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Theorem mbfi1fseqlem1 22248
Description: Lemma for mbfi1fseq 22254. (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 461 . . . . . . . . . 10  |-  ( ( m  e.  NN  /\  y  e.  RR )  ->  y  e.  RR )
3 ffvelrn 6030 . . . . . . . . . 10  |-  ( ( F : RR --> ( 0 [,) +oo )  /\  y  e.  RR )  ->  ( F `  y
)  e.  ( 0 [,) +oo ) )
41, 2, 3syl2an 477 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( F `  y
)  e.  ( 0 [,) +oo ) )
5 elrege0 11652 . . . . . . . . 9  |-  ( ( F `  y )  e.  ( 0 [,) +oo )  <->  ( ( F `
 y )  e.  RR  /\  0  <_ 
( F `  y
) ) )
64, 5sylib 196 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( ( F `  y )  e.  RR  /\  0  <_  ( F `  y ) ) )
76simpld 459 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( F `  y
)  e.  RR )
8 2nn 10714 . . . . . . . . . 10  |-  2  e.  NN
9 nnnn0 10823 . . . . . . . . . 10  |-  ( m  e.  NN  ->  m  e.  NN0 )
10 nnexpcl 12182 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  m  e.  NN0 )  -> 
( 2 ^ m
)  e.  NN )
118, 9, 10sylancr 663 . . . . . . . . 9  |-  ( m  e.  NN  ->  (
2 ^ m )  e.  NN )
1211ad2antrl 727 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( 2 ^ m
)  e.  NN )
1312nnred 10571 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( 2 ^ m
)  e.  RR )
147, 13remulcld 9641 . . . . . 6  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( ( F `  y )  x.  (
2 ^ m ) )  e.  RR )
15 reflcl 11936 . . . . . 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 10605 . . . 4  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( ( |_ `  ( ( F `  y )  x.  (
2 ^ m ) ) )  /  (
2 ^ m ) )  e.  RR )
1812nnnn0d 10873 . . . . . . . . 9  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( 2 ^ m
)  e.  NN0 )
1918nn0ge0d 10876 . . . . . . . 8  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <_  ( 2 ^ m ) )
20 mulge0 10091 . . . . . . . 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 1226 . . . . . . 7  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <_  ( ( F `  y )  x.  ( 2 ^ m
) ) )
22 flge0nn0 11957 . . . . . . 7  |-  ( ( ( ( F `  y )  x.  (
2 ^ m ) )  e.  RR  /\  0  <_  ( ( F `
 y )  x.  ( 2 ^ m
) ) )  -> 
( |_ `  (
( F `  y
)  x.  ( 2 ^ m ) ) )  e.  NN0 )
2314, 21, 22syl2anc 661 . . . . . 6  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( |_ `  (
( F `  y
)  x.  ( 2 ^ m ) ) )  e.  NN0 )
2423nn0ge0d 10876 . . . . 5  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <_  ( |_ `  ( ( F `  y )  x.  (
2 ^ m ) ) ) )
2512nngt0d 10600 . . . . 5  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <  ( 2 ^ m ) )
26 divge0 10432 . . . . 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 1229 . . . 4  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
0  <_  ( ( |_ `  ( ( F `
 y )  x.  ( 2 ^ m
) ) )  / 
( 2 ^ m
) ) )
28 elrege0 11652 . . . 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 664 . . 3  |-  ( (
ph  /\  ( m  e.  NN  /\  y  e.  RR ) )  -> 
( ( |_ `  ( ( F `  y )  x.  (
2 ^ m ) ) )  /  (
2 ^ m ) )  e.  ( 0 [,) +oo ) )
3029ralrimivva 2878 . 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 6866 . 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 196 1  |-  ( ph  ->  J : ( NN 
X.  RR ) --> ( 0 [,) +oo )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   class class class wbr 4456    X. cxp 5006   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   RRcr 9508   0cc0 9509    x. cmul 9514   +oocpnf 9642    < clt 9645    <_ cle 9646    / cdiv 10227   NNcn 10556   2c2 10606   NN0cn0 10816   [,)cico 11556   |_cfl 11930   ^cexp 12169  MblFncmbf 22149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-ico 11560  df-fl 11932  df-seq 12111  df-exp 12170
This theorem is referenced by:  mbfi1fseqlem5  22252
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