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Theorem stoweidlem32 27781
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 5525 . . . . . . . . . . . . . 14  |-  ( t  =  s  ->  (
( G `  i
) `  t )  =  ( ( G `
 i ) `  s ) )
54sumeq2sdv 12177 . . . . . . . . . . . . 13  |-  ( t  =  s  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  s )
)
65cbvmptv 4111 . . . . . . . . . . . 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 2303 . . . . . . . . . . 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 5525 . . . . . . . . . . . 12  |-  ( s  =  t  ->  (
( G `  i
) `  s )  =  ( ( G `
 i ) `  t ) )
109sumeq2sdv 12177 . . . . . . . . . . 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 11034 . . . . . . . . . . . 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 5663 . . . . . . . . . . . . . . . . 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 2343 . . . . . . . . . . . . . . . . . . 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 5375 . . . . . . . . . . . . . . . . . 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 2849 . . . . . . . . . . . . . . . 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 5663 . . . . . . . . . . . 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 12207 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t )  e.  RR )
388, 11, 12, 37fvmptd 5606 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  =  sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  t
) )
3938, 37eqeltrd 2357 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( F `  t )  e.  RR )
40 recn 8827 . . . . . . . 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 2284 . . . . . . . . . . . . 13  |-  ( s  =  t  ->  Y  =  Y )
4443cbvmptv 4111 . . . . . . . . . . . 12  |-  ( s  e.  T  |->  Y )  =  ( t  e.  T  |->  Y )
4542, 44eqtr4i 2306 . . . . . . . . . . 11  |-  H  =  ( s  e.  T  |->  Y )
4645a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  t  e.  T )  ->  H  =  ( s  e.  T  |->  Y ) )
47 eqidd 2284 . . . . . . . . . 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 5606 . . . . . . . . 9  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  =  Y )
5150, 49eqeltrd 2357 . . . . . . . 8  |-  ( (
ph  /\  t  e.  T )  ->  ( H `  t )  e.  RR )
52 recn 8827 . . . . . . . 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 8823 . . . . . 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 5876 . . . . 5  |-  ( (
ph  /\  t  e.  T )  ->  (
( H `  t
)  x.  ( F `
 t ) )  =  ( Y  x.  sum_ i  e.  ( 1 ... M ) ( ( G `  i
) `  t )
) )
5856, 57eqtr2d 2316 . . . 4  |-  ( (
ph  /\  t  e.  T )  ->  ( Y  x.  sum_ i  e.  ( 1 ... M
) ( ( G `
 i ) `  t ) )  =  ( ( F `  t )  x.  ( H `  t )
) )
592, 58mpteq2da 4105 . . 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 2327 . 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 27769 . . . 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 27753 . . . . . 6  |-  ( (
ph  /\  Y  e.  RR )  ->  ( t  e.  T  |->  Y )  e.  A )
6865, 67syl 15 . . . . 5  |-  ( ph  ->  ( t  e.  T  |->  Y )  e.  A
)
6942, 68syl5eqel 2367 . . . 4  |-  ( ph  ->  H  e.  A )
7061, 64, 693jca 1132 . . 3  |-  ( ph  ->  ( ph  /\  F  e.  A  /\  H  e.  A ) )
71 nfcv 2419 . . . . 5  |-  F/_ t
f
72 nfmpt1 4109 . . . . . 6  |-  F/_ t
( t  e.  T  |-> 
sum_ i  e.  ( 1 ... M ) ( ( G `  i ) `  t
) )
733, 72nfcxfr 2416 . . . . 5  |-  F/_ t F
7471, 73nfeq 2426 . . . 4  |-  F/ t  f  =  F
75 nfcv 2419 . . . . 5  |-  F/_ t
g
76 nfmpt1 4109 . . . . . 6  |-  F/_ t
( t  e.  T  |->  Y )
7742, 76nfcxfr 2416 . . . . 5  |-  F/_ t H
7875, 77nfeq 2426 . . . 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 27755 . . 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 2357 1  |-  ( ph  ->  P  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   F/wnf 1531    = wceq 1623    e. wcel 1684    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   RRcr 8736   1c1 8738    + caddc 8740    x. cmul 8742   NNcn 9746   ...cfz 10782   sum_csu 12158
This theorem is referenced by:  stoweidlem44  27793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159
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