Theorem trilpolemisumle 13231

Description: Lemma for trilpo 13236. 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 9172	. 2 |- ( ph -> M e. ZZ )
4	1	eleq2i 2206	. . . . 5 |- ( i e. Z <-> i e. ( ZZ>= ` M ) )
5	4	biimpi 119	. . . 4 |- ( i e. Z -> i e. ( ZZ>= ` M ) )
6		eluznn 9394	. . . 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 2139	. . . 4 |- ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) = ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) )
9		oveq2 5782	. . . . . 6 |- ( n = i -> ( 2 ^ n ) = ( 2 ^ i ) )
10	9	oveq2d 5790	. . . . 5 |- ( n = i -> ( 1 / ( 2 ^ n ) ) = ( 1 / ( 2 ^ i ) ) )
11		fveq2 5421	. . . . 5 |- ( n = i -> ( F ` n ) = ( F ` i ) )
12	10, 11	oveq12d 5792	. . . 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 9446	. . . . . . . . 9 |- 2 e. RR+
15	14	a1i 9	. . . . . . . 8 |- ( ( ph /\ i e. NN ) -> 2 e. RR+ )
16	13	nnzd 9172	. . . . . . . 8 |- ( ( ph /\ i e. NN ) -> i e. ZZ )
17	15, 16	rpexpcld 10448	. . . . . . 7 |- ( ( ph /\ i e. NN ) -> ( 2 ^ i ) e. RR+ )
18	17	rpreccld 9494	. . . . . 6 |- ( ( ph /\ i e. NN ) -> ( 1 / ( 2 ^ i ) ) e. RR+ )
19	18	rpred 9483	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( 1 / ( 2 ^ i ) ) e. RR )
20		trilpolemgt1.f	. . . . . . 7 |- ( ph -> F : NN --> { 0 , 1 } )
21		0re 7766	. . . . . . . . 9 |- 0 e. RR
22		1re 7765	. . . . . . . . 9 |- 1 e. RR
23		prssi 3678	. . . . . . . . 9 |- ( ( 0 e. RR /\ 1 e. RR ) -> { 0 , 1 } C_ RR )
24	21, 22, 23	mp2an 422	. . . . . . . 8 |- { 0 , 1 } C_ RR
25	24	a1i 9	. . . . . . 7 |- ( ph -> { 0 , 1 } C_ RR )
26	20, 25	fssd 5285	. . . . . 6 |- ( ph -> F : NN --> RR )
27	26	ffvelrnda 5555	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( F ` i ) e. RR )
28	19, 27	remulcld 7796	. . . 4 |- ( ( ph /\ i e. NN ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) e. RR )
29	8, 12, 13, 28	fvmptd3 5514	. . 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 2139	. . . 4 |- ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) = ( n e. NN |-> ( 1 / ( 2 ^ n ) ) )
33	32, 10, 13, 18	fvmptd3 5514	. . 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 5790	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) = ( ( 1 / ( 2 ^ i ) ) x. 0 ) )
38	18	rpcnd 9485	. . . . . . . 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 8155	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( ( 1 / ( 2 ^ i ) ) x. 0 ) = 0 )
41	37, 40	eqtrd 2172	. . . . 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 9487	. . . . 5 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> 0 <_ ( 1 / ( 2 ^ i ) ) )
44	41, 43	eqbrtrd 3950	. . . 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 5790	. . . . . 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 7783	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( ( 1 / ( 2 ^ i ) ) x. 1 ) = ( 1 / ( 2 ^ i ) ) )
49	46, 48	eqtrd 2172	. . . . 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 8276	. . . . 5 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( 1 / ( 2 ^ i ) ) <_ ( 1 / ( 2 ^ i ) ) )
52	49, 51	eqbrtrd 3950	. . . 4 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) <_ ( 1 / ( 2 ^ i ) ) )
53	20	ffvelrnda 5555	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( F ` i ) e. { 0 , 1 } )
54		elpri 3550	. . . . 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 787	. . 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 13229	. . 3 |- ( ph -> seq 1 ( + , ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ) e. dom ~~> )
59		nnuz 9361	. . . 4 |- NN = ( ZZ>= ` 1 )
60	29, 28	eqeltrd 2216	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ` i ) e. RR )
61	60	recnd 7794	. . . 4 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ` i ) e. CC )
