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Theorem stoweidlem33 27782
Description: If a set of real functions from a common domain is closed under addition, multiplication and it contains constants, then it is closed under subtraction. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem33.1  |-  F/_ t F
stoweidlem33.2  |-  F/_ t G
stoweidlem33.3  |-  F/ t
ph
stoweidlem33.4  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
stoweidlem33.5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
stoweidlem33.6  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
stoweidlem33.7  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
Assertion
Ref Expression
stoweidlem33  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  e.  A )
Distinct variable groups:    f, g,
t, A    f, F, g    f, G, g    T, f, g, t    ph, f,
g    x, t, A    x, T    ph, x
Allowed substitution hints:    ph( t)    F( x, t)    G( x, t)

Proof of Theorem stoweidlem33
StepHypRef Expression
1 stoweidlem33.3 . 2  |-  F/ t
ph
2 stoweidlem33.1 . 2  |-  F/_ t F
3 stoweidlem33.2 . 2  |-  F/_ t G
4 eqid 2283 . 2  |-  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  =  ( t  e.  T  |->  ( ( F `
 t )  -  ( G `  t ) ) )
5 eqid 2283 . 2  |-  ( t  e.  T  |->  -u 1
)  =  ( t  e.  T  |->  -u 1
)
6 eqid 2283 . 2  |-  ( t  e.  T  |->  ( ( ( t  e.  T  |-> 
-u 1 ) `  t )  x.  ( G `  t )
) )  =  ( t  e.  T  |->  ( ( ( t