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Theorem stoweidlem40 27656
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 448 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  t  e.  T )
4 1re 9046 . . . . . . . . 9  |-  1  e.  RR
54a1i 11 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  1  e.  RR )
6 stoweidlem40.8 . . . . . . . . . 10  |-  ( ph  ->  P : T --> RR )
76fnvinran 27552 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( P `  t )  e.  RR )
8 stoweidlem40.13 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN )
98nnnn0d 10230 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
109adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  N  e.  NN0 )
117, 10reexpcld 11495 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
) ^ N )  e.  RR )
125, 11resubcld 9421 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  e.  RR )
13 stoweidlem40.4 . . . . . . . 8  |-  F  =  ( t  e.  T  |->  ( 1  -  (
( P `  t
) ^ N ) ) )
1413fvmpt2 5771 . . . . . . 7  |-  ( ( t  e.  T  /\  ( 1  -  (
( P `  t
) ^ N ) )  e.  RR )  ->  ( F `  t )  =  ( 1  -  ( ( P `  t ) ^ N ) ) )
153, 12, 14syl2anc 643 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  =  ( 1  -  ( ( P `  t ) ^ N
) ) )
1615eqcomd 2409 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  =  ( F `  t ) )
1716oveq1d 6055 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  (
( 1  -  (
( P `  t
) ^ N ) ) ^ M )  =  ( ( F `
 t ) ^ M ) )
182, 17mpteq2da 4254 . . 3  |-  ( ph  ->  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N
) ) ^ M
) )  =  ( t  e.  T  |->  ( ( F `  t
) ^ M ) ) )
191, 18syl5eq 2448 . 2  |-  ( ph  ->  Q  =  ( t  e.  T  |->  ( ( F `  t ) ^ M ) ) )
20 nfmpt1 4258 . . . 4  |-  F/_ t
( t  e.  T  |->  ( 1  -  (
( P `  t
) ^ N ) ) )
2113, 20nfcxfr 2537 . . 3  |-  F/_ t F
22 stoweidlem40.9 . . 3  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
23 stoweidlem40.11 . . 3  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
24 stoweidlem40.12 . . 3  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
25 stoweidlem40.5 . . . . . . . . . . 11  |-  G  =  ( t  e.  T  |->  1 )
2625fvmpt2 5771 . . . . . . . . . 10  |-  ( ( t  e.  T  /\  1  e.  RR )  ->  ( G `  t
)  =  1 )
274, 26mpan2 653 . . . . . . . . 9  |-  ( t  e.  T  ->  ( G `  t )  =  1 )
2827eqcomd 2409 . . . . . . . 8  |-  ( t  e.  T  ->  1  =  ( G `  t ) )
2928adantl 453 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  1  =  ( G `  t ) )
30 stoweidlem40.6 . . . . . . . . . 10  |-  H  =  ( t  e.  T  |->  ( ( P `  t ) ^ N
) )
3130fvmpt2 5771 . . . . . . . . 9  |-  ( ( t  e.  T  /\  ( ( P `  t ) ^ N
)  e.  RR )  ->  ( H `  t )  =  ( ( P `  t
) ^ N ) )
323, 11, 31syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  =  ( ( P `
 t ) ^ N ) )
3332eqcomd 2409 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
) ^ N )  =  ( H `  t ) )
3429, 33oveq12d 6058 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  =  ( ( G `
 t )  -  ( H `  t ) ) )
352, 34mpteq2da 4254 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  ( 1  -  (
( P `  t
) ^ N ) ) )  =  ( t  e.  T  |->  ( ( G `  t
)  -  ( H `
 t ) ) ) )
3613, 35syl5eq 2448 . . . 4  |-  ( ph  ->  F  =  ( t  e.  T  |->  ( ( G `  t )  -  ( H `  t ) ) ) )
3724stoweidlem4 27620 . . . . . . 7  |-  ( (
ph  /\  1  e.  RR )  ->  ( t  e.  T  |->  1 )  e.  A )
384, 37mpan2 653 . . . . . 6  |-  ( ph  ->  ( t  e.  T  |->  1 )  e.  A
)
3925, 38syl5eqel 2488 . . . . 5  |-  ( ph  ->  G  e.  A )
40 stoweidlem40.1 . . . . . . 7  |-  F/_ t P
41 stoweidlem40.7 . . . . . . 7  |-  ( ph  ->  P  e.  A )
4240, 2, 22, 23, 24, 41, 9stoweidlem19 27635 . . . . . 6  |-  ( ph  ->  ( t  e.  T  |->  ( ( P `  t ) ^ N
) )  e.  A
)
4330, 42syl5eqel 2488 . . . . 5  |-  ( ph  ->  H  e.  A )
44 nfmpt1 4258 . . . . . . 7  |-  F/_ t
( t  e.  T  |->  1 )
4525, 44nfcxfr 2537 . . . . . 6  |-  F/_ t G
46 nfmpt1 4258 . . . . . . 7  |-  F/_ t
( t  e.  T  |->  ( ( P `  t ) ^ N
) )
4730, 46nfcxfr 2537 . . . . . 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, 22, 48, 23, 24stoweidlem33 27649 . . . . 5  |-  ( (
ph  /\  G  e.  A  /\  H  e.  A
)  ->  ( t  e.  T  |->  ( ( G `  t )  -  ( H `  t ) ) )  e.  A )
5039, 43, 49mpd3an23 1281 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  -  ( H `  t )
) )  e.  A
)
5136, 50eqeltrd 2478 . . 3  |-  ( ph  ->  F  e.  A )
52 stoweidlem40.14 . . . 4  |-  ( ph  ->  M  e.  NN )
5352nnnn0d 10230 . . 3  |-  ( ph  ->  M  e.  NN0 )
5421, 2, 22, 23, 24, 51, 53stoweidlem19 27635 . 2  |-  ( ph  ->  ( t  e.  T  |->  ( ( F `  t ) ^ M
) )  e.  A
)
5519, 54eqeltrd 2478 1  |-  ( ph  ->  Q  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   F/wnf 1550    = wceq 1649    e. wcel 1721   F/_wnfc 2527    e. cmpt 4226   -->wf 5409   ` cfv 5413  (class class class)co 6040   RRcr 8945   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247   NNcn 9956   NN0cn0 10177   ^cexp 11337
This theorem is referenced by:  stoweidlem45  27661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-seq 11279  df-exp 11338
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