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Theorem stoweidlem10 27759
Description: Lemma for stoweid 27812. 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 9110 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
213ad2ant1 976 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  -u A  e.  RR )
3 simp2 956 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  N  e.  NN0 )
4 simpr 447 . . . . . 6  |-  ( ( A  e.  RR  /\  A  <_  1 )  ->  A  <_  1 )
5 simpl 443 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  <_  1 )  ->  A  e.  RR )
6 1re 8837 . . . . . . . . 9  |-  1  e.  RR
76a1i 10 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  <_  1 )  -> 
1  e.  RR )
85, 7jca 518 . . . . . . 7  |-  ( ( A  e.  RR  /\  A  <_  1 )  -> 
( A  e.  RR  /\  1  e.  RR ) )
9 leneg 9277 . . . . . . 7  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  <_  1  <->  -u 1  <_  -u A ) )
108, 9syl 15 . . . . . 6  |-  ( ( A  e.  RR  /\  A  <_  1 )  -> 
( A  <_  1  <->  -u 1  <_  -u A ) )
114, 10mpbid 201 . . . . 5  |-  ( ( A  e.  RR  /\  A  <_  1 )  ->  -u 1  <_  -u A )
12113adant2 974 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  -u 1  <_ 
-u A )
132, 3, 123jca 1132 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  ( -u A  e.  RR  /\  N  e.  NN0  /\  -u 1  <_ 
-u A ) )
14 bernneq 11227 . . 3  |-  ( (
-u A  e.  RR  /\  N  e.  NN0  /\  -u 1  <_  -u A )  ->  ( 1  +  ( -u A  x.  N ) )  <_ 
( ( 1  + 
-u A ) ^ N ) )
1513, 14syl 15 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  (
1  +  ( -u A  x.  N )
)  <_  ( (
1  +  -u A
) ^ N ) )
16 recn 8827 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
17163ad2ant1 976 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  A  e.  CC )
18 nn0cn 9975 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  CC )
19183ad2ant2 977 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  N  e.  CC )
20 ax-1cn 8795 . . . . 5  |-  1  e.  CC
2120a1i 10 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  1  e.  CC )
2217, 19, 213jca 1132 . . 3  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC ) )
23 mulneg1 9216 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( -u A  x.  N )  =  -u ( A  x.  N
) )
2423oveq2d 5874 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( 1  +  (
-u A  x.  N
) )  =  ( 1  +  -u ( A  x.  N )
) )
25243adant3 975 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  (
1  +  ( -u A  x.  N )
)  =  ( 1  +  -u ( A  x.  N ) ) )
26 simp3 957 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  1  e.  CC )
27 3simpa 952 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  ( A  e.  CC  /\  N  e.  CC ) )
28 mulcl 8821 . . . . . . 7  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( A  x.  N
)  e.  CC )
2927, 28syl 15 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  ( A  x.  N )  e.  CC )
3026, 29jca 518 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  (
1  e.  CC  /\  ( A  x.  N
)  e.  CC ) )
31 negsub 9095 . . . . 5  |-  ( ( 1  e.  CC  /\  ( A  x.  N
)  e.  CC )  ->  ( 1  + 
-u ( A  x.  N ) )  =  ( 1  -  ( A  x.  N )
) )
3230, 31syl 15 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  (
1  +  -u ( A  x.  N )
)  =  ( 1  -  ( A  x.  N ) ) )
33 mulcom 8823 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( A  x.  N
)  =  ( N  x.  A ) )
3433oveq2d 5874 . . . . 5  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( 1  -  ( A  x.  N )
)  =  ( 1  -  ( N  x.  A ) ) )
35343adant3 975 . . . 4  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  (
1  -  ( A  x.  N ) )  =  ( 1  -  ( N  x.  A
) ) )
3625, 32, 353eqtrd 2319 . . 3  |-  ( ( A  e.  CC  /\  N  e.  CC  /\  1  e.  CC )  ->  (
1  +  ( -u A  x.  N )
)  =  ( 1  -  ( N  x.  A ) ) )
3722, 36syl 15 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  (
1  +  ( -u A  x.  N )
)  =  ( 1  -  ( N  x.  A ) ) )
3820a1i 10 . . . . . 6  |-  ( A  e.  RR  ->  1  e.  CC )
3938, 16jca 518 . . . . 5  |-  ( A  e.  RR  ->  (
1  e.  CC  /\  A  e.  CC )
)
40 negsub 9095 . . . . 5  |-  ( ( 1  e.  CC  /\  A  e.  CC )  ->  ( 1  +  -u A )  =  ( 1  -  A ) )
4139, 40syl 15 . . . 4  |-  ( A  e.  RR  ->  (
1  +  -u A
)  =  ( 1  -  A ) )
4241oveq1d 5873 . . 3  |-  ( A  e.  RR  ->  (
( 1  +  -u A ) ^ N
)  =  ( ( 1  -  A ) ^ N ) )
43423ad2ant1 976 . 2  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  (
( 1  +  -u A ) ^ N
)  =  ( ( 1  -  A ) ^ N ) )
4415, 37, 433brtr3d 4052 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023  (class class class)co 5858   CCcc 8735   RRcr 8736   1c1 8738    + caddc 8740    x. cmul 8742    <_ cle 8868    - cmin 9037   -ucneg 9038   NN0cn0 9965   ^cexp 11104
This theorem is referenced by:  stoweidlem24  27773
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-exp 11105
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