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Theorem stoweidlem32 27449
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 . . . . . . . . . . 11  |-  F  =  ( t  e.  T  |-> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  t
) )
4 fveq2 5668 . . . . . . . . . . . . 13  |-  ( t  =  s  ->  (
( G `  i
) `  t )  =  ( ( G `
 i ) `  s ) )
54sumeq2sdv 12425 . . . . . . . . . . . 12  |-  ( t  =  s  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )
)
65cbvmptv 4241 . . . . . . . . . . 11  |-  ( t  e.  T  |->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t ) )  =  ( s  e.  T  |-> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  s
) )
73, 6eqtri 2407 . . . . . . . . . 10  |-  F  =  ( s  e.  T  |-> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  s
) )
87a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  F  =  ( s  e.  T  |->  sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  s
) ) )
9 fveq2 5668 . . . . . . . . . . 11  |-  ( s  =  t  ->  (
( G `  i
) `  s )  =  ( ( G `
 i ) `  t ) )
109sumeq2sdv 12425 . . . . . . . . . 10  |-  ( s  =  t  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
)
1110adantl 453 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  T )  /\  s  =  t )  ->  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  t
) )
12 simpr 448 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  t  e.  T )
13 fzfid 11239 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  (
1 ... M )  e. 
Fin )
14 simpl 444 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ph )
15 stoweidlem32.7 . . . . . . . . . . . . . 14  |-  ( ph  ->  G : ( 1 ... M ) --> A )
1615fnvinran 27353 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G `  i )  e.  A )
17 eleq1 2447 . . . . . . . . . . . . . . . . 17  |-  ( f  =  ( G `  i )  ->  (
f  e.  A  <->  ( G `  i )  e.  A
) )
1817anbi2d 685 . . . . . . . . . . . . . . . 16  |-  ( f  =  ( G `  i )  ->  (
( ph  /\  f  e.  A )  <->  ( ph  /\  ( G `  i
)  e.  A ) ) )
19 feq1 5516 . . . . . . . . . . . . . . . 16  |-  ( f  =  ( G `  i )  ->  (
f : T --> RR  <->  ( G `  i ) : T --> RR ) )
2018, 19imbi12d 312 . . . . . . . . . . . . . . 15  |-  ( f  =  ( G `  i )  ->  (
( ( ph  /\  f  e.  A )  ->  f : T --> RR )  <-> 
( ( ph  /\  ( G `  i )  e.  A )  -> 
( G `  i
) : T --> RR ) ) )
21 stoweidlem32.11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
2220, 21vtoclg 2954 . . . . . . . . . . . . . 14  |-  ( ( G `  i )  e.  A  ->  (
( ph  /\  ( G `  i )  e.  A )  ->  ( G `  i ) : T --> RR ) )
2316, 22syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( ph  /\  ( G `  i )  e.  A )  ->  ( G `  i ) : T --> RR ) )
2414, 16, 23mp2and 661 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G `  i ) : T --> RR )
2524adantlr 696 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  ( G `  i ) : T --> RR )
26 simplr 732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  t  e.  T )
2725, 26ffvelrnd 5810 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  t )  e.  RR )
2813, 27fsumrecl 12455 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t )  e.  RR )
298, 11, 12, 28fvmptd 5749 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  t
) )
3029, 28eqeltrd 2461 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
3130recnd 9047 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  CC )
32 stoweidlem32.4 . . . . . . . . . . 11  |-  H  =  ( t  e.  T  |->  Y )
33 eqidd 2388 . . . . . . . . . . . 12  |-  ( s  =  t  ->  Y  =  Y )
3433cbvmptv 4241 . . . . . . . . . . 11  |-  ( s  e.  T  |->  Y )  =  ( t  e.  T  |->  Y )
3532, 34eqtr4i 2410 . . . . . . . . . 10  |-  H  =  ( s  e.  T  |->  Y )
3635a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  H  =  ( s  e.  T  |->  Y ) )
37 eqidd 2388 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  T )  /\  s  =  t )  ->  Y  =  Y )
38 stoweidlem32.6 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  RR )
3938adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  Y  e.  RR )
4036, 37, 12, 39fvmptd 5749 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  =  Y )
4140, 39eqeltrd 2461 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  RR )
4241recnd 9047 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  CC )
4331, 42mulcomd 9042 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( F `  t
)  x.  ( H `
 t ) )  =  ( ( H `
 t )  x.  ( F `  t
) ) )
4440, 29oveq12d 6038 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  x.  ( F `
 t ) )  =  ( Y  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
4543, 44eqtr2d 2420 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  ( Y  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t ) )  =  ( ( F `  t )  x.  ( H `  t )
) )
462, 45mpteq2da 4235 . . 3  |-  ( ph  ->  ( t  e.  T  |->  ( Y  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )  =  ( t  e.  T  |->  ( ( F `  t
)  x.  ( H `
 t ) ) ) )
471, 46syl5eq 2431 . 2  |-  ( ph  ->  P  =  ( t  e.  T  |->  ( ( F `  t )  x.  ( H `  t ) ) ) )
48 stoweidlem32.5 . . . 4  |-  ( ph  ->  M  e.  NN )
49 stoweidlem32.8 . . . 4  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
502, 3, 48, 15, 49, 21stoweidlem20 27437 . . 3  |-  ( ph  ->  F  e.  A )
51 stoweidlem32.10 . . . . . 6  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
5251stoweidlem4 27421 . . . . 5  |-  ( (
ph  /\  Y  e.  RR )  ->  ( t  e.  T  |->  Y )  e.  A )
5338, 52mpdan 650 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  Y )  e.  A
)
5432, 53syl5eqel 2471 . . 3  |-  ( ph  ->  H  e.  A )
55 nfmpt1 4239 . . . . . 6  |-  F/_ t
( t  e.  T  |-> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  t
) )
563, 55nfcxfr 2520 . . . . 5  |-  F/_ t F
5756nfeq2 2534 . . . 4  |-  F/ t  f  =  F
58 nfmpt1 4239 . . . . . 6  |-  F/_ t
( t  e.  T  |->  Y )
5932, 58nfcxfr 2520 . . . . 5  |-  F/_ t H
6059nfeq2 2534 . . . 4  |-  F/ t  g  =  H
61 stoweidlem32.9 . . . 4  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
6257, 60, 61stoweidlem6 27423 . . 3  |-  ( (
ph  /\  F  e.  A  /\  H  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  x.  ( H `  t ) ) )  e.  A )
6350, 54, 62mpd3an23 1281 . 2  |-  ( ph  ->  ( t  e.  T  |->  ( ( F `  t )  x.  ( H `  t )
) )  e.  A
)
6447, 63eqeltrd 2461 1  |-  ( ph  ->  P  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   F/wnf 1550    = wceq 1649    e. wcel 1717    e. cmpt 4207   -->wf 5390   ` cfv 5394  (class class class)co 6020   RRcr 8922   1c1 8924    + caddc 8926    x. cmul 8928   NNcn 9932   ...cfz 10975   sum_csu 12406
This theorem is referenced by:  stoweidlem44  27461
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-fzo 11066  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-sum 12407
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