Theorem trilpolemisumle 16642

Description: Lemma for trilpo 16647. 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 9600	. 2 |- ( ph -> M e. ZZ )
4	1	eleq2i 2298	. . . . 5 |- ( i e. Z <-> i e. ( ZZ>= ` M ) )
5	4	biimpi 120	. . . 4 |- ( i e. Z -> i e. ( ZZ>= ` M ) )
6		eluznn 9833	. . . 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 2231	. . . 4 |- ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) = ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) )
9		oveq2 6025	. . . . . 6 |- ( n = i -> ( 2 ^ n ) = ( 2 ^ i ) )
10	9	oveq2d 6033	. . . . 5 |- ( n = i -> ( 1 / ( 2 ^ n ) ) = ( 1 / ( 2 ^ i ) ) )
11		fveq2 5639	. . . . 5 |- ( n = i -> ( F ` n ) = ( F ` i ) )
12	10, 11	oveq12d 6035	. . . 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 9892	. . . . . . . . 9 |- 2 e. RR+
15	14	a1i 9	. . . . . . . 8 |- ( ( ph /\ i e. NN ) -> 2 e. RR+ )
16	13	nnzd 9600	. . . . . . . 8 |- ( ( ph /\ i e. NN ) -> i e. ZZ )
17	15, 16	rpexpcld 10958	. . . . . . 7 |- ( ( ph /\ i e. NN ) -> ( 2 ^ i ) e. RR+ )
18	17	rpreccld 9941	. . . . . 6 |- ( ( ph /\ i e. NN ) -> ( 1 / ( 2 ^ i ) ) e. RR+ )
19	18	rpred 9930	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( 1 / ( 2 ^ i ) ) e. RR )
20		trilpolemgt1.f	. . . . . . 7 |- ( ph -> F : NN --> { 0 , 1 } )
21		0re 8178	. . . . . . . . 9 |- 0 e. RR
22		1re 8177	. . . . . . . . 9 |- 1 e. RR
23		prssi 3831	. . . . . . . . 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 5495	. . . . . 6 |- ( ph -> F : NN --> RR )
27	26	ffvelcdmda 5782	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( F ` i ) e. RR )
28	19, 27	remulcld 8209	. . . 4 |- ( ( ph /\ i e. NN ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) e. RR )
29	8, 12, 13, 28	fvmptd3 5740	. . 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 2231	. . . 4 |- ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) = ( n e. NN |-> ( 1 / ( 2 ^ n ) ) )
33	32, 10, 13, 18	fvmptd3 5740	. . 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 6033	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) = ( ( 1 / ( 2 ^ i ) ) x. 0 ) )
38	18	rpcnd 9932	. . . . . . . 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 8571	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> ( ( 1 / ( 2 ^ i ) ) x. 0 ) = 0 )
41	37, 40	eqtrd 2264	. . . . 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 9934	. . . . 5 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 0 ) -> 0 <_ ( 1 / ( 2 ^ i ) ) )
44	41, 43	eqbrtrd 4110	. . . 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 6033	. . . . . 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 8195	. . . . . 6 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( ( 1 / ( 2 ^ i ) ) x. 1 ) = ( 1 / ( 2 ^ i ) ) )
49	46, 48	eqtrd 2264	. . . . 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 8693	. . . . 5 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( 1 / ( 2 ^ i ) ) <_ ( 1 / ( 2 ^ i ) ) )
52	49, 51	eqbrtrd 4110	. . . 4 |- ( ( ( ph /\ i e. NN ) /\ ( F ` i ) = 1 ) -> ( ( 1 / ( 2 ^ i ) ) x. ( F ` i ) ) <_ ( 1 / ( 2 ^ i ) ) )
53	20	ffvelcdmda 5782	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( F ` i ) e. { 0 , 1 } )
54		elpri 3692	. . . . 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 805	. . 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 16640	. . 3 |- ( ph -> seq 1 ( + , ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ) e. dom ~~> )
59		nnuz 9791	. . . 4 |- NN = ( ZZ>= ` 1 )
60	29, 28	eqeltrd 2308	. . . . 5 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ` i ) e. RR )
61	60	recnd 8207	. . . 4 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( ( 1 / ( 2 ^ n ) ) x. ( F ` n ) ) ) ` i ) e. CC )
