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Theorem stoweidlem40 27789
Description: This lemma proves that qn is in the subalgebra, as in the prove of Lemma 1 in [BrosowskiDeutsh] p. 90. Q is used to represent qn in the paper, N is used to represent n in the paper, and M is used to represent k^n in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem40.1  |-  F/_ t P
stoweidlem40.2  |-  F/ t
ph
stoweidlem40.3  |-  Q  =  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N
) ) ^ M
) )
stoweidlem40.4  |-  F  =  ( t  e.  T  |->  ( 1  -  (
( P `  t
) ^ N ) ) )
stoweidlem40.5  |-  G  =  ( t  e.  T  |->  1 )
stoweidlem40.6  |-  H  =  ( t  e.  T  |->  ( ( P `  t ) ^ N
) )
stoweidlem40.7  |-  ( ph  ->  P  e.  A )
stoweidlem40.8  |-  ( ph  ->  P : T --> RR )
stoweidlem40.9  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
stoweidlem40.10  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
stoweidlem40.11  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
stoweidlem40.12  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
stoweidlem40.13  |-  ( ph  ->  N  e.  NN )
stoweidlem40.14  |-  ( ph  ->  M  e.  NN )
Assertion
Ref Expression
stoweidlem40  |-  ( ph  ->  Q  e.  A )
Distinct variable groups:    f, g,
t, A    f, F, g    f, G, g    f, H, g    P, f, g    T, f, g, t    ph, f,
g    x, t, A    t, M    t, N    x, T    ph, x
Allowed substitution hints:    ph( t)    P( x, t)    Q( x, t, f, g)    F( x, t)    G( x, t)    H( x, t)    M( x, f, g)    N( x, f, g)

Proof of Theorem stoweidlem40
StepHypRef Expression
1 stoweidlem40.3 . . 3  |-  Q  =  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N
) ) ^ M
) )
2 stoweidlem40.2 . . . 4  |-  F/ t
ph
3 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  t  e.  T )
4 1re 8837 . . . . . . . . . . 11  |-  1  e.  RR
54a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  1  e.  RR )
6 stoweidlem40.8 . . . . . . . . . . . . . . 15  |-  ( ph  ->  P : T --> RR )
76adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  t  e.  T )  ->  P : T --> RR )
87, 3jca 518 . . . . . . . . . . . . 13  |-  ( (
ph  /\  t  e.  T )  ->  ( P : T --> RR  /\  t  e.  T )
)
9 ffvelrn 5663 . . . . . . . . . . . . 13  |-  ( ( P : T --> RR  /\  t  e.  T )  ->  ( P `  t
)  e.  RR )
108, 9syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  T )  ->  ( P `  t )  e.  RR )
11 stoweidlem40.13 . . . . . . . . . . . . . 14  |-  ( ph  ->  N  e.  NN )
12 nnnn0 9972 . . . . . . . . . . . . . 14  |-  ( N  e.  NN  ->  N  e.  NN0 )
1311, 12syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  NN0 )
1413adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  t  e.  T )  ->  N  e.  NN0 )
1510, 14jca 518 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
)  e.  RR  /\  N  e.  NN0 ) )
16 reexpcl 11120 . . . . . . . . . . 11  |-  ( ( ( P `  t
)  e.  RR  /\  N  e.  NN0 )  -> 
( ( P `  t ) ^ N
)  e.  RR )
1715, 16syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
) ^ N )  e.  RR )
185, 17jca 518 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  (
1  e.  RR  /\  ( ( P `  t ) ^ N
)  e.  RR ) )
19 resubcl 9111 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  ( ( P `  t ) ^ N
)  e.  RR )  ->  ( 1  -  ( ( P `  t ) ^ N
) )  e.  RR )
2018, 19syl 15 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  e.  RR )
213, 20jca 518 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
t  e.  T  /\  ( 1  -  (
( P `  t
) ^ N ) )  e.  RR ) )
22 stoweidlem40.4 . . . . . . . 8  |-  F  =  ( t  e.  T  |->  ( 1  -  (
( P `  t
) ^ N ) ) )
2322fvmpt2 5608 . . . . . . 7  |-  ( ( t  e.  T  /\  ( 1  -  (
( P `  t
) ^ N ) )  e.  RR )  ->  ( F `  t )  =  ( 1  -  ( ( P `  t ) ^ N ) ) )
2421, 23syl 15 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  =  ( 1  -  ( ( P `  t ) ^ N
) ) )
2524eqcomd 2288 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  =  ( F `  t ) )
2625oveq1d 5873 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  (
( 1  -  (
( P `  t
) ^ N ) ) ^ M )  =  ( ( F `
 t ) ^ M ) )
