Theorem iscmet3lem3 23088

Description: Lemma for iscmet3 23091. (Contributed by Mario Carneiro, 15-Oct-2015.)

Hypothesis

Ref	Expression
iscmet3.1	|- Z = ( ZZ>= ` M )

Assertion

Ref	Expression
iscmet3lem3	|- ( ( M e. ZZ /\ R e. RR+ ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( ( 1 / 2 ) ^ k ) < R )

Distinct variable groups:   j, k, R   j, Z, k   j, M, k

Proof of Theorem iscmet3lem3

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscmet3.1	. . 3 |- Z = ( ZZ>= ` M )
2		simpl 473	. . 3 |- ( ( M e. ZZ /\ R e. RR+ ) -> M e. ZZ )
3		simpr 477	. . 3 |- ( ( M e. ZZ /\ R e. RR+ ) -> R e. RR+ )
4		eluzelz 11697	. . . . . 6 |- ( k e. ( ZZ>= ` M ) -> k e. ZZ )
5	4, 1	eleq2s 2719	. . . . 5 |- ( k e. Z -> k e. ZZ )
6	5	adantl 482	. . . 4 |- ( ( ( M e. ZZ /\ R e. RR+ ) /\ k e. Z ) -> k e. ZZ )
7		oveq2 6658	. . . . 5 |- ( n = k -> ( ( 1 / 2 ) ^ n ) = ( ( 1 / 2 ) ^ k ) )
8		eqid 2622	. . . . 5 |- ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) ) = ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) )
9		ovex 6678	. . . . 5 |- ( ( 1 / 2 ) ^ k ) e. _V
10	7, 8, 9	fvmpt 6282	. . . 4 |- ( k e. ZZ -> ( ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) ) ` k ) = ( ( 1 / 2 ) ^ k ) )
11	6, 10	syl 17	. . 3 |- ( ( ( M e. ZZ /\ R e. RR+ ) /\ k e. Z ) -> ( ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) ) ` k ) = ( ( 1 / 2 ) ^ k ) )
12		nn0uz 11722	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
13	12	reseq2i 5393	. . . . . 6 |- ( ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) ) |` NN0 ) = ( ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) ) |` ( ZZ>= ` 0 ) )
14		nn0ssz 11398	. . . . . . 7 |- NN0 C_ ZZ
15		resmpt 5449	. . . . . . 7 |- ( NN0 C_ ZZ -> ( ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) ) |` NN0 ) = ( n e. NN0 |-> ( ( 1 / 2 ) ^ n ) ) )
16	14, 15	ax-mp 5	. . . . . 6 |- ( ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) ) |` NN0 ) = ( n e. NN0 |-> ( ( 1 / 2 ) ^ n ) )
17	13, 16	eqtr3i 2646	. . . . 5 |- ( ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) ) |` ( ZZ>= ` 0 ) ) = ( n e. NN0 |-> ( ( 1 / 2 ) ^ n ) )
18		halfcn 11247	. . . . . . 7 |- ( 1 / 2 ) e. CC
19	18	a1i 11	. . . . . 6 |- ( ( M e. ZZ /\ R e. RR+ ) -> ( 1 / 2 ) e. CC )
20		halfre 11246	. . . . . . . . 9 |- ( 1 / 2 ) e. RR
21		1rp 11836	. . . . . . . . . . 11 |- 1 e. RR+
22		rphalfcl 11858	. . . . . . . . . . 11 |- ( 1 e. RR+ -> ( 1 / 2 ) e. RR+ )
23	21, 22	ax-mp 5	. . . . . . . . . 10 |- ( 1 / 2 ) e. RR+
24		rpge0 11845	. . . . . . . . . 10 |- ( ( 1 / 2 ) e. RR+ -> 0 <_ ( 1 / 2 ) )
25	23, 24	ax-mp 5	. . . . . . . . 9 |- 0 <_ ( 1 / 2 )
26		absid 14036	. . . . . . . . 9 |- ( ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) ) -> ( abs ` ( 1 / 2 ) ) = ( 1 / 2 ) )
27	20, 25, 26	mp2an 708	. . . . . . . 8 |- ( abs ` ( 1 / 2 ) ) = ( 1 / 2 )
28		halflt1 11250	. . . . . . . 8 |- ( 1 / 2 ) < 1
29	27, 28	eqbrtri 4674	. . . . . . 7 |- ( abs ` ( 1 / 2 ) ) < 1
30	29	a1i 11	. . . . . 6 |- ( ( M e. ZZ /\ R e. RR+ ) -> ( abs ` ( 1 / 2 ) ) < 1 )
31	19, 30	expcnv 14596	. . . . 5 |- ( ( M e. ZZ /\ R e. RR+ ) -> ( n e. NN0 |-> ( ( 1 / 2 ) ^ n ) ) ~~> 0 )
