- ( ( ph /\ i e. NN ) -> ( 1 / ( 2 ^ i ) ) e. CC )
39	38	adantr 274	. . . . . . 7 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( 1 / ( 2 ^ i ) ) e. CC )
40	39	mul01d 8312	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( ( 1 / ( 2 ^ i ) ) x. 0 ) = 0 )
41	37, 40	eqtrd 2203	. . . . 5 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) = 0 )
42	18	adantr 274	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( 1 / ( 2 ^ i ) ) e. RR+ )
43	42	rpge0d 9657	. . . . 5 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> 0 <_ ( 1 / ( 2 ^ i ) ) )
44	41, 43	eqbrtrd 4011	. . . 4 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) <_ ( 1 / ( 2 ^ i ) ) )
45		simpr 109	. . . . . . 7 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( F ` i ) = 1 )
46	45	oveq2d 5869	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) = ( ( 1 / ( 2 ^ i ) ) x. 1 ) )
47	38	adantr 274	. . . . . . 7 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( 1 / ( 2 ^ i ) ) e. CC )
48	47	mulid1d 7937	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( ( 1 / ( 2 ^ i ) ) x. 1 ) = ( 1 / ( 2 ^ i ) ) )
49	46, 48	eqtrd 2203	. . . . 5 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) = ( 1 / ( 2 ^ i ) ) )
50	19	adantr 274	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( 1 / ( 2 ^ i ) ) e. RR )
51	50	leidd 8433	. . . . 5 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( 1 / ( 2 ^ i ) ) <_ ( 1 / ( 2 ^ i ) ) )
52	49, 51	eqbrtrd 4011	. . . 4 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) <_ ( 1 / ( 2 ^ i ) ) )
53	20	ffvelrnda 5631	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( F ` i ) e. { 0 , 1 } )
54		elpri 3606	. . . . 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 793	. . 3 |- ( ( ph /\ i e. NN ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) <_ ( 1 / ( 2 ^ i ) ) )
57	7, 56	syldan 280	. 2 |- ( ( ph /\ i e. Z ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) <_ ( 1 / ( 2 ^ i ) ) )
58	20, 8	trilpolemclim 14068	. . 3 |- ( ph -> seq 1 ( + , ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ) e. dom ~~> )
59		nnuz 9522	. . . 4 |- NN = ( ZZ>= ` 1 )
60	29, 28	eqeltrd 2247	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ` i ) e. RR )
61	60	recnd 7948	. . . 4 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ` i ) e. CC )
62	59, 2, 61	iserex 11302	. . 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 146	. 2 |- ( ph -> seq M ( + , ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ) e. dom ~~> )
64		seqex 10403	. . . 4 |- seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) e. _V
65		rpreccl 9637	. . . . . . . 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 9239	. . . . . 6 |- ( ph -> 1 e. ZZ )
69	67, 68	rpexpcld 10633	. . . . 5 |- ( ph -> ( ( 1 / 2 ) ^ 1 ) e. RR+ )
70		1mhlfehlf 9096	. . . . . . 7 |- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 )
71	70, 66	eqeltri 2243	. . . . . 6 |- ( 1 - ( 1 / 2 ) ) e. RR+
72	71	a1i 9	. . . . 5 |- ( ph -> ( 1 - ( 1 / 2 ) ) e. RR+ )
73	69, 72	rpdivcld 9671	. . . 4 |- ( ph -> ( ( ( 1 / 2 ) ^ 1 ) / ( 1 - ( 1 / 2 ) ) ) e. RR+ )
74		halfcn 9092	. . . . . 6 |- ( 1 / 2 ) e. CC
75	74	a1i 9	. . . . 5 |- ( ph -> ( 1 / 2 ) e. CC )
76		halfge0 9094	. . . . . . . 8 |- 0 <_ ( 1 / 2 )
77		halfre 9091	. . . . . . . . 9 |- ( 1 / 2 ) e. RR
78	77	absidi 11090	. . . . . . . 8 |- ( 0 <_ ( 1 / 2 ) -> ( abs ` ( 1 / 2 ) ) = ( 1 / 2 ) )
79	76, 78	ax-mp 5	. . . . . . 7 |- ( abs ` ( 1 / 2 ) ) = ( 1 / 2 )
80		halflt1 9095	. . . . . . 7 |- ( 1 / 2 ) < 1
81	79, 80	eqbrtri 4010	. . . . . 6 |- ( abs ` ( 1 / 2 ) ) < 1
82	81	a1i 9	. . . . 5 |- ( ph -> ( abs ` ( 1 / 2 ) ) < 1 )
83		1nn0 9151	. . . . . 6 |- 1 e. NN0
84	83	a1i 9	. . . . 5 |- ( ph -> 1 e. NN0 )
85		oveq2 5861	. . . . . . . 8 |- ( n = j -> ( 2 ^ n ) = ( 2 ^ j ) )
86	85	oveq2d 5869	. . . . . . 7 |- ( n = j -> ( 1 / ( 2 ^ n ) ) = ( 1 / ( 2 ^ j ) ) )
87		elnnuz 9523	. . . . . . . . 9 |- ( j e. NN <-> j e. ( ZZ>= ` 1 ) )
88	87	biimpri 132	. . . . . . . 8 |- ( j e. ( ZZ>= ` 1 ) -> j e. NN )
89	88	adantl 275	. . . . . . 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 9333	. . . . . . . . 9 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> j e. ZZ )
92	90, 91	rpexpcld 10633	. . . . . . . 8 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( 2 ^ j ) e. RR+ )
93	92	rpreccld 9664	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( 1 / ( 2 ^ j ) ) e. RR+ )
94	32, 86, 89, 93	fvmptd3 5589	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` j ) = ( 1 / ( 2 ^ j ) ) )
95		2cnd 8951	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> 2 e. CC )
96	90	rpap0d 9659	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> 2 # 0 )
97	95, 96, 91	exprecapd 10617	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
98	94, 97	eqtr4d 2206	. . . . 5 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` j ) = ( ( 1 / 2 ) ^ j ) )
99	75, 82, 84, 98	geolim2 11475	. . . 4 |- ( ph -> seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) ~~> ( ( ( 1 / 2 ) ^ 1 ) / ( 1 - ( 1 / 2 ) ) ) )
100		breldmg 4817	. . . 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 1337	. . 3 |- ( ph -> seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) e. dom ~~> )
102	33, 38	eqeltrd 2247	. . . 4 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` i ) e. CC )
103	59, 2, 102	iserex 11302	. . 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 146	. 2 |- ( ph -> seq M ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) e. dom ~~> )
105	1, 3, 30, 31, 34, 35, 57, 63, 104	isumle 11458	1 |- ( ph -> sum_ i e. Z ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) <_ sum_ i e. Z ( 1 / ( 2 ^ i ) ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 703   = wceq 1348   e. wcel 2141   _Vcvv 2730   C_ wss 3121   {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   < clt 7954   <_ cle 7955   - cmin 8090   / cdiv 8589   NNcn 8878   2c2 8929   NN0cn0 9135   ZZ>=cuz 9487   RR+crp 9610   seqcseq 10401   ^cexp 10475   abscabs 10961   ~~> cli 11241   sum_csu 11316

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:  trilpolemgt1  14071  trilpolemeq1  14072