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Theorem mbfi1fseqlem1 19070
Description: Lemma for mbfi1fseq 19076. (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
StepHypRef Expression
1 mbfi1fseq.2 . . . . . . . . . 10  |-  ( ph  ->  F : RR --> ( 0 [,)  +oo ) )
2 simpr 447 . . . . . . . . . 10  |-  ( ( m  e.  NN  /\  y  e.  RR )  ->  y  e.  RR )
3 ffvelrn 5663 . . . . . . . . . 10  |-  ( ( F : RR --> ( 0 [,)  +oo )  /\  y  e.  RR )  ->  ( F `  y )  e.  ( 0 [,)  +oo ) )
41, 2, 3syl2an 463 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( F `  y
)  e.  ( 0 [,)  +oo ) )
5 elrege0 10746 . . . . . . . . 9  |-  ( ( F `  y )  e.  ( 0 [,) 
+oo )  <->  ( ( F `  y )  e.  RR  /\  0  <_ 
( F `  y
) ) )
64, 5sylib 188 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( ( F `  y )  e.  RR  /\  0  <_  ( F `  y ) ) )
76simpld 445 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( F `  y
)  e.  RR )
8 2nn 9877 . . . . . . . . . 10  |-  2  e.  NN
9 nnnn0 9972 . . . . . . . . . 10  |-  ( m  e.  NN  ->  m  e.  NN0 )
10 nnexpcl 11116 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  m  e.  NN0 )  -> 
( 2 ^ m
)  e.  NN )
118, 9, 10sylancr 644 . . . . . . . . 9  |-  ( m  e.  NN  ->  (
2 ^ m )  e.  NN )
1211ad2antrl 708 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( 2 ^ m
)  e.  NN )
1312nnred 9761 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( 2 ^ m
)  e.  RR )
147, 13remulcld 8863 . . . . . 6  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( ( F `  y )  x.  (
2 ^ m ) )  e.  RR )
15 reflcl 10928 . . . . . 6  |-  ( ( ( F `  y
)  x.  ( 2 ^ m ) )  e.  RR  ->  ( |_ `  ( ( F `
 y )  x.  ( 2 ^ m
) ) )  e.  RR )
1614, 15syl 15 . . . . 5  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( |_ `  (
( F `  y
)  x.  ( 2 ^ m ) ) )  e.  RR )
1716, 12nndivred 9794 . . . 4  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( ( |_ `  ( ( F `  y )  x.  (
2 ^ m ) ) )  /  (
2 ^ m ) )  e.  RR )
1812nnnn0d 10018 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( 2 ^ m
)  e.  NN0 )
1918nn0ge0d 10021 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <_  ( 2 ^ m ) )
20 mulge0 9291 . . . . . . . 8  |-  ( ( ( ( F `  y )  e.  RR  /\  0  <_  ( F `  y ) )  /\  ( ( 2 ^ m )  e.  RR  /\  0  <_  ( 2 ^ m ) ) )  ->  0  <_  ( ( F `  y
)  x.  ( 2 ^ m ) ) )
216, 13, 19, 20syl12anc 1180 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <_  ( ( F `  y )  x.  ( 2 ^ m
) ) )
22 flge0nn0 10948 . . . . . . 7  |-  ( ( ( ( F `  y )  x.  (
2 ^ m ) )  e.  RR  /\  0  <_  ( ( F `
 y )  x.  ( 2 ^ m
) ) )  -> 
( |_ `  (
( F `  y
)  x.  ( 2 ^ m ) ) )  e.  NN0 )
2314, 21, 22syl2anc 642 . . . . . 6  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( |_ `  (
( F `  y
)  x.  ( 2 ^ m ) ) )  e.  NN0 )
2423nn0ge0d 10021 . . . . 5  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <_  ( |_ `  ( ( F `  y )  x.  (
2 ^ m ) ) ) )
2512nngt0d 9789 . . . . 5  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <  ( 2 ^ m ) )
26 divge0 9625 . . . . 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 ) ) )
2716, 24, 13, 25, 26syl22anc 1183 . . . 4  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <_  ( ( |_ `  ( ( F `
 y )  x.  ( 2 ^ m
) ) )  / 
( 2 ^ m
) ) )
28 elrege0 10746 . . . 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 ) ) ) )
2917, 27, 28sylanbrc 645 . . 3  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( ( |_ `  ( ( F `  y )  x.  (
2 ^ m ) ) )  /  (
2 ^ m ) )  e.  ( 0 [,)  +oo ) )
3029ralrimivva 2635 . 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 ) ) )
3231fmpt2 6191 . 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 ) )
3330, 32sylib 188 1  |-  ( ph  ->  J : ( NN 
X.  RR ) --> ( 0 [,)  +oo )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   RRcr 8736   0cc0 8737    x. cmul 8742    +oocpnf 8864    < clt 8867    <_ cle 8868    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   [,)cico 10658   |_cfl 10924   ^cexp 11104  MblFncmbf 18969
This theorem is referenced by:  mbfi1fseqlem5  19074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-ico 10662  df-fl 10925  df-seq 11047  df-exp 11105
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