272, 26mpteq2da 4105 . . 3  |-  ( ph  ->  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N
) ) ^ M
) )  =  ( t  e.  T  |->  ( ( F `  t
) ^ M ) ) )
281, 27syl5eq 2327 . 2  |-  ( ph  ->  Q  =  ( t  e.  T  |->  ( ( F `  t ) ^ M ) ) )
29 nfmpt1 4109 . . . 4  |-  F/_ t
( t  e.  T  |->  ( 1  -  (
( P `  t
) ^ N ) ) )
3022, 29nfcxfr 2416 . . 3  |-  F/_ t F
31 stoweidlem40.9 . . 3  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
32 stoweidlem40.11 . . 3  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
33 stoweidlem40.12 . . 3  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
34 id 19 . . . . . . . . . . 11  |-  ( t  e.  T  ->  t  e.  T )
354a1i 10 . . . . . . . . . . 11  |-  ( t  e.  T  ->  1  e.  RR )
3634, 35jca 518 . . . . . . . . . 10  |-  ( t  e.  T  ->  (
t  e.  T  /\  1  e.  RR )
)
37 stoweidlem40.5 . . . . . . . . . . 11  |-  G  =  ( t  e.  T  |->  1 )
3837fvmpt2 5608 . . . . . . . . . 10  |-  ( ( t  e.  T  /\  1  e.  RR )  ->  ( G `  t
)  =  1 )
3936, 38syl 15 . . . . . . . . 9  |-  ( t  e.  T  ->  ( G `  t )  =  1 )
4039eqcomd 2288 . . . . . . . 8  |-  ( t  e.  T  ->  1  =  ( G `  t ) )
4140adantl 452 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  1  =  ( G `  t ) )
423, 17jca 518 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  (
t  e.  T  /\  ( ( P `  t ) ^ N
)  e.  RR ) )
43 stoweidlem40.6 . . . . . . . . . 10  |-  H  =  ( t  e.  T  |->  ( ( P `  t ) ^ N
) )
4443fvmpt2 5608 . . . . . . . . 9  |-  ( ( t  e.  T  /\  ( ( P `  t ) ^ N
)  e.  RR )  ->  ( H `  t )  =  ( ( P `  t
) ^ N ) )
4542, 44syl 15 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  =  ( ( P `
 t ) ^ N ) )
4645eqcomd 2288 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
) ^ N )  =  ( H `  t ) )
4741, 46oveq12d 5876 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  =  ( ( G `
 t )  -  ( H `  t ) ) )
482, 47mpteq2da 4105 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  ( 1  -  (
( P `  t
) ^ N ) ) )  =  ( t  e.  T  |->  ( ( G `  t
)  -  ( H `
 t ) ) ) )
4922, 48syl5eq 2327 . . . 4  |-  ( ph  ->  F  =  ( t  e.  T  |->  ( ( G `  t )  -  ( H `  t ) ) ) )
50 id 19 . . . . . 6  |-  ( ph  ->  ph )
514jctr 526 . . . . . . . 8  |-  ( ph  ->  ( ph  /\  1  e.  RR ) )
5233stoweidlem4 27753 . . . . . . . 8  |-  ( (
ph  /\  1  e.  RR )  ->  ( t  e.  T  |->  1 )  e.  A )
5351, 52syl 15 . . . . . . 7  |-  ( ph  ->  ( t  e.  T  |->  1 )  e.  A
)
5437, 53syl5eqel 2367 . . . . . 6  |-  ( ph  ->  G  e.  A )
55 stoweidlem40.1 . . . . . . . 8  |-  F/_ t P
56 stoweidlem40.7 . . . . . . . 8  |-  ( ph  ->  P  e.  A )
5755, 2, 31, 32, 33, 56, 13stoweidlem19 27768 . . . . . . 7  |-  ( ph  ->  ( t  e.  T  |->  ( ( P `  t ) ^ N
) )  e.  A
)
5843, 57syl5eqel 2367 . . . . . 6  |-  ( ph  ->  H  e.  A )
5950, 54, 583jca 1132 . . . . 5  |-  ( ph  ->  ( ph  /\  G  e.  A  /\  H  e.  A ) )
60 nfmpt1 4109 . . . . . . 7  |-  F/_ t
( t  e.  T  |->  1 )
6137, 60nfcxfr 2416 . . . . . 6  |-  F/_ t G
62 nfmpt1 4109 . . . . . . 7  |-  F/_ t
( t  e.  T  |->  ( ( P `  t ) ^ N
) )
6343, 62nfcxfr 2416 . . . . . 6  |-  F/_ t H
64 stoweidlem40.10 . . . . . 6  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
6561, 63, 2, 31, 64, 32, 33stoweidlem33 27782 . . . . 5  |-  ( (
ph  /\  G  e.  A  /\  H  e.  A
)  ->  ( t  e.  T  |->  ( ( G `  t )  -  ( H `  t ) ) )  e.  A )
6659, 65syl 15 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  -  ( H `  t )
) )  e.  A
)
6749, 66eqeltrd 2357 . . 3  |-  ( ph  ->  F  e.  A )
68 stoweidlem40.14 . . . 4  |-  ( ph  ->  M  e.  NN )
69 nnnn0 9972 . . . 4  |-  ( M  e.  NN  ->  M  e.  NN0 )
7068, 69syl 15 . . 3  |-  ( ph  ->  M  e.  NN0 )
7130, 2, 31, 32, 33, 67, 70stoweidlem19 27768 . 2  |-  ( ph  ->  ( t  e.  T  |->  ( ( F `  t ) ^ M
) )  e.  A
)
7228, 71eqeltrd 2357 1  |-  ( ph  ->  Q  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   NNcn 9746   NN0cn0 9965   ^cexp 11104
This theorem is referenced by:  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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