Theorem trilpolemisumle 13406

Description: Lemma for trilpo 13411. 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 9196	. 2 |- ( ph -> M e. ZZ )
4	1	eleq2i 2207	. . . . 5 |- ( i e. Z <-> i e. ( ZZ>= ` M ) )
5	4	biimpi 119	. . . 4 |- ( i e. Z -> i e. ( ZZ>= ` M ) )
6		eluznn 9421	. . . 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 2140	. . . 4 |- ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) = ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) )
9		oveq2 5790	. . . . . 6 |- ( n = i -> ( 2 ^ n ) = ( 2 ^ i ) )
10	9	oveq2d 5798	. . . . 5 |- ( n = i -> ( 1 / ( 2 ^ n ) ) = ( 1 / ( 2 ^ i ) ) )
11		fveq2 5429	. . . . 5 |- ( n = i -> ( F ` n ) = ( F ` i ) )
12	10, 11	oveq12d 5800	. . . 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 9475	. . . . . . . . 9 |- 2 e. RR+
15	14	a1i 9	. . . . . . . 8 |- ( ( ph /\ i e. NN ) -> 2 e. RR+ )
16	13	nnzd 9196	. . . . . . . 8 |- ( ( ph /\ i e. NN ) -> i e. ZZ )
17	15, 16	rpexpcld 10479	. . . . . . 7 |- ( ( ph /\ i e. NN ) -> ( 2 ^ i ) e. RR+ )
18	17	rpreccld 9524	. . . . . 6 |- ( ( ph /\ i e. NN ) -> ( 1 / ( 2 ^ i ) ) e. RR+ )
19	18	rpred 9513	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( 1 / ( 2 ^ i ) ) e. RR )
20		trilpolemgt1.f	. . . . . . 7 |- ( ph -> F : NN --> { 0 , 1 } )
21		0re 7790	. . . . . . . . 9 |- 0 e. RR
22		1re 7789	. . . . . . . . 9 |- 1 e. RR
23		prssi 3686	. . . . . . . . 9 |- ( ( 0 e. RR /\ 1 e. RR ) -> { 0 , 1 } C_ RR )
24	21, 22, 23	mp2an 423	. . . . . . . 8 |- { 0 , 1 } C_ RR
25	24	a1i 9	. . . . . . 7 |- ( ph -> { 0 , 1 } C_ RR )
26	20, 25	fssd 5293	. . . . . 6 |- ( ph -> F : NN --> RR )
27	26	ffvelrnda 5563	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( F ` i ) e. RR )
28	19, 27	remulcld 7820	. . . 4 |- ( ( ph /\ i e. NN ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) e. RR )
29	8, 12, 13, 28	fvmptd3 5522	. . 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 2140	. . . 4 |- ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) = ( n e. NN |-> ( 1 / ( 2 ^ n ) ) )
33	32, 10, 13, 18	fvmptd3 5522	. . 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 5798	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) = ( ( 1 / ( 2 ^ i ) ) x. 0 ) )
38	18	rpcnd 9515	. . . . . . . 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 8179	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( ( 1 / ( 2 ^ i ) ) x. 0 ) = 0 )
41	37, 40	eqtrd 2173	. . . . 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 9517	. . . . 5 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> 0 <_ ( 1 / ( 2 ^ i ) ) )
44	41, 43	eqbrtrd 3958	. . . 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 5798	. . . . . 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 7807	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( ( 1 / ( 2 ^ i ) ) x. 1 ) = ( 1 / ( 2 ^ i ) ) )
49	46, 48	eqtrd 2173	. . . . 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 8300	. . . . 5 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( 1 / ( 2 ^ i ) ) <_ ( 1 / ( 2 ^ i ) ) )
52	49, 51	eqbrtrd 3958	. . . 4 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) <_ ( 1 / ( 2 ^ i ) ) )
53	20	ffvelrnda 5563	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( F ` i ) e. { 0 , 1 } )
54		elpri 3555	. . . . 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 788	. . 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 13404	. . 3 |- ( ph -> seq 1 ( + , ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ) e. dom ~~> )
59		nnuz 9385	. . . 4 |- NN = ( ZZ>= ` 1 )
60	29, 28	eqeltrd 2217	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ` i ) e. RR )
61	60	recnd 7818	. . . 4 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ` i ) e. CC )
