Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem32 Unicode version

Theorem stoweidlem32 27884
Description: If a set A of real functions from a common domain T is a subalgebra and it contains constants, then it is closed under the sum of a finite number of functions, indexed by G and finally scaled by a real Y. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem32.1  |-  F/ t
ph
stoweidlem32.2  |-  P  =  ( t  e.  T  |->  ( Y  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
stoweidlem32.3  |-  F  =  ( t  e.  T  |-> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  t
) )
stoweidlem32.4  |-  H  =  ( t  e.  T  |->  Y )
stoweidlem32.5  |-  ( ph  ->  M  e.  NN )
stoweidlem32.6  |-  ( ph  ->  Y  e.  RR )
stoweidlem32.7  |-  ( ph  ->  G : ( 1 ... M ) --> A )
stoweidlem32.8  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
stoweidlem32.9  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
stoweidlem32.10  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
stoweidlem32.11  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
Assertion
Ref Expression
stoweidlem32  |-  ( ph  ->  P  e.  A )
Distinct variable groups:    f, g,
i, t, G    A, f, g    f, F, g    T, f, g, i, t    ph, f, g, i    g, H    i, M, t    t, Y, x    x, T    x, A    x, Y    ph, x
Allowed substitution hints:    ph( t)    A( t, i)    P( x, t, f, g, i)    F( x, t, i)    G( x)    H( x, t, f, i)    M( x, f, g)    Y( f, g, i)

Proof of Theorem stoweidlem32
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 stoweidlem32.2 . . 3  |-  P  =  ( t  e.  T  |->  ( Y  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
2 stoweidlem32.1 . . . 4  |-  F/ t
ph
3 stoweidlem32.3 . . . . . . . . . . . 12  |-  F  =  ( t  e.  T  |-> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  t
) )
4 fveq2 5541 . . . . . . . . . . . . . 14  |-  ( t  =  s  ->  (
( G `  i
) `  t )  =  ( ( G `
 i ) `  s ) )
54sumeq2sdv 12193 . . . . . . . . . . . . 13  |-  ( t  =  s  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )
)
65cbvmptv 4127 . . . . . . . . . . . 12  |-  ( t  e.  T  |->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t ) )  =  ( s  e.  T  |-> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  s
) )
73, 6eqtri 2316 . . . . . . . . . . 11  |-  F  =  ( s  e.  T  |-> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  s
) )
87a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  F  =  ( s  e.  T  |->  sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  s
) ) )
9 fveq2 5541 . . . . . . . . . . . 12  |-  ( s  =  t  ->  (
( G `  i
) `  s )  =  ( ( G `
 i ) `  t ) )
109sumeq2sdv 12193 . . . . . . . . . . 11  |-  ( s  =  t  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
)
1110adantl 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  T )  /\  s  =  t )  ->  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  t
) )
12 simpr 447 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  t  e.  T )
13 fzfi 11050 . . . . . . . . . . . 12  |-  ( 1 ... M )  e. 
Fin
1413a1i 10 . . . . . . . . . . 11  |-  ( (
ph  /\  t  e.  T )  ->  (
1 ... M )  e. 
Fin )
15 simpl 443 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ph )
16 stoweidlem32.7 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  G : ( 1 ... M ) --> A )
1716adantr 451 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  G : ( 1 ... M ) --> A )
18 simpr 447 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  i  e.  ( 1 ... M
) )
1917, 18jca 518 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G : ( 1 ... M ) --> A  /\  i  e.  ( 1 ... M ) ) )
20 ffvelrn 5679 . . . . . . . . . . . . . . . . 17  |-  ( ( G : ( 1 ... M ) --> A  /\  i  e.  ( 1 ... M ) )  ->  ( G `  i )  e.  A
)
2119, 20syl 15 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G `  i )  e.  A )
2215, 21jca 518 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( ph  /\  ( G `  i )  e.  A
) )
23 eleq1 2356 . . . . . . . . . . . . . . . . . . 19  |-  ( f  =  ( G `  i )  ->  (
f  e.  A  <->  ( G `  i )  e.  A
) )
2423anbi2d 684 . . . . . . . . . . . . . . . . . 18  |-  ( f  =  ( G `  i )  ->  (
( ph  /\  f  e.  A )  <->  ( ph  /\  ( G `  i
)  e.  A ) ) )
25 feq1 5391 . . . . . . . . . . . . . . . . . 18  |-  ( f  =  ( G `  i )  ->  (
f : T --> RR  <->  ( G `  i ) : T --> RR ) )
2624, 25imbi12d 311 . . . . . . . . . . . . . . . . 17  |-  ( f  =  ( G `  i )  ->  (
( ( ph  /\  f  e.  A )  ->  f : T --> RR )  <-> 
( ( ph  /\  ( G `  i )  e.  A )  -> 
( G `  i
) : T --> RR ) ) )
27 stoweidlem32.11 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
2827a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( f  e.  A  ->  (
( ph  /\  f  e.  A )  ->  f : T --> RR ) )
2926, 28vtoclga 2862 . . . . . . . . . . . . . . . 16  |-  ( ( G `  i )  e.  A  ->  (
( ph  /\  ( G `  i )  e.  A )  ->  ( G `  i ) : T --> RR ) )
3021, 29syl 15 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( ph  /\  ( G `  i )  e.  A )  ->  ( G `  i ) : T --> RR ) )