62	59, 2, 61	iserex 11899	. . 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 10710	. . . 4 |- seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) e. _V
65		rpreccl 9914	. . . . . . . 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 9505	. . . . . 6 |- ( ph -> 1 e. ZZ )
69	67, 68	rpexpcld 10958	. . . . 5 |- ( ph -> ( ( 1 / 2 ) ^ 1 ) e. RR+ )
70		1mhlfehlf 9361	. . . . . . 7 |- ( 1 - ( 1 / 2 ) ) = ( 1 / 2 )
71	70, 66	eqeltri 2304	. . . . . 6 |- ( 1 - ( 1 / 2 ) ) e. RR+
72	71	a1i 9	. . . . 5 |- ( ph -> ( 1 - ( 1 / 2 ) ) e. RR+ )
73	69, 72	rpdivcld 9948	. . . 4 |- ( ph -> ( ( ( 1 / 2 ) ^ 1 ) / ( 1 - ( 1 / 2 ) ) ) e. RR+ )
74		halfcn 9357	. . . . . 6 |- ( 1 / 2 ) e. CC
75	74	a1i 9	. . . . 5 |- ( ph -> ( 1 / 2 ) e. CC )
76		halfge0 9359	. . . . . . . 8 |- 0 <_ ( 1 / 2 )
77		halfre 9356	. . . . . . . . 9 |- ( 1 / 2 ) e. RR
78	77	absidi 11686	. . . . . . . 8 |- ( 0 <_ ( 1 / 2 ) -> ( abs ` ( 1 / 2 ) ) = ( 1 / 2 ) )
79	76, 78	ax-mp 5	. . . . . . 7 |- ( abs ` ( 1 / 2 ) ) = ( 1 / 2 )
80		halflt1 9360	. . . . . . 7 |- ( 1 / 2 ) < 1
81	79, 80	eqbrtri 4109	. . . . . 6 |- ( abs ` ( 1 / 2 ) ) < 1
82	81	a1i 9	. . . . 5 |- ( ph -> ( abs ` ( 1 / 2 ) ) < 1 )
83		1nn0 9417	. . . . . 6 |- 1 e. NN0
84	83	a1i 9	. . . . 5 |- ( ph -> 1 e. NN0 )
85		oveq2 6025	. . . . . . . 8 |- ( n = j -> ( 2 ^ n ) = ( 2 ^ j ) )
86	85	oveq2d 6033	. . . . . . 7 |- ( n = j -> ( 1 / ( 2 ^ n ) ) = ( 1 / ( 2 ^ j ) ) )
87		elnnuz 9792	. . . . . . . . 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 9600	. . . . . . . . 9 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> j e. ZZ )
92	90, 91	rpexpcld 10958	. . . . . . . 8 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( 2 ^ j ) e. RR+ )
93	92	rpreccld 9941	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( 1 / ( 2 ^ j ) ) e. RR+ )
94	32, 86, 89, 93	fvmptd3 5740	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` j ) = ( 1 / ( 2 ^ j ) ) )
95		2cnd 9215	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> 2 e. CC )
96	90	rpap0d 9936	. . . . . . 7 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> 2 # 0 )
97	95, 96, 91	exprecapd 10942	. . . . . 6 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( ( 1 / 2 ) ^ j ) = ( 1 / ( 2 ^ j ) ) )
98	94, 97	eqtr4d 2267	. . . . 5 |- ( ( ph /\ j e. ( ZZ>= ` 1 ) ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` j ) = ( ( 1 / 2 ) ^ j ) )
99	75, 82, 84, 98	geolim2 12072	. . . 4 |- ( ph -> seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) ~~> ( ( ( 1 / 2 ) ^ 1 ) / ( 1 - ( 1 / 2 ) ) ) )
100		breldmg 4937	. . . 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 1378	. . 3 |- ( ph -> seq 1 ( + , ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ) e. dom ~~> )
102	33, 38	eqeltrd 2308	. . . 4 |- ( ( ph /\ i e. NN ) -> ( ( n e. NN |-> ( 1 / ( 2 ^ n ) ) ) ` i ) e. CC )
103	59, 2, 102	iserex 11899	. . 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 12055	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 715   = wceq 1397   e. wcel 2202   _Vcvv 2802   C_ wss 3200   {cpr 3670   class class class wbr 4088   |-> cmpt 4150   dom cdm 4725   -->wf 5322   ` cfv 5326  (class class class)co 6017   CCcc 8029   RRcr 8030   0cc0 8031   1c1 8032   + caddc 8034   x. cmul 8036   < clt 8213   <_ cle 8214   - cmin 8349   / cdiv 8851   NNcn 9142   2c2 9193   NN0cn0 9401   ZZ>=cuz 9754   RR+crp 9887   seqcseq 10708   ^cexp 10799   abscabs 11557   ~~> cli 11838   sum_csu 11913

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-oadd 6585  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-ico 10128  df-fz 10243  df-fzo 10377  df-seqfrec 10709  df-exp 10800  df-ihash 11037  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-clim 11839  df-sumdc 11914

This theorem is referenced by:  trilpolemgt1  16643  trilpolemeq1  16644