Theorem mbfi1fseqlem1 23482

Description: Lemma for mbfi1fseq 23488. (Contributed by Mario Carneiro, 16-Aug-2014.)

Hypotheses

Ref	Expression
mbfi1fseq.1	|- ( ph -> F e. MblFn )
mbfi1fseq.2	|- ( ph -> F : RR --> ( 0 [,) +oo ) )
mbfi1fseq.3	|- J = ( m e. NN , y e. RR |-> ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) / ( 2 ^ m ) ) )

Assertion

Ref	Expression
mbfi1fseqlem1	|- ( ph -> J : ( NN X. RR ) --> ( 0 [,) +oo ) )

Distinct variable groups:   y, m, F   m, J   ph, m, y

Allowed substitution hint:   J( y)

Proof of Theorem mbfi1fseqlem1

Step	Hyp	Ref	Expression
1		mbfi1fseq.2	. . . . . . . . . 10 |- ( ph -> F : RR --> ( 0 [,) +oo ) )
2		simpr 477	. . . . . . . . . 10 |- ( ( m e. NN /\ y e. RR ) -> y e. RR )
3		ffvelrn 6357	. . . . . . . . . 10 |- ( ( F : RR --> ( 0 [,) +oo ) /\ y e. RR ) -> ( F ` y ) e. ( 0 [,) +oo ) )
4	1, 2, 3	syl2an 494	. . . . . . . . 9 |- ( ( ph /\ ( m e. NN /\ y e. RR ) ) -> ( F ` y ) e. ( 0 [,) +oo ) )
5		elrege0 12278	. . . . . . . . 9 |- ( ( F ` y ) e. ( 0 [,) +oo ) <-> ( ( F ` y ) e. RR /\ 0 <_ ( F ` y ) ) )
6	4, 5	sylib 208	. . . . . . . 8 |- ( ( ph /\ ( m e. NN /\ y e. RR ) ) -> ( ( F ` y ) e. RR /\ 0 <_ ( F ` y ) ) )
7	6	simpld 475	. . . . . . 7 |- ( ( ph /\ ( m e. NN /\ y e. RR ) ) -> ( F ` y ) e. RR )
8		2nn 11185	. . . . . . . . . 10 |- 2 e. NN
9		nnnn0 11299	. . . . . . . . . 10 |- ( m e. NN -> m e. NN0 )
10		nnexpcl 12873	. . . . . . . . . 10 |- ( ( 2 e. NN /\ m e. NN0 ) -> ( 2 ^ m ) e. NN )
11	8, 9, 10	sylancr 695	. . . . . . . . 9 |- ( m e. NN -> ( 2 ^ m ) e. NN )
12	11	ad2antrl 764	. . . . . . . 8 |- ( ( ph /\ ( m e. NN /\ y e. RR ) ) -> ( 2 ^ m ) e. NN )
13	12	nnred 11035	. . . . . . 7 |- ( ( ph /\ ( m e. NN /\ y e. RR ) ) -> ( 2 ^ m ) e. RR )
14	7, 13	remulcld 10070	. . . . . 6 |- ( ( ph /\ ( m e. NN /\ y e. RR ) ) -> ( ( F ` y ) x. ( 2 ^ m ) ) e. RR )
15		reflcl 12597	. . . . . 6 |- ( ( ( F ` y ) x. ( 2 ^ m ) ) e. RR -> ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) e. RR )
16	14, 15	syl 17	. . . . 5 |- ( ( ph /\ ( m e. NN /\ y e. RR ) ) -> ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) e. RR )
17	16, 12	nndivred 11069	. . . 4 |- ( ( ph /\ ( m e. NN /\ y e. RR ) ) -> ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) / ( 2 ^ m ) ) e. RR )
18	12	nnnn0d 11351	. . . . . . . . 9 |- ( ( ph /\ ( m e. NN /\ y e. RR ) ) -> ( 2 ^ m ) e. NN0 )
19	18	nn0ge0d 11354	. . . . . . . 8 |- ( ( ph /\ ( m e. NN /\ y e. RR ) ) -> 0 <_ ( 2 ^ m ) )
20		mulge0 10546	. . . . . . . 8 |- ( ( ( ( F ` y ) e. RR /\ 0 <_ ( F ` y ) ) /\ ( ( 2 ^ m ) e. RR /\ 0 <_ ( 2 ^ m ) ) ) -> 0 <_ ( ( F ` y ) x. ( 2 ^ m ) ) )
21	6, 13, 19, 20	syl12anc 1324	. . . . . . 7 |- ( ( ph /\ ( m e. NN /\ y e. RR ) ) -> 0 <_ ( ( F ` y ) x. ( 2 ^ m ) ) )
22		flge0nn0 12621	. . . . . . 7 |- ( ( ( ( F ` y ) x. ( 2 ^ m ) ) e. RR /\ 0 <_ ( ( F ` y ) x. ( 2 ^ m ) ) ) -> ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) e. NN0 )
23	14, 21, 22	syl2anc 693	. . . . . 6 |- ( ( ph /\ ( m e. NN /\ y e. RR ) ) -> ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) e. NN0 )
24	23	nn0ge0d 11354	. . . . 5 |- ( ( ph /\ ( m e. NN /\ y e. RR ) ) -> 0 <_ ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) )
25	12	nngt0d 11064	. . . . 5 |- ( ( ph /\ ( m e. NN /\ y e. RR ) ) -> 0 < ( 2 ^ m ) )
26		divge0 10892	. . . . 5 |- ( ( ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) e. RR /\ 0 <_ ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) ) /\ ( ( 2 ^ m ) e. RR /\ 0 < ( 2 ^ m ) ) ) -> 0 <_ ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) / ( 2 ^ m ) ) )
27	16, 24, 13, 25, 26	syl22anc 1327	. . . 4 |- ( ( ph /\ ( m e. NN /\ y e. RR ) ) -> 0 <_ ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) / ( 2 ^ m ) ) )
28		elrege0 12278	. . . 4 |- ( ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) / ( 2 ^ m ) ) e. ( 0 [,) +oo ) <-> ( ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) / ( 2 ^ m ) ) e. RR /\ 0 <_ ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) / ( 2 ^ m ) ) ) )
29	17, 27, 28	sylanbrc 698	. . 3 |- ( ( ph /\ ( m e. NN /\ y e. RR ) ) -> ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) / ( 2 ^ m ) ) e. ( 0 [,) +oo ) )
30	29	ralrimivva 2971	. 2 |- ( ph -> A. m e. NN A. y e. RR ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) / ( 2 ^ m ) ) e. ( 0 [,) +oo ) )
31		mbfi1fseq.3	. . 3 |- J = ( m e. NN , y e. RR |-> ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) / ( 2 ^ m ) ) )
32	31	fmpt2 7237	. 2 |- ( A. m e. NN A. y e. RR ( ( |_ ` ( ( F ` y ) x. ( 2 ^ m ) ) ) / ( 2 ^ m ) ) e. ( 0 [,) +oo ) <-> J : ( NN X. RR ) --> ( 0 [,) +oo ) )
33	30, 32	sylib 208	1 |- ( ph -> J : ( NN X. RR ) --> ( 0 [,) +oo ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   0cc0 9936   x. cmul 9941   +oocpnf 10071   < clt 10074   <_ cle 10075   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   [,)cico 12177   |_cfl 12591   ^cexp 12860  MblFncmbf 23383

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-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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  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-n0 11293  df-z 11378  df-uz 11688  df-ico 12181  df-fl 12593  df-seq 12802  df-exp 12861

This theorem is referenced by:  mbfi1fseqlem5  23486