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

Theorem stoweidlem32 27792
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 5527 . . . . . . . . . . . . . 14  |-  ( t  =  s  ->  (
( G `  i
) `  t )  =  ( ( G `
 i ) `  s ) )
54sumeq2sdv 12179 . . . . . . . . . . . . 13  |-  ( t  =  s  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )
)
65cbvmptv 4113 . . . . . . . . . . . 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 2305 . . . . . . . . . . 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 5527 . . . . . . . . . . . 12  |-  ( s  =  t  ->  (
( G `  i
) `  s )  =  ( ( G `
 i ) `  t ) )
109sumeq2sdv 12179 . . . . . . . . . . 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 11036 . . . . . . . . . . . 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 5665 . . . . . . . . . . . . . . . . 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 2345 . . . . . . . . . . . . . . . . . . 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 5377 . . . . . . . . . . . . . . . . . 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 2851 . . . . . . . . . . . . . . . 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 5665 . . . . . . . . . . . 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 12209 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t )  e.  RR )
388, 11, 12, 37fvmptd 5608 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  t
) )
3938, 37eqeltrd 2359 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
40 recn 8829 . . . . . . . 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 2286 . . . . . . . . . . . . 13  |-  ( s  =  t  ->  Y  =  Y )
4443cbvmptv 4113 . . . . . . . . . . . 12  |-  ( s  e.  T  |->  Y )  =  ( t  e.  T  |->  Y )
4542, 44eqtr4i 2308 . . . . . . . . . . 11  |-  H  =  ( s  e.  T  |->  Y )
4645a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  H  =  ( s  e.  T  |->  Y ) )
47 eqidd 2286 . . . . . . . . . 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 5608 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  =  Y )
5150, 49eqeltrd 2359 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  RR )
52 recn 8829 . . . . . . . 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 8825 . . . . . 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 5878 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  x.  ( F `
 t ) )  =  ( Y  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
5856, 57eqtr2d 2318 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  ( Y  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t ) )  =  ( ( F `  t )  x.  ( H `  t )
) )
592, 58mpteq2da 4107 . . 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 2329 . 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 27780 . . . 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 27764 . . . . . 6  |-  ( (
ph  /\  Y  e.  RR )  ->  ( t  e.  T  |->  Y )  e.  A )
6865, 67syl 15 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  Y )  e.  A
)
6942, 68syl5eqel 2369 . . . 4  |-  ( ph  ->  H  e.  A )
7061, 64, 693jca 1132 . . 3  |-  ( ph  ->  ( ph  /\  F  e.  A  /\  H  e.  A ) )
71 nfcv 2421 . . . . 5  |-  F/_ t
f
72 nfmpt1 4111 . . . . . 6  |-  F/_ t
( t  e.  T  |-> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  t
) )
733, 72nfcxfr 2418 . . . . 5  |-  F/_ t F
7471, 73nfeq 2428 . . . 4  |-  F/ t  f  =  F
75 nfcv 2421 . . . . 5  |-  F/_ t
g
76 nfmpt1 4111 . . . . . 6  |-  F/_ t
( t  e.  T  |->  Y )
7742, 76nfcxfr 2418 . . . . 5  |-  F/_ t H
7875, 77nfeq 2428 . . . 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 27766 . . 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 2359 1  |-  ( ph  ->  P  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   F/wnf 1533    = wceq 1625    e. wcel 1686    e. cmpt 4079   -->wf 5253   ` cfv 5257  (class class class)co 5860   Fincfn 6865   CCcc 8737   RRcr 8738   1c1 8740    + caddc 8742    x. cmul 8744   NNcn 9748   ...cfz 10784   sum_csu 12160
This theorem is referenced by:  stoweidlem44  27804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-oi 7227  df-card 7574  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-n0 9968  df-z 10027  df-uz 10233  df-rp 10357  df-fz 10785  df-fzo 10873  df-seq 11049  df-exp 11107  df-hash 11340  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-clim 11964  df-sum 12161
  Copyright terms: Public domain W3C validator