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Theorem stoweidlem10 27082
Description: Lemma for stoweid 27135. 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 9197 . . . . 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 8924 . . . . . . . . 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 9364 . . . . . . 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 11317 . . 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 8914 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
17163ad2ant1 976 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  A  e.  CC )
18 nn0cn 10064 . . . . 5  |-  ( N  e.  NN0  ->  N  e.  CC )
19183ad2ant2 977 . . . 4  |-  ( ( A  e.  RR  /\  N  e.  NN0  /\  A  <_  1 )  ->  N  e.  CC )
20 ax-1cn 8882 . . . . 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 9303 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( -u A  x.  N )  =  -u ( A  x.  N
) )
2423oveq2d 5958 . . . . 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 8908 . . . . . . 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 9182 . . . . 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 8910 . . . . . 6  |-  ( ( A  e.  CC  /\  N  e.  CC )  ->  ( A  x.  N
)  =  ( N  x.  A ) )
3433oveq2d 5958 . . . . 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 2394 . . 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 9182 . . . . 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 5957 . . 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 4131 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 1642    e. wcel 1710   class class class wbr 4102  (class class class)co 5942   CCcc 8822   RRcr 8823   1c1 8825    + caddc 8827    x. cmul 8829    <_ cle 8955    - cmin 9124   -ucneg 9125   NN0cn0 10054   ^cexp 11194
This theorem is referenced by:  stoweidlem24  27096
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-n0 10055  df-z 10114  df-uz 10320  df-seq 11136  df-exp 11195
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