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Theorem stoweidlem40 37895
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 463 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  t  e.  T )
4 1red 9655 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  1  e.  RR )
5 stoweidlem40.8 . . . . . . . . . 10  |-  ( ph  ->  P : T --> RR )
65fnvinran 37329 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( P `  t )  e.  RR )
7 stoweidlem40.13 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN )
87nnnn0d 10922 . . . . . . . . . 10  |-  ( ph  ->  N  e.  NN0 )
98adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  N  e.  NN0 )
106, 9reexpcld 12430 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
) ^ N )  e.  RR )
114, 10resubcld 10044 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  e.  RR )
12 stoweidlem40.4 . . . . . . . 8  |-  F  =  ( t  e.  T  |->  ( 1  -  (
( P `  t
) ^ N ) ) )
1312fvmpt2 5955 . . . . . . 7  |-  ( ( t  e.  T  /\  ( 1  -  (
( P `  t
) ^ N ) )  e.  RR )  ->  ( F `  t )  =  ( 1  -  ( ( P `  t ) ^ N ) ) )
143, 11, 13syl2anc 666 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  =  ( 1  -  ( ( P `  t ) ^ N
) ) )
1514eqcomd 2456 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  =  ( F `  t ) )
1615oveq1d 6303 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  (
( 1  -  (
( P `  t
) ^ N ) ) ^ M )  =  ( ( F `
 t ) ^ M ) )
172, 16mpteq2da 4487 . . 3  |-  ( ph  ->  ( t  e.  T  |->  ( ( 1  -  ( ( P `  t ) ^ N
) ) ^ M
) )  =  ( t  e.  T  |->  ( ( F `  t
) ^ M ) ) )
181, 17syl5eq 2496 . 2  |-  ( ph  ->  Q  =  ( t  e.  T  |->  ( ( F `  t ) ^ M ) ) )
19 nfmpt1 4491 . . . 4  |-  F/_ t
( t  e.  T  |->  ( 1  -  (
( P `  t
) ^ N ) ) )
2012, 19nfcxfr 2589 . . 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 9639 . . . . . . . . . 10  |-  1  e.  RR
25 stoweidlem40.5 . . . . . . . . . . 11  |-  G  =  ( t  e.  T  |->  1 )
2625fvmpt2 5955 . . . . . . . . . 10  |-  ( ( t  e.  T  /\  1  e.  RR )  ->  ( G `  t
)  =  1 )
2724, 26mpan2 676 . . . . . . . . 9  |-  ( t  e.  T  ->  ( G `  t )  =  1 )
2827eqcomd 2456 . . . . . . . 8  |-  ( t  e.  T  ->  1  =  ( G `  t ) )
2928adantl 468 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  1  =  ( G `  t ) )
30 stoweidlem40.6 . . . . . . . . . 10  |-  H  =  ( t  e.  T  |->  ( ( P `  t ) ^ N
) )
3130fvmpt2 5955 . . . . . . . . 9  |-  ( ( t  e.  T  /\  ( ( P `  t ) ^ N
)  e.  RR )  ->  ( H `  t )  =  ( ( P `  t
) ^ N ) )
323, 10, 31syl2anc 666 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  =  ( ( P `
 t ) ^ N ) )
3332eqcomd 2456 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  (
( P `  t
) ^ N )  =  ( H `  t ) )
3429, 33oveq12d 6306 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
1  -  ( ( P `  t ) ^ N ) )  =  ( ( G `
 t )  -  ( H `  t ) ) )
352, 34mpteq2da 4487 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  ( 1  -  (
( P `  t
) ^ N ) ) )  =  ( t  e.  T  |->  ( ( G `  t
)  -  ( H `
 t ) ) ) )
3612, 35syl5eq 2496 . . . 4  |-  ( ph  ->  F  =  ( t  e.  T  |->  ( ( G `  t )  -  ( H `  t ) ) ) )
3723stoweidlem4 37858 . . . . . . 7  |-  ( (
ph  /\  1  e.  RR )  ->  ( t  e.  T  |->  1 )  e.  A )
3824, 37mpan2 676 . . . . . 6  |-  ( ph  ->  ( t  e.  T  |->  1 )  e.  A
)
3925, 38syl5eqel 2532 . . . . 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 37873 . . . . . 6  |-  ( ph  ->  ( t  e.  T  |->  ( ( P `  t ) ^ N
) )  e.  A
)
4330, 42syl5eqel 2532 . . . . 5  |-  ( ph  ->  H  e.  A )
44 nfmpt1 4491 . . . . . . 7  |-  F/_ t
( t  e.  T  |->  1 )
4525, 44nfcxfr 2589 . . . . . 6  |-  F/_ t G
46 nfmpt1 4491 . . . . . . 7  |-  F/_ t
( t  e.  T  |->  ( ( P `  t ) ^ N
) )
4730, 46nfcxfr 2589 . . . . . 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 37888 . . . . 5  |-  ( (
ph  /\  G  e.  A  /\  H  e.  A
)  ->  ( t  e.  T  |->  ( ( G `  t )  -  ( H `  t ) ) )  e.  A )
5039, 43, 49mpd3an23 1365 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  ( ( G `  t )  -  ( H `  t )
) )  e.  A
)
5136, 50eqeltrd 2528 . . 3  |-  ( ph  ->  F  e.  A )
52 stoweidlem40.14 . . . 4  |-  ( ph  ->  M  e.  NN )
5352nnnn0d 10922 . . 3  |-  ( ph  ->  M  e.  NN0 )
5420, 2, 21, 22, 23, 51, 53stoweidlem19 37873 . 2  |-  ( ph  ->  ( t  e.  T  |->  ( ( F `  t ) ^ M
) )  e.  A
)
5518, 54eqeltrd 2528 1  |-  ( ph  ->  Q  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 984    = wceq 1443   F/wnf 1666    e. wcel 1886   F/_wnfc 2578    |-> cmpt 4460   -->wf 5577   ` cfv 5581  (class class class)co 6288   RRcr 9535   1c1 9537    + caddc 9539    x. cmul 9541    - cmin 9857   NNcn 10606   NN0cn0 10866   ^cexp 12269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-n0 10867  df-z 10935  df-uz 11157  df-seq 12211  df-exp 12270
This theorem is referenced by:  stoweidlem45  37900
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