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Theorem stoweidlem12 27761
Description: Lemma for stoweid 27812. This Lemma is used by other three Lemmas. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem12.1  |-  Q  =  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N
) ) ^ ( K ^ N ) ) )
stoweidlem12.2  |-  ( ph  ->  P : T --> RR )
stoweidlem12.3  |-  ( ph  ->  N  e.  NN0 )
stoweidlem12.4  |-  ( ph  ->  K  e.  NN0 )
Assertion
Ref Expression
stoweidlem12  |-  ( (
ph  /\  t  e.  T )  ->  ( Q `  t )  =  ( ( 1  -  ( ( P `
 t ) ^ N ) ) ^
( K ^ N
) ) )
Distinct variable group:    t, T
Allowed substitution hints:    ph( t)    P( t)    Q( t)    K( t)    N( t)

Proof of Theorem stoweidlem12
StepHypRef Expression
1 simpr 447 . . 3  |-  ( (
ph  /\  t  e.  T )  ->  t  e.  T )
2 1re 8837 . . . . . . 7  |-  1  e.  RR
32a1i 10 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  1  e.  RR )
4 stoweidlem12.2 . . . . . . . . . . 11  |-  ( ph  ->  P : T --> RR )
54adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  P : T --> RR )
65, 1jca 518 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( P : T --> RR  /\  t  e.  T )
)
7 ffvelrn 5663 . . . . . . . . 9  |-  ( ( P : T --> RR  /\  t  e.  T )  ->  ( P `  t
)  e.  RR )
86, 7syl 15 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( P `  t )  e.  RR )
9 stoweidlem12.3 . . . . . . . . 9  |-  ( ph  ->  N  e.  NN0 )
109adantr 451 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  N  e.  NN0 )
118, 10jca 518 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
)  e.  RR  /\  N  e.  NN0 ) )
12 reexpcl 11120 . . . . . . 7  |-  ( ( ( P `  t
)  e.  RR  /\  N  e.  NN0 )  -> 
( ( P `  t ) ^ N
)  e.  RR )
1311, 12syl 15 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
) ^ N )  e.  RR )
143, 13resubcld 9211 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  e.  RR )
15 stoweidlem12.4 . . . . . . . 8  |-  ( ph  ->  K  e.  NN0 )
1615, 9jca 518 . . . . . . 7  |-  ( ph  ->  ( K  e.  NN0  /\  N  e.  NN0 )
)
1716adantr 451 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( K  e.  NN0  /\  N  e.  NN0 ) )
18 nn0expcl 11117 . . . . . 6  |-  ( ( K  e.  NN0  /\  N  e.  NN0 )  -> 
( K ^ N
)  e.  NN0 )
1917, 18syl 15 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  ( K ^ N )  e. 
NN0 )
2014, 19jca 518 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  (
( 1  -  (
( P `  t
) ^ N ) )  e.  RR  /\  ( K ^ N )  e.  NN0 ) )
21 reexpcl 11120 . . . 4  |-  ( ( ( 1  -  (
( P `  t
) ^ N ) )  e.  RR  /\  ( K ^ N )  e.  NN0 )  -> 
( ( 1  -  ( ( P `  t ) ^ N
) ) ^ ( K ^ N ) )  e.  RR )
2220, 21syl 15 . . 3  |-  ( (
ph  /\  t  e.  T )  ->  (
( 1  -  (
( P `  t
) ^ N ) ) ^ ( K ^ N ) )  e.  RR )
231, 22jca 518 . 2  |-  ( (
ph  /\  t  e.  T )  ->  (
t  e.  T  /\  ( ( 1  -  ( ( P `  t ) ^ N
) ) ^ ( K ^ N ) )  e.  RR ) )
24 stoweidlem12.1 . . 3  |-  Q  =  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N
) ) ^ ( K ^ N ) ) )
2524fvmpt2 5608 . 2  |-  ( ( t  e.  T  /\  ( ( 1  -  ( ( P `  t ) ^ N
) ) ^ ( K ^ N ) )  e.  RR )  -> 
( Q `  t
)  =  ( ( 1  -  ( ( P `  t ) ^ N ) ) ^ ( K ^ N ) ) )
2623, 25syl 15 1  |-  ( (
ph  /\  t  e.  T )  ->  ( Q `  t )  =  ( ( 1  -  ( ( P `
 t ) ^ N ) ) ^
( K ^ N
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   1c1 8738    - cmin 9037   NN0cn0 9965   ^cexp 11104
This theorem is referenced by:  stoweidlem24  27773  stoweidlem25  27774  stoweidlem45  27794
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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