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Theorem stoweidlem10 31746
Description: Lemma for stoweid 31799. 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 9887 . . . 4  |-  ( A  e.  RR  ->  -u A  e.  RR )
213ad2ant1 1018 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  -u A  e.  RR )
3 simp2 998 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  N  e.  NN0 )
4 simpr 461 . . . . 5  |-  ( ( A  e.  RR  /\  A  <_  1 )  ->  A  <_  1 )
5 simpl 457 . . . . . 6  |-  ( ( A  e.  RR  /\  A  <_  1 )  ->  A  e.  RR )
6 1red 9614 . . . . . 6  |-  ( ( A  e.  RR  /\  A  <_  1 )  -> 
1  e.  RR )
75, 6lenegd 10138 . . . . 5  |-  ( ( A  e.  RR  /\  A  <_  1 )  -> 
( A  <_  1  <->  -u 1  <_  -u A ) )
84, 7mpbid 210 . . . 4  |-  ( ( A  e.  RR  /\  A  <_  1 )  ->  -u 1  <_  -u A )
983adant2 1016 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  -u 1  <_ 
-u A )
10 bernneq 12274 . . 3  |-  ( (
-u A  e.  RR  /\  N  e.  NN0  /\  -u 1  <_  -u A )  ->  ( 1  +  ( -u A  x.  N ) )  <_ 
( ( 1  + 
-u A ) ^ N ) )
112, 3, 9, 10syl3anc 1229 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  (
1  +  ( -u A  x.  N )
)  <_  ( (
1  +  -u A
) ^ N ) )
12 recn 9585 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
13123ad2ant1 1018 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  A  e.  CC )
14 nn0cn 10812 . . . 4  |-  ( N  e.  NN0  ->  N  e.  CC )
15143ad2ant2 1019 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  N  e.  CC )
16 1cnd 9615 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  1  e.  CC )
17 mulneg1 10000 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( -u A  x.  N )  =  -u ( A  x.  N
) )
1817oveq2d 6297 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( 1  +  (
-u A  x.  N
) )  =  ( 1  +  -u ( A  x.  N )
) )
19183adant3 1017 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  (
1  +  ( -u A  x.  N )
)  =  ( 1  +  -u ( A  x.  N ) ) )
20 simp3 999 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  1  e.  CC )
21 mulcl 9579 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( A  x.  N
)  e.  CC )
22213adant3 1017 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  ( A  x.  N )  e.  CC )
2320, 22negsubd 9942 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  (
1  +  -u ( A  x.  N )
)  =  ( 1  -  ( A  x.  N ) ) )
24 mulcom 9581 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( A  x.  N
)  =  ( N  x.  A ) )
2524oveq2d 6297 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( 1  -  ( A  x.  N )
)  =  ( 1  -  ( N  x.  A ) ) )
26253adant3 1017 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  (
1  -  ( A  x.  N ) )  =  ( 1  -  ( N  x.  A
) ) )
2719, 23, 263eqtrd 2488 . . 3  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  (
1  +  ( -u A  x.  N )
)  =  ( 1  -  ( N  x.  A ) ) )
2813, 15, 16, 27syl3anc 1229 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  (
1  +  ( -u A  x.  N )
)  =  ( 1  -  ( N  x.  A ) ) )
29 1cnd 9615 . . . . 5  |-  ( A  e.  RR  ->  1  e.  CC )
3029, 12negsubd 9942 . . . 4  |-  ( A  e.  RR  ->  (
1  +  -u A
)  =  ( 1  -  A ) )
3130oveq1d 6296 . . 3  |-  ( A  e.  RR  ->  (
( 1  +  -u A ) ^ N
)  =  ( ( 1  -  A ) ^ N ) )
32313ad2ant1 1018 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  (
( 1  +  -u A ) ^ N
)  =  ( ( 1  -  A ) ^ N ) )
3311, 28, 323brtr3d 4466 1  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  (
1  -  ( N  x.  A ) )  <_  ( ( 1  -  A ) ^ N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   class class class wbr 4437  (class class class)co 6281   CCcc 9493   RRcr 9494   1c1 9496    + caddc 9498    x. cmul 9500    <_ cle 9632    - cmin 9810   -ucneg 9811   NN0cn0 10802   ^cexp 12148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-n0 10803  df-z 10872  df-uz 11093  df-seq 12090  df-exp 12149
This theorem is referenced by:  stoweidlem24  31760
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