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Theorem mbfi1fseqlem1 19607
Description: Lemma for mbfi1fseq 19613. (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 448 . . . . . . . . . 10  |-  ( ( m  e.  NN  /\  y  e.  RR )  ->  y  e.  RR )
3 ffvelrn 5868 . . . . . . . . . 10  |-  ( ( F : RR --> ( 0 [,)  +oo )  /\  y  e.  RR )  ->  ( F `  y )  e.  ( 0 [,)  +oo ) )
41, 2, 3syl2an 464 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( F `  y
)  e.  ( 0 [,)  +oo ) )
5 elrege0 11007 . . . . . . . . 9  |-  ( ( F `  y )  e.  ( 0 [,) 
+oo )  <->  ( ( F `  y )  e.  RR  /\  0  <_ 
( F `  y
) ) )
64, 5sylib 189 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( ( F `  y )  e.  RR  /\  0  <_  ( F `  y ) ) )
76simpld 446 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( F `  y
)  e.  RR )
8 2nn 10133 . . . . . . . . . 10  |-  2  e.  NN
9 nnnn0 10228 . . . . . . . . . 10  |-  ( m  e.  NN  ->  m  e.  NN0 )
10 nnexpcl 11394 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  m  e.  NN0 )  -> 
( 2 ^ m
)  e.  NN )
118, 9, 10sylancr 645 . . . . . . . . 9  |-  ( m  e.  NN  ->  (
2 ^ m )  e.  NN )
1211ad2antrl 709 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( 2 ^ m
)  e.  NN )
1312nnred 10015 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( 2 ^ m
)  e.  RR )
147, 13remulcld 9116 . . . . . 6  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( ( F `  y )  x.  (
2 ^ m ) )  e.  RR )
15 reflcl 11205 . . . . . 6  |-  ( ( ( F `  y
)  x.  ( 2 ^ m ) )  e.  RR  ->  ( |_ `  ( ( F `
 y )  x.  ( 2 ^ m
) ) )  e.  RR )
1614, 15syl 16 . . . . 5  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( |_ `  (
( F `  y
)  x.  ( 2 ^ m ) ) )  e.  RR )
1716, 12nndivred 10048 . . . 4  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( ( |_ `  ( ( F `  y )  x.  (
2 ^ m ) ) )  /  (
2 ^ m ) )  e.  RR )
1812nnnn0d 10274 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( 2 ^ m
)  e.  NN0 )
1918nn0ge0d 10277 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <_  ( 2 ^ m ) )
20 mulge0 9545 . . . . . . . 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 1182 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <_  ( ( F `  y )  x.  ( 2 ^ m
) ) )
22 flge0nn0 11225 . . . . . . 7  |-  ( ( ( ( F `  y )  x.  (
2 ^ m ) )  e.  RR  /\  0  <_  ( ( F `
 y )  x.  ( 2 ^ m
) ) )  -> 
( |_ `  (
( F `  y
)  x.  ( 2 ^ m ) ) )  e.  NN0 )
2314, 21, 22syl2anc 643 . . . . . 6  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( |_ `  (
( F `  y
)  x.  ( 2 ^ m ) ) )  e.  NN0 )
2423nn0ge0d 10277 . . . . 5  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <_  ( |_ `  ( ( F `  y )  x.  (
2 ^ m ) ) ) )
2512nngt0d 10043 . . . . 5  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <  ( 2 ^ m ) )
26 divge0 9879 . . . . 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 1185 . . . 4  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <_  ( ( |_ `  ( ( F `
 y )  x.  ( 2 ^ m
) ) )  / 
( 2 ^ m
) ) )
28 elrege0 11007 . . . 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 646 . . 3  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( ( |_ `  ( ( F `  y )  x.  (
2 ^ m ) ) )  /  (
2 ^ m ) )  e.  ( 0 [,)  +oo ) )
3029ralrimivva 2798 . 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 6418 . 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 189 1  |-  ( ph  ->  J : ( NN 
X.  RR ) --> ( 0 [,)  +oo )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   class class class wbr 4212    X. cxp 4876   -->wf 5450   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   RRcr 8989   0cc0 8990    x. cmul 8995    +oocpnf 9117    < clt 9120    <_ cle 9121    / cdiv 9677   NNcn 10000   2c2 10049   NN0cn0 10221   [,)cico 10918   |_cfl 11201   ^cexp 11382  MblFncmbf 19506
This theorem is referenced by:  mbfi1fseqlem5  19611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-ico 10922  df-fl 11202  df-seq 11324  df-exp 11383
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