62	59, 2, 61	iserex 11108	. . 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 10220	. . . 4 |- seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) e. _V
65		rpreccl 9468	. . . . . . . 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 9081	. . . . . 6 |- ( ph -> 1 e. ZZ )
69	67, 68	rpexpcld 10448	. . . . 5 |- ( ph -> ( ( 1 / 2 ) ^ 1 ) e. RR+ )
70		1mhlfehlf 8938	. . . . . . 7 |- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 )
71	70, 66	eqeltri 2212	. . . . . 6 |- ( 1 - ( 1 / 2 ) ) e. RR+
72	71	a1i 9	. . . . 5 |- ( ph -> ( 1 - ( 1 / 2 ) ) e. RR+ )
73	69, 72	rpdivcld 9501	. . . 4 |- ( ph -> ( ( ( 1 / 2 ) ^ 1 ) / ( 1 - ( 1 / 2 ) ) ) e. RR+ )
74		halfcn 8934	. . . . . 6 |- ( 1 / 2 ) e. CC
75	74	a1i 9	. . . . 5 |- ( ph -> ( 1 / 2 ) e. CC )
76		halfge0 8936	. . . . . . . 8 |- 0 <_ ( 1 / 2 )
77		halfre 8933	. . . . . . . . 9 |- ( 1 / 2 ) e. RR
78	77	absidi 10898	. . . . . . . 8 |- ( 0 <_ ( 1 / 2 ) -> ( abs ` ( 1 / 2 ) ) = ( 1 / 2 ) )
79	76, 78	ax-mp 5	. . . . . . 7 |- ( abs ` ( 1 / 2 ) ) = ( 1 / 2 )
80		halflt1 8937	. . . . . . 7 |- ( 1 / 2 ) < 1
81	79, 80	eqbrtri 3949	. . . . . 6 |- ( abs ` ( 1 / 2 ) ) < 1
82	81	a1i 9	. . . . 5 |- ( ph -> ( abs ` ( 1 / 2 ) ) < 1 )
83		1nn0 8993	. . . . . 6 |- 1 e. NN0
84	83	a1i 9	. . . . 5 |- ( ph -> 1 e. NN0 )
85		oveq2 5782	. . . . . . . 8 |- ( n = j -> ( 2 ^ n ) = ( 2 ^ j ) )
86	85	oveq2d 5790	. . . . . . 7 |- ( n = j -> ( 1 / ( 2 ^ n ) ) = ( 1 / ( 2 ^ j ) ) )
87		elnnuz 9362	. . . . . . . . 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 9172	. . . . . . . . 9 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> j e. ZZ )
92	90, 91	rpexpcld 10448	. . . . . . . 8 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( 2 ^ j ) e. RR+ )
93	92	rpreccld 9494	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( 1 / ( 2 ^ j ) ) e. RR+ )
94	32, 86, 89, 93	fvmptd3 5514	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` j ) = ( 1 / ( 2 ^ j ) ) )
95		2cnd 8793	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> 2 e. CC )
96	90	rpap0d 9489	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> 2 # 0 )
97	95, 96, 91	exprecapd 10432	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
98	94, 97	eqtr4d 2175	. . . . 5 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` j ) = ( ( 1 / 2 ) ^ j ) )
99	75, 82, 84, 98	geolim2 11281	. . . 4 |- ( ph -> seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) ~~> ( ( ( 1 / 2 ) ^ 1 ) / ( 1 - ( 1 / 2 ) ) ) )
100		breldmg 4745	. . . 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 1320	. . 3 |- ( ph -> seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) e. dom ~~> )
102	33, 38	eqeltrd 2216	. . . 4 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` i ) e. CC )
103	59, 2, 102	iserex 11108	. . 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 11264	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 697   = wceq 1331   e. wcel 1480   _Vcvv 2686   C_ wss 3071   {cpr 3528   class class class wbr 3929   |-> cmpt 3989   dom cdm 4539   -->wf 5119   ` cfv 5123  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620   1c1 7621   + caddc 7623   x. cmul 7625   < clt 7800   <_ cle 7801   - cmin 7933   / cdiv 8432   NNcn 8720   2c2 8771   NN0cn0 8977   ZZ>=cuz 9326   RR+crp 9441   seqcseq 10218   ^cexp 10292   abscabs 10769   ~~> cli 11047   sum_csu 11122

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-en 6635  df-dom 6636  df-fin 6637  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-ico 9677  df-fz 9791  df-fzo 9920  df-seqfrec 10219  df-exp 10293  df-ihash 10522  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-clim 11048  df-sumdc 11123

This theorem is referenced by:  trilpolemgt1  13232  trilpolemeq1  13233