32	17, 31	syl5eqbr 4688	. . . 4 |- ( ( M e. ZZ /\ R e. RR+ ) -> ( ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) ) |` ( ZZ>= ` 0 ) ) ~~> 0 )
33		0z 11388	. . . . 5 |- 0 e. ZZ
34		zex 11386	. . . . . . 7 |- ZZ e. _V
35	34	mptex 6486	. . . . . 6 |- ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) ) e. _V
36	35	a1i 11	. . . . 5 |- ( ( M e. ZZ /\ R e. RR+ ) -> ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) ) e. _V )
37		climres 14306	. . . . 5 |- ( ( 0 e. ZZ /\ ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) ) e. _V ) -> ( ( ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) ) |` ( ZZ>= ` 0 ) ) ~~> 0 <-> ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) ) ~~> 0 ) )
38	33, 36, 37	sylancr 695	. . . 4 |- ( ( M e. ZZ /\ R e. RR+ ) -> ( ( ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) ) |` ( ZZ>= ` 0 ) ) ~~> 0 <-> ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) ) ~~> 0 ) )
39	32, 38	mpbid 222	. . 3 |- ( ( M e. ZZ /\ R e. RR+ ) -> ( n e. ZZ |-> ( ( 1 / 2 ) ^ n ) ) ~~> 0 )
40	1, 2, 3, 11, 39	climi0 14243	. 2 |- ( ( M e. ZZ /\ R e. RR+ ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( 1 / 2 ) ^ k ) ) < R )
41	1	uztrn2 11705	. . . . . 6 |- ( ( j e. Z /\ k e. ( ZZ>= ` j ) ) -> k e. Z )
42		rpexpcl 12879	. . . . . . . . 9 |- ( ( ( 1 / 2 ) e. RR+ /\ k e. ZZ ) -> ( ( 1 / 2 ) ^ k ) e. RR+ )
43	23, 6, 42	sylancr 695	. . . . . . . 8 |- ( ( ( M e. ZZ /\ R e. RR+ ) /\ k e. Z ) -> ( ( 1 / 2 ) ^ k ) e. RR+ )
44		rpre 11839	. . . . . . . . 9 |- ( ( ( 1 / 2 ) ^ k ) e. RR+ -> ( ( 1 / 2 ) ^ k ) e. RR )
45		rpge0 11845	. . . . . . . . 9 |- ( ( ( 1 / 2 ) ^ k ) e. RR+ -> 0 <_ ( ( 1 / 2 ) ^ k ) )
46	44, 45	absidd 14161	. . . . . . . 8 |- ( ( ( 1 / 2 ) ^ k ) e. RR+ -> ( abs ` ( ( 1 / 2 ) ^ k ) ) = ( ( 1 / 2 ) ^ k ) )
47	43, 46	syl 17	. . . . . . 7 |- ( ( ( M e. ZZ /\ R e. RR+ ) /\ k e. Z ) -> ( abs ` ( ( 1 / 2 ) ^ k ) ) = ( ( 1 / 2 ) ^ k ) )
48	47	breq1d 4663	. . . . . 6 |- ( ( ( M e. ZZ /\ R e. RR+ ) /\ k e. Z ) -> ( ( abs ` ( ( 1 / 2 ) ^ k ) ) < R <-> ( ( 1 / 2 ) ^ k ) < R ) )
49	41, 48	sylan2 491	. . . . 5 |- ( ( ( M e. ZZ /\ R e. RR+ ) /\ ( j e. Z /\ k e. ( ZZ>= ` j ) ) ) -> ( ( abs ` ( ( 1 / 2 ) ^ k ) ) < R <-> ( ( 1 / 2 ) ^ k ) < R ) )
50	49	anassrs 680	. . . 4 |- ( ( ( ( M e. ZZ /\ R e. RR+ ) /\ j e. Z ) /\ k e. ( ZZ>= ` j ) ) -> ( ( abs ` ( ( 1 / 2 ) ^ k ) ) < R <-> ( ( 1 / 2 ) ^ k ) < R ) )
51	50	ralbidva 2985	. . 3 |- ( ( ( M e. ZZ /\ R e. RR+ ) /\ j e. Z ) -> ( A. k e. ( ZZ>= ` j ) ( abs ` ( ( 1 / 2 ) ^ k ) ) < R <-> A. k e. ( ZZ>= ` j ) ( ( 1 / 2 ) ^ k ) < R ) )
52	51	rexbidva 3049	. 2 |- ( ( M e. ZZ /\ R e. RR+ ) -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( abs ` ( ( 1 / 2 ) ^ k ) ) < R <-> E. j e. Z A. k e. ( ZZ>= ` j ) ( ( 1 / 2 ) ^ k ) < R ) )
53	40, 52	mpbid 222	1 |- ( ( M e. ZZ /\ R e. RR+ ) -> E. j e. Z A. k e. ( ZZ>= ` j ) ( ( 1 / 2 ) ^ k ) < R )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   |` cres 5116   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   <_ cle 10075   / cdiv 10684   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   ^cexp 12860   abscabs 13974   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220

This theorem is referenced by:  iscmet3lem1  23089  iscmet3lem2  23090