3122, 30mpd 14 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G `  i ) : T --> RR )
3231adantlr 695 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  ( G `  i ) : T --> RR )
33 simplr 731 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  t  e.  T )
3432, 33jca 518 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) : T --> RR  /\  t  e.  T )
)
35 ffvelrn 5679 . . . . . . . . . . . 12  |-  ( ( ( G `  i
) : T --> RR  /\  t  e.  T )  ->  ( ( G `  i ) `  t
)  e.  RR )
3634, 35syl 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  t )  e.  RR )
3714, 36fsumrecl 12223 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t )  e.  RR )
388, 11, 12, 37fvmptd 5622 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  t
) )
3938, 37eqeltrd 2370 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
40 recn 8843 . . . . . . . 8  |-  ( ( F `  t )  e.  RR  ->  ( F `  t )  e.  CC )
4139, 40syl 15 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  CC )
42 stoweidlem32.4 . . . . . . . . . . . 12  |-  H  =  ( t  e.  T  |->  Y )
43 eqidd 2297 . . . . . . . . . . . . 13  |-  ( s  =  t  ->  Y  =  Y )
4443cbvmptv 4127 . . . . . . . . . . . 12  |-  ( s  e.  T  |->  Y )  =  ( t  e.  T  |->  Y )
4542, 44eqtr4i 2319 . . . . . . . . . . 11  |-  H  =  ( s  e.  T  |->  Y )
4645a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  H  =  ( s  e.  T  |->  Y ) )
47 eqidd 2297 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  T )  /\  s  =  t )  ->  Y  =  Y )
48 stoweidlem32.6 . . . . . . . . . . 11  |-  ( ph  ->  Y  e.  RR )
4948adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  Y  e.  RR )
5046, 47, 12, 49fvmptd 5622 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  =  Y )
5150, 49eqeltrd 2370 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  RR )
52 recn 8843 . . . . . . . 8  |-  ( ( H `  t )  e.  RR  ->  ( H `  t )  e.  CC )
5351, 52syl 15 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  CC )
5441, 53jca 518 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  (
( F `  t
)  e.  CC  /\  ( H `  t )  e.  CC ) )
55 mulcom 8839 . . . . . 6  |-  ( ( ( F `  t
)  e.  CC  /\  ( H `  t )  e.  CC )  -> 
( ( F `  t )  x.  ( H `  t )
)  =  ( ( H `  t )  x.  ( F `  t ) ) )
5654, 55syl 15 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( F `  t
)  x.  ( H `
 t ) )  =  ( ( H `
 t )  x.  ( F `  t
) ) )
5750, 38oveq12d 5892 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  x.  ( F `
 t ) )  =  ( Y  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
5856, 57eqtr2d 2329 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  ( Y  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t ) )  =  ( ( F `  t )  x.  ( H `  t )
) )
592, 58mpteq2da 4121 . . 3  |-  ( ph  ->  ( t  e.  T  |->  ( Y  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )  =  ( t  e.  T  |->  ( ( F `  t
)  x.  ( H `
 t ) ) ) )
601, 59syl5eq 2340 . 2  |-  ( ph  ->  P  =  ( t  e.  T  |->  ( ( F `  t )  x.  ( H `  t ) ) ) )
61 id 19 . . . 4  |-  ( ph  ->  ph )
62 stoweidlem32.5 . . . . 5  |-  ( ph  ->  M  e.  NN )
63 stoweidlem32.8 . . . . 5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
642, 3, 62, 16, 63, 27stoweidlem20 27872 . . . 4  |-  ( ph  ->  F  e.  A )
6561, 48jca 518 . . . . . 6  |-  ( ph  ->  ( ph  /\  Y  e.  RR ) )
66 stoweidlem32.10 . . . . . . 7  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
6766stoweidlem4 27856 . . . . . 6  |-  ( (
ph  /\  Y  e.  RR )  ->  ( t  e.  T  |->  Y )  e.  A )
6865, 67syl 15 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  Y )  e.  A
)
6942, 68syl5eqel 2380 . . . 4  |-  ( ph  ->  H  e.  A )
7061, 64, 693jca 1132 . . 3  |-  ( ph  ->  ( ph  /\  F  e.  A  /\  H  e.  A ) )
71 nfcv 2432 . . . . 5  |-  F/_ t
f
72 nfmpt1 4125 . . . . . 6  |-  F/_ t
( t  e.  T  |-> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  t
) )
733, 72nfcxfr 2429 . . . . 5  |-  F/_ t F
7471, 73nfeq 2439 . . . 4  |-  F/ t  f  =  F
75 nfcv 2432 . . . . 5  |-  F/_ t
g
76 nfmpt1 4125 . . . . . 6  |-  F/_ t
( t  e.  T  |->  Y )
7742, 76nfcxfr 2429 . . . . 5  |-  F/_ t H
7875, 77nfeq 2439 . . . 4  |-  F/ t  g  =  H
79 stoweidlem32.9 . . . 4  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
8074, 78, 79stoweidlem6 27858 . . 3  |-  ( (
ph  /\  F  e.  A  /\  H  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  x.  ( H `  t ) ) )  e.  A )
8170, 80syl 15 . 2  |-  ( ph  ->  ( t  e.  T  |->  ( ( F `  t )  x.  ( H `  t )
) )  e.  A
)
8260, 81eqeltrd 2370 1  |-  ( ph  ->  P  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   F/wnf 1534    = wceq 1632    e. wcel 1696    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874   Fincfn 6879   CCcc 8751   RRcr 8752   1c1 8754    + caddc 8756    x. cmul 8758   NNcn 9762   ...cfz 10798   sum_csu 12174
This theorem is referenced by:  stoweidlem44  27896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175
  Copyright terms: Public domain W3C validator