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Theorem stoweidlem10 27861
Description: Lemma for stoweid 27914. 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 9126 . . . . 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 8853 . . . . . . . . 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 9293 . . . . . . 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 11243 . . 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 8843 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
17163ad2ant1 976 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  A  e.  CC )
18 nn0cn 9991 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  CC )
19183ad2ant2 977 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  N  e.  CC )
20 ax-1cn 8811 . . . . 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 9232 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( -u A  x.  N )  =  -u ( A  x.  N
) )
2423oveq2d 5890 . . . . 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 8837 . . . . . . 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 9111 . . . . 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 8839 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( A  x.  N
)  =  ( N  x.  A ) )
3433oveq2d 5890 . . . . 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 2332 . . 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 9111 . . . . 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 5889 . . 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 4068 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 1632    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   CCcc 8751   RRcr 8752   1c1 8754    + caddc 8756    x. cmul 8758    <_ cle 8884    - cmin 9053   -ucneg 9054   NN0cn0 9981   ^cexp 11120
This theorem is referenced by:  stoweidlem24  27875
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
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-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-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-seq 11063  df-exp 11121
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