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Theorem stoweidlem40 30003
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 9515 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  1  e.  RR )
5 stoweidlem40.8 . . . . . . . . . 10  |-  ( ph  ->  P : T --> RR )
65fnvinran 29904 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( P `  t )  e.  RR )
7 stoweidlem40.13 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN )
87nnnn0d 10750 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
98adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  N  e.  NN0 )
106, 9reexpcld 12145 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
) ^ N )  e.  RR )
114, 10resubcld 9890 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  e.  RR )
12 stoweidlem40.4 . . . . . . . 8  |-  F  =  ( t  e.  T  |->  ( 1  -  (
( P `  t
) ^ N ) ) )
1312fvmpt2 5893 . . . . . . 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 2462 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  =  ( F `  t ) )
1615oveq1d 6218 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  (
( 1  -  (
( P `  t
) ^ N ) ) ^ M )  =  ( ( F `
 t ) ^ M ) )
172, 16mpteq2da 4488 . . 3  |-  ( ph  ->  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N
) ) ^ M
) )  =  ( t  e.  T  |->  ( ( F `  t
) ^ M ) ) )
181, 17syl5eq 2507 . 2  |-  ( ph  ->  Q  =  ( t  e.  T  |->  ( ( F `  t ) ^ M ) ) )
19 nfmpt1 4492 . . . 4  |-  F/_ t
( t  e.  T  |->  ( 1  -  (
( P `  t
) ^ N ) ) )
2012, 19nfcxfr 2614 . . 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 9499 . . . . . . . . . 10  |-  1  e.  RR
25 stoweidlem40.5 . . . . . . . . . . 11  |-  G  =  ( t  e.  T  |->  1 )
2625fvmpt2 5893 . . . . . . . . . 10  |-  ( ( t  e.  T  /\  1  e.  RR )  ->  ( G `  t
)  =  1 )
2724, 26mpan2 671 . . . . . . . . 9  |-  ( t  e.  T  ->  ( G `  t )  =  1 )
2827eqcomd 2462 . . . . . . . 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 5893 . . . . . . . . 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 2462 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
) ^ N )  =  ( H `  t ) )
3429, 33oveq12d 6221 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  =  ( ( G `
 t )  -  ( H `  t ) ) )
352, 34mpteq2da 4488 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  ( 1  -  (
( P `  t
) ^ N ) ) )  =  ( t  e.  T  |->  ( ( G `  t
)  -  ( H `
 t ) ) ) )
3612, 35syl5eq 2507 . . . 4  |-  ( ph  ->  F  =  ( t  e.  T  |->  ( ( G `  t )  -  ( H `  t ) ) ) )
3723stoweidlem4 29967 . . . . . . 7  |-  ( (
ph  /\  1  e.  RR )  ->  ( t  e.  T  |->  1 )  e.  A )
3824, 37mpan2 671 . . . . . 6  |-  ( ph  ->  ( t  e.  T  |->  1 )  e.  A
)
3925, 38syl5eqel 2546 . . . . 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 29982 . . . . . 6  |-  ( ph  ->  ( t  e.  T  |->  ( ( P `  t ) ^ N
) )  e.  A
)
4330, 42syl5eqel 2546 . . . . 5  |-  ( ph  ->  H  e.  A )
44 nfmpt1 4492 . . . . . . 7  |-  F/_ t
( t  e.  T  |->  1 )
4525, 44nfcxfr 2614 . . . . . 6  |-  F/_ t G
46 nfmpt1 4492 . . . . . . 7  |-  F/_ t
( t  e.  T  |->  ( ( P `  t ) ^ N
) )
4730, 46nfcxfr 2614 . . . . . 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 29996 . . . . 5  |-  ( (
ph  /\  G  e.  A  /\  H  e.  A
)  ->  ( t  e.  T  |->  ( ( G `  t )  -  ( H `  t ) ) )  e.  A )
5039, 43, 49mpd3an23 1317 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  -  ( H `  t )
) )  e.  A
)
5136, 50eqeltrd 2542 . . 3  |-  ( ph  ->  F  e.  A )
52 stoweidlem40.14 . . . 4  |-  ( ph  ->  M  e.  NN )
5352nnnn0d 10750 . . 3  |-  ( ph  ->  M  e.  NN0 )
5420, 2, 21, 22, 23, 51, 53stoweidlem19 29982 . 2  |-  ( ph  ->  ( t  e.  T  |->  ( ( F `  t ) ^ M
) )  e.  A
)
5518, 54eqeltrd 2542 1  |-  ( ph  ->  Q  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370   F/wnf 1590    e. wcel 1758   F/_wnfc 2602    |-> cmpt 4461   -->wf 5525   ` cfv 5529  (class class class)co 6203   RRcr 9395   1c1 9397    + caddc 9399    x. cmul 9401    - cmin 9709   NNcn 10436   NN0cn0 10693   ^cexp 11985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-n0 10694  df-z 10761  df-uz 10976  df-seq 11927  df-exp 11986
This theorem is referenced by:  stoweidlem45  30008
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