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Theorem stoweidlem40 31654
Description: This lemma proves that qn is in the subalgebra, as in the proof 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 461 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  t  e.  T )
4 1red 9623 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  1  e.  RR )
5 stoweidlem40.8 . . . . . . . . . 10  |-  ( ph  ->  P : T --> RR )
65fnvinran 31282 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( P `  t )  e.  RR )
7 stoweidlem40.13 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN )
87nnnn0d 10864 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
98adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  N  e.  NN0 )
106, 9reexpcld 12307 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
) ^ N )  e.  RR )
114, 10resubcld 9999 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  e.  RR )
12 stoweidlem40.4 . . . . . . . 8  |-  F  =  ( t  e.  T  |->  ( 1  -  (
( P `  t
) ^ N ) ) )
1312fvmpt2 5964 . . . . . . 7  |-  ( ( t  e.  T  /\  ( 1  -  (
( P `  t
) ^ N ) )  e.  RR )  ->  ( F `  t )  =  ( 1  -  ( ( P `  t ) ^ N ) ) )
143, 11, 13syl2anc 661 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  =  ( 1  -  ( ( P `  t ) ^ N
) ) )
1514eqcomd 2475 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  =  ( F `  t ) )
1615oveq1d 6310 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  (
( 1  -  (
( P `  t
) ^ N ) ) ^ M )  =  ( ( F `
 t ) ^ M ) )
172, 16mpteq2da 4538 . . 3  |-  ( ph  ->  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N
) ) ^ M
) )  =  ( t  e.  T  |->  ( ( F `  t
) ^ M ) ) )
181, 17syl5eq 2520 . 2  |-  ( ph  ->  Q  =  ( t  e.  T  |->  ( ( F `  t ) ^ M ) ) )
19 nfmpt1 4542 . . . 4  |-  F/_ t
( t  e.  T  |->  ( 1  -  (
( P `  t
) ^ N ) ) )
2012, 19nfcxfr 2627 . . 3  |-  F/_ t F
21 stoweidlem40.9 . . 3  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
22 stoweidlem40.11 . . 3  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
23 stoweidlem40.12 . . 3  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
24 1re 9607 . . . . . . . . . 10  |-  1  e.  RR
25 stoweidlem40.5 . . . . . . . . . . 11  |-  G  =  ( t  e.  T  |->  1 )
2625fvmpt2 5964 . . . . . . . . . 10  |-  ( ( t  e.  T  /\  1  e.  RR )  ->  ( G `  t
)  =  1 )
2724, 26mpan2 671 . . . . . . . . 9  |-  ( t  e.  T  ->  ( G `  t )  =  1 )
2827eqcomd 2475 . . . . . . . 8  |-  ( t  e.  T  ->  1  =  ( G `  t ) )
2928adantl 466 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  1  =  ( G `  t ) )
30 stoweidlem40.6 . . . . . . . . . 10  |-  H  =  ( t  e.  T  |->  ( ( P `  t ) ^ N
) )
3130fvmpt2 5964 . . . . . . . . 9  |-  ( ( t  e.  T  /\  ( ( P `  t ) ^ N
)  e.  RR )  ->  ( H `  t )  =  ( ( P `  t
) ^ N ) )
323, 10, 31syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  =  ( ( P `
 t ) ^ N ) )
3332eqcomd 2475 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
) ^ N )  =  ( H `  t ) )
3429, 33oveq12d 6313 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  =  ( ( G `
 t )  -  ( H `  t ) ) )
352, 34mpteq2da 4538 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  ( 1  -  (
( P `  t
) ^ N ) ) )  =  ( t  e.  T  |->  ( ( G `  t
)  -  ( H `
 t ) ) ) )
3612, 35syl5eq 2520 . . . 4  |-  ( ph  ->  F  =  ( t  e.  T  |->  ( ( G `  t )  -  ( H `  t ) ) ) )
3723stoweidlem4 31618 . . . . . . 7  |-  ( (
ph  /\  1  e.  RR )  ->  ( t  e.  T  |->  1 )  e.  A )
3824, 37mpan2 671 . . . . . 6  |-  ( ph  ->  ( t  e.  T  |->  1 )  e.  A
)
3925, 38syl5eqel 2559 . . . . 5  |-  ( ph  ->  G  e.  A )
40 stoweidlem40.1 . . . . . . 7  |-  F/_ t P
41 stoweidlem40.7 . . . . . . 7  |-  ( ph  ->  P  e.  A )
4240, 2, 21, 22, 23, 41, 8stoweidlem19 31633 . . . . . 6  |-  ( ph  ->  ( t  e.  T  |->  ( ( P `  t ) ^ N
) )  e.  A
)
4330, 42syl5eqel 2559 . . . . 5  |-  ( ph  ->  H  e.  A )
44 nfmpt1 4542 . . . . . . 7  |-  F/_ t
( t  e.  T  |->  1 )
4525, 44nfcxfr 2627 . . . . . 6  |-  F/_ t G
46 nfmpt1 4542 . . . . . . 7  |-  F/_ t
( t  e.  T  |->  ( ( P `  t ) ^ N
) )
4730, 46nfcxfr 2627 . . . . . 6  |-  F/_ t H
48 stoweidlem40.10 . . . . . 6  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
4945, 47, 2, 21, 48, 22, 23stoweidlem33 31647 . . . . 5  |-  ( (
ph  /\  G  e.  A  /\  H  e.  A
)  ->  ( t  e.  T  |->  ( ( G `  t )  -  ( H `  t ) ) )  e.  A )
5039, 43, 49mpd3an23 1326 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  -  ( H `  t )
) )  e.  A
)
5136, 50eqeltrd 2555 . . 3  |-  ( ph  ->  F  e.  A )
52 stoweidlem40.14 . . . 4  |-  ( ph  ->  M  e.  NN )
5352nnnn0d 10864 . . 3  |-  ( ph  ->  M  e.  NN0 )
5420, 2, 21, 22, 23, 51, 53stoweidlem19 31633 . 2  |-  ( ph  ->  ( t  e.  T  |->  ( ( F `  t ) ^ M
) )  e.  A
)
5518, 54eqeltrd 2555 1  |-  ( ph  ->  Q  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379   F/wnf 1599    e. wcel 1767   F/_wnfc 2615    |-> cmpt 4511   -->wf 5590   ` cfv 5594  (class class class)co 6295   RRcr 9503   1c1 9505    + caddc 9507    x. cmul 9509    - cmin 9817   NNcn 10548   NN0cn0 10807   ^cexp 12146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-seq 12088  df-exp 12147
This theorem is referenced by:  stoweidlem45  31659
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