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Theorem stoweidlem32 37893
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 5865 . . . . . . . . . . . . 13  |-  ( t  =  s  ->  (
( G `  i
) `  t )  =  ( ( G `
 i ) `  s ) )
54sumeq2sdv 13770 . . . . . . . . . . . 12  |-  ( t  =  s  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )
)
65cbvmptv 4495 . . . . . . . . . . 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 2473 . . . . . . . . . 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 5865 . . . . . . . . . . 11  |-  ( s  =  t  ->  (
( G `  i
) `  s )  =  ( ( G `
 i ) `  t ) )
109sumeq2sdv 13770 . . . . . . . . . 10  |-  ( s  =  t  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  s )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
)
1110adantl 468 . . . . . . . . 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 463 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  t  e.  T )
13 fzfid 12186 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  (
1 ... M )  e. 
Fin )
14 simpl 459 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ph )
15 stoweidlem32.7 . . . . . . . . . . . . . 14  |-  ( ph  ->  G : ( 1 ... M ) --> A )
1615fnvinran 37335 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G `  i )  e.  A )
17 eleq1 2517 . . . . . . . . . . . . . . . . 17  |-  ( f  =  ( G `  i )  ->  (
f  e.  A  <->  ( G `  i )  e.  A
) )
1817anbi2d 710 . . . . . . . . . . . . . . . 16  |-  ( f  =  ( G `  i )  ->  (
( ph  /\  f  e.  A )  <->  ( ph  /\  ( G `  i
)  e.  A ) ) )
19 feq1 5710 . . . . . . . . . . . . . . . 16  |-  ( f  =  ( G `  i )  ->  (
f : T --> RR  <->  ( G `  i ) : T --> RR ) )
2018, 19imbi12d 322 . . . . . . . . . . . . . . 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 3107 . . . . . . . . . . . . . 14  |-  ( ( G `  i )  e.  A  ->  (
( ph  /\  ( G `  i )  e.  A )  ->  ( G `  i ) : T --> RR ) )
2316, 22syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  (
( ph  /\  ( G `  i )  e.  A )  ->  ( G `  i ) : T --> RR ) )
2414, 16, 23mp2and 685 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  ( 1 ... M
) )  ->  ( G `  i ) : T --> RR )
2524adantlr 721 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  ( G `  i ) : T --> RR )
26 simplr 762 . . . . . . . . . . 11  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  t  e.  T )
2725, 26ffvelrnd 6023 . . . . . . . . . 10  |-  ( ( ( ph  /\  t  e.  T )  /\  i  e.  ( 1 ... M
) )  ->  (
( G `  i
) `  t )  e.  RR )
2813, 27fsumrecl 13800 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t )  e.  RR )
298, 11, 12, 28fvmptd 5954 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  t
) )
3029, 28eqeltrd 2529 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
3130recnd 9669 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  CC )
32 stoweidlem32.4 . . . . . . . . . . 11  |-  H  =  ( t  e.  T  |->  Y )
33 eqidd 2452 . . . . . . . . . . . 12  |-  ( s  =  t  ->  Y  =  Y )
3433cbvmptv 4495 . . . . . . . . . . 11  |-  ( s  e.  T  |->  Y )  =  ( t  e.  T  |->  Y )
3532, 34eqtr4i 2476 . . . . . . . . . 10  |-  H  =  ( s  e.  T  |->  Y )
3635a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  H  =  ( s  e.  T  |->  Y ) )
37 eqidd 2452 . . . . . . . . 9  |-  ( ( ( ph  /\  t  e.  T )  /\  s  =  t )  ->  Y  =  Y )
38 stoweidlem32.6 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  RR )
3938adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  Y  e.  RR )
4036, 37, 12, 39fvmptd 5954 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  =  Y )
4140, 39eqeltrd 2529 . . . . . . 7  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  RR )
4241recnd 9669 . . . . . 6  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  CC )
4331, 42mulcomd 9664 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( F `  t
)  x.  ( H `
 t ) )  =  ( ( H `
 t )  x.  ( F `  t
) ) )
4440, 29oveq12d 6308 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  x.  ( F `
 t ) )  =  ( Y  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
4543, 44eqtr2d 2486 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  ( Y  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t ) )  =  ( ( F `  t )  x.  ( H `  t )
) )
462, 45mpteq2da 4488 . . 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 2497 . 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 37880 . . 3  |-  ( ph  ->  F  e.  A )
51 stoweidlem32.10 . . . . . 6  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
5251stoweidlem4 37864 . . . . 5  |-  ( (
ph  /\  Y  e.  RR )  ->  ( t  e.  T  |->  Y )  e.  A )
5338, 52mpdan 674 . . . 4  |-  ( ph  ->  ( t  e.  T  |->  Y )  e.  A
)
5432, 53syl5eqel 2533 . . 3  |-  ( ph  ->  H  e.  A )
55 nfmpt1 4492 . . . . . 6  |-  F/_ t
( t  e.  T  |-> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  t
) )
563, 55nfcxfr 2590 . . . . 5  |-  F/_ t F
5756nfeq2 2607 . . . 4  |-  F/ t  f  =  F
58 nfmpt1 4492 . . . . . 6  |-  F/_ t
( t  e.  T  |->  Y )
5932, 58nfcxfr 2590 . . . . 5  |-  F/_ t H
6059nfeq2 2607 . . . 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 37866 . . 3  |-  ( (
ph  /\  F  e.  A  /\  H  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  x.  ( H `  t ) ) )  e.  A )
6350, 54, 62mpd3an23 1366 . 2  |-  ( ph  ->  ( t  e.  T  |->  ( ( F `  t )  x.  ( H `  t )
) )  e.  A
)
6447, 63eqeltrd 2529 1  |-  ( ph  ->  P  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444   F/wnf 1667    e. wcel 1887    |-> cmpt 4461   -->wf 5578   ` cfv 5582  (class class class)co 6290   RRcr 9538   1c1 9540    + caddc 9542    x. cmul 9544   NNcn 10609   ...cfz 11784   sum_csu 13752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753
This theorem is referenced by:  stoweidlem44  37905
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