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Theorem mbfi1fseqlem1 19086
Description: Lemma for mbfi1fseq 19092. (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 5679 . . . . . . . . . 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 10762 . . . . . . . . 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 9893 . . . . . . . . . 10  |-  2  e.  NN
9 nnnn0 9988 . . . . . . . . . 10  |-  ( m  e.  NN  ->  m  e.  NN0 )
10 nnexpcl 11132 . . . . . . . . . 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 9777 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( 2 ^ m
)  e.  RR )
147, 13remulcld 8879 . . . . . 6  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( ( F `  y )  x.  (
2 ^ m ) )  e.  RR )
15 reflcl 10944 . . . . . 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 9810 . . . 4  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( ( |_ `  ( ( F `  y )  x.  (
2 ^ m ) ) )  /  (
2 ^ m ) )  e.  RR )
1812nnnn0d 10034 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( 2 ^ m
)  e.  NN0 )
1918nn0ge0d 10037 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <_  ( 2 ^ m ) )
20 mulge0 9307 . . . . . . . 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 10964 . . . . . . 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 10037 . . . . 5  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <_  ( |_ `  ( ( F `  y )  x.  (
2 ^ m ) ) ) )
2512nngt0d 9805 . . . . 5  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <  ( 2 ^ m ) )
26 divge0 9641 . . . . 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 10762 . . . 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 2648 . 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 6207 . 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 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039    X. cxp 4703   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   RRcr 8752   0cc0 8753    x. cmul 8758    +oocpnf 8880    < clt 8883    <_ cle 8884    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   [,)cico 10674   |_cfl 10940   ^cexp 11120  MblFncmbf 18985
This theorem is referenced by:  mbfi1fseqlem5  19090
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-ico 10678  df-fl 10941  df-seq 11063  df-exp 11121
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