62	59, 2, 61	iserex 11140	. . 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 10251	. . . 4 |- seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) e. _V
65		rpreccl 9497	. . . . . . . 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 9105	. . . . . 6 |- ( ph -> 1 e. ZZ )
69	67, 68	rpexpcld 10479	. . . . 5 |- ( ph -> ( ( 1 / 2 ) ^ 1 ) e. RR+ )
70		1mhlfehlf 8962	. . . . . . 7 |- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 )
71	70, 66	eqeltri 2213	. . . . . 6 |- ( 1 - ( 1 / 2 ) ) e. RR+
72	71	a1i 9	. . . . 5 |- ( ph -> ( 1 - ( 1 / 2 ) ) e. RR+ )
73	69, 72	rpdivcld 9531	. . . 4 |- ( ph -> ( ( ( 1 / 2 ) ^ 1 ) / ( 1 - ( 1 / 2 ) ) ) e. RR+ )
74		halfcn 8958	. . . . . 6 |- ( 1 / 2 ) e. CC
75	74	a1i 9	. . . . 5 |- ( ph -> ( 1 / 2 ) e. CC )
76		halfge0 8960	. . . . . . . 8 |- 0 <_ ( 1 / 2 )
77		halfre 8957	. . . . . . . . 9 |- ( 1 / 2 ) e. RR
78	77	absidi 10930	. . . . . . . 8 |- ( 0 <_ ( 1 / 2 ) -> ( abs ` ( 1 / 2 ) ) = ( 1 / 2 ) )
79	76, 78	ax-mp 5	. . . . . . 7 |- ( abs ` ( 1 / 2 ) ) = ( 1 / 2 )
80		halflt1 8961	. . . . . . 7 |- ( 1 / 2 ) < 1
81	79, 80	eqbrtri 3957	. . . . . 6 |- ( abs ` ( 1 / 2 ) ) < 1
82	81	a1i 9	. . . . 5 |- ( ph -> ( abs ` ( 1 / 2 ) ) < 1 )
83		1nn0 9017	. . . . . 6 |- 1 e. NN0
84	83	a1i 9	. . . . 5 |- ( ph -> 1 e. NN0 )
85		oveq2 5790	. . . . . . . 8 |- ( n = j -> ( 2 ^ n ) = ( 2 ^ j ) )
86	85	oveq2d 5798	. . . . . . 7 |- ( n = j -> ( 1 / ( 2 ^ n ) ) = ( 1 / ( 2 ^ j ) ) )
87		elnnuz 9386	. . . . . . . . 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 9196	. . . . . . . . 9 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> j e. ZZ )
92	90, 91	rpexpcld 10479	. . . . . . . 8 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( 2 ^ j ) e. RR+ )
93	92	rpreccld 9524	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( 1 / ( 2 ^ j ) ) e. RR+ )
94	32, 86, 89, 93	fvmptd3 5522	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` j ) = ( 1 / ( 2 ^ j ) ) )
95		2cnd 8817	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> 2 e. CC )
96	90	rpap0d 9519	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> 2 # 0 )
97	95, 96, 91	exprecapd 10463	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
98	94, 97	eqtr4d 2176	. . . . 5 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` j ) = ( ( 1 / 2 ) ^ j ) )
99	75, 82, 84, 98	geolim2 11313	. . . 4 |- ( ph -> seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) ~~> ( ( ( 1 / 2 ) ^ 1 ) / ( 1 - ( 1 / 2 ) ) ) )
100		breldmg 4753	. . . 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 1321	. . 3 |- ( ph -> seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) e. dom ~~> )
102	33, 38	eqeltrd 2217	. . . 4 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` i ) e. CC )
103	59, 2, 102	iserex 11140	. . 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 11296	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 698   = wceq 1332   e. wcel 1481   _Vcvv 2689   C_ wss 3076   {cpr 3533   class class class wbr 3937   |-> cmpt 3997   dom cdm 4547   -->wf 5127   ` cfv 5131  (class class class)co 5782   CCcc 7642   RRcr 7643   0cc0 7644   1c1 7645   + caddc 7647   x. cmul 7649   < clt 7824   <_ cle 7825   - cmin 7957   / cdiv 8456   NNcn 8744   2c2 8795   NN0cn0 9001   ZZ>=cuz 9350   RR+crp 9470   seqcseq 10249   ^cexp 10323   abscabs 10801   ~~> cli 11079   sum_csu 11154

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-frec 6296  df-1o 6321  df-oadd 6325  df-er 6437  df-en 6643  df-dom 6644  df-fin 6645  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-ico 9707  df-fz 9822  df-fzo 9951  df-seqfrec 10250  df-exp 10324  df-ihash 10554  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-clim 11080  df-sumdc 11155

This theorem is referenced by:  trilpolemgt1  13407  trilpolemeq1  13408