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Theorem stoweidlem10 27634
Description: Lemma for stoweid 27687. This lemma is used by Lemma 1 in [BrosowskiDeutsh] p. 90, this lemma is an application of Bernoulli's inequality. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Assertion
Ref Expression
stoweidlem10  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  (
1  -  ( N  x.  A ) )  <_  ( ( 1  -  A ) ^ N ) )

Proof of Theorem stoweidlem10
StepHypRef Expression
1 renegcl 9328 . . . 4  |-  ( A  e.  RR  ->  -u A  e.  RR )
213ad2ant1 978 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  -u A  e.  RR )
3 simp2 958 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  N  e.  NN0 )
4 simpr 448 . . . . 5  |-  ( ( A  e.  RR  /\  A  <_  1 )  ->  A  <_  1 )
5 simpl 444 . . . . . 6  |-  ( ( A  e.  RR  /\  A  <_  1 )  ->  A  e.  RR )
6 1re 9054 . . . . . . 7  |-  1  e.  RR
76a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  A  <_  1 )  -> 
1  e.  RR )
85, 7lenegd 9569 . . . . 5  |-  ( ( A  e.  RR  /\  A  <_  1 )  -> 
( A  <_  1  <->  -u 1  <_  -u A ) )
94, 8mpbid 202 . . . 4  |-  ( ( A  e.  RR  /\  A  <_  1 )  ->  -u 1  <_  -u A )
1093adant2 976 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  -u 1  <_ 
-u A )
11 bernneq 11468 . . 3  |-  ( (
-u A  e.  RR  /\  N  e.  NN0  /\  -u 1  <_  -u A )  ->  ( 1  +  ( -u A  x.  N ) )  <_ 
( ( 1  + 
-u A ) ^ N ) )
122, 3, 10, 11syl3anc 1184 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  (
1  +  ( -u A  x.  N )
)  <_  ( (
1  +  -u A
) ^ N ) )
13 recn 9044 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
14133ad2ant1 978 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  A  e.  CC )
15 nn0cn 10195 . . . 4  |-  ( N  e.  NN0  ->  N  e.  CC )
16153ad2ant2 979 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  N  e.  CC )
17 ax-1cn 9012 . . . 4  |-  1  e.  CC
1817a1i 11 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  1  e.  CC )
19 mulneg1 9434 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( -u A  x.  N )  =  -u ( A  x.  N
) )
2019oveq2d 6064 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( 1  +  (
-u A  x.  N
) )  =  ( 1  +  -u ( A  x.  N )
) )
21203adant3 977 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  (
1  +  ( -u A  x.  N )
)  =  ( 1  +  -u ( A  x.  N ) ) )
22 simp3 959 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  1  e.  CC )
23 mulcl 9038 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( A  x.  N
)  e.  CC )
24233adant3 977 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  ( A  x.  N )  e.  CC )
2522, 24negsubd 9381 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  (
1  +  -u ( A  x.  N )
)  =  ( 1  -  ( A  x.  N ) ) )
26 mulcom 9040 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( A  x.  N
)  =  ( N  x.  A ) )
2726oveq2d 6064 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( 1  -  ( A  x.  N )
)  =  ( 1  -  ( N  x.  A ) ) )
28273adant3 977 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  (
1  -  ( A  x.  N ) )  =  ( 1  -  ( N  x.  A
) ) )
2921, 25, 283eqtrd 2448 . . 3  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  (
1  +  ( -u A  x.  N )
)  =  ( 1  -  ( N  x.  A ) ) )
3014, 16, 18, 29syl3anc 1184 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  (
1  +  ( -u A  x.  N )
)  =  ( 1  -  ( N  x.  A ) ) )
3117a1i 11 . . . . 5  |-  ( A  e.  RR  ->  1  e.  CC )
3231, 13negsubd 9381 . . . 4  |-  ( A  e.  RR  ->  (
1  +  -u A
)  =  ( 1  -  A ) )
3332oveq1d 6063 . . 3  |-  ( A  e.  RR  ->  (
( 1  +  -u A ) ^ N
)  =  ( ( 1  -  A ) ^ N ) )
34333ad2ant1 978 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  (
( 1  +  -u A ) ^ N
)  =  ( ( 1  -  A ) ^ N ) )
3512, 30, 343brtr3d 4209 1  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  (
1  -  ( N  x.  A ) )  <_  ( ( 1  -  A ) ^ N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4180  (class class class)co 6048   CCcc 8952   RRcr 8953   1c1 8955    + caddc 8957    x. cmul 8959    <_ cle 9085    - cmin 9255   -ucneg 9256   NN0cn0 10185   ^cexp 11345
This theorem is referenced by:  stoweidlem24  27648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-n0 10186  df-z 10247  df-uz 10453  df-seq 11287  df-exp 11346
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