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Theorem stoweidlem32 29832
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 5696 . . . . . . . . . . . . 13  |-  ( t  =  s  ->  (
( G `  i
) `  t )  =  ( ( G `
 i ) `  s ) )
54sumeq2sdv 13186 . . . . . . . . . . . 12  |-  ( t  =  s  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )
)
65cbvmptv 4388 . . . . . . . . . . 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 2463 . . . . . . . . . 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 5696 . . . . . . . . . . 11  |-  ( s  =  t  ->  (
( G `  i
) `  s )  =  ( ( G `
 i ) `  t ) )
109sumeq2sdv 13186 . . . . . . . . . 10  |-  ( s  =  t  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
)
1110adantl 466 . . . . . . . . 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 461 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  t  e.  T )
13 fzfid 11800 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  (
1 ... M )  e. 
Fin )
14 simpl 457 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ph )
15 stoweidlem32.7 . . . . . . . . . . . . . 14  |-  ( ph  ->  G : ( 1 ... M ) --> A )
1615fnvinran 29741 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G `  i )  e.  A )
17 eleq1 2503 . . . . . . . . . . . . . . . . 17  |-  ( f  =  ( G `  i )  ->  (
f  e.  A  <->  ( G `  i )  e.  A
) )
1817anbi2d 703 . . . . . . . . . . . . . . . 16  |-  ( f  =  ( G `  i )  ->  (
( ph  /\  f  e.  A )  <->  ( ph  /\  ( G `  i
)  e.  A ) ) )
19 feq1 5547 . . . . . . . . . . . . . . . 16  |-  ( f  =  ( G `  i )  ->  (
f : T --> RR  <->  ( G `  i ) : T --> RR ) )
2018, 19imbi12d 320 . . . . . . . . . . . . . . 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 3035 . . . . . . . . . . . . . 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 679 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G `  i ) : T --> RR )
2524adantlr 714 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  ( G `  i ) : T --> RR )
26 simplr 754 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  t  e.  T )
2725, 26ffvelrnd 5849 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  t )  e.  RR )
2813, 27fsumrecl 13216 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t )  e.  RR )
298, 11, 12, 28fvmptd 5784 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  t
) )
3029, 28eqeltrd 2517 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
3130recnd 9417 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  CC )
32 stoweidlem32.4 . . . . . . . . . . 11  |-  H  =  ( t  e.  T  |->  Y )
33 eqidd 2444 . . . . . . . . . . . 12  |-  ( s  =  t  ->  Y  =  Y )
3433cbvmptv 4388 . . . . . . . . . . 11  |-  ( s  e.  T  |->  Y )  =  ( t  e.  T  |->  Y )
3532, 34eqtr4i 2466 . . . . . . . . . 10  |-  H  =  ( s  e.  T  |->  Y )
3635a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  H  =  ( s  e.  T  |->  Y ) )
37 eqidd 2444 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  T )  /\  s  =  t )  ->  Y  =  Y )
38 stoweidlem32.6 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  RR )
3938adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  Y  e.  RR )
4036, 37, 12, 39fvmptd 5784 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  =  Y )
4140, 39eqeltrd 2517 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  RR )
4241recnd 9417 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  CC )
4331, 42mulcomd 9412 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( F `  t
)  x.  ( H `
 t ) )  =  ( ( H `
 t )  x.  ( F `  t
) ) )
4440, 29oveq12d 6114 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  x.  ( F `
 t ) )  =  ( Y  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
4543, 44eqtr2d 2476 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  ( Y  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t ) )  =  ( ( F `  t )  x.  ( H `  t )
) )
462, 45mpteq2da 4382 . . 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 2487 . 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 29820 . . 3  |-  ( ph  ->  F  e.  A )
51 stoweidlem32.10 . . . . . 6  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
5251stoweidlem4 29804 . . . . 5  |-  ( (
ph  /\  Y  e.  RR )  ->  ( t  e.  T  |->  Y )  e.  A )
5338, 52mpdan 668 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  Y )  e.  A
)
5432, 53syl5eqel 2527 . . 3  |-  ( ph  ->  H  e.  A )
55 nfmpt1 4386 . . . . . 6  |-  F/_ t
( t  e.  T  |-> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  t
) )
563, 55nfcxfr 2581 . . . . 5  |-  F/_ t F
5756nfeq2 2595 . . . 4  |-  F/ t  f  =  F
58 nfmpt1 4386 . . . . . 6  |-  F/_ t
( t  e.  T  |->  Y )
5932, 58nfcxfr 2581 . . . . 5  |-  F/_ t H
6059nfeq2 2595 . . . 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 29806 . . 3  |-  ( (
ph  /\  F  e.  A  /\  H  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  x.  ( H `  t ) ) )  e.  A )
6350, 54, 62mpd3an23 1316 . 2  |-  ( ph  ->  ( t  e.  T  |->  ( ( F `  t )  x.  ( H `  t )
) )  e.  A
)
6447, 63eqeltrd 2517 1  |-  ( ph  ->  P  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369   F/wnf 1589    e. wcel 1756    e. cmpt 4355   -->wf 5419   ` cfv 5423  (class class class)co 6096   RRcr 9286   1c1 9288    + caddc 9290    x. cmul 9292   NNcn 10327   ...cfz 11442   sum_csu 13168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-fzo 11554  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169
This theorem is referenced by:  stoweidlem44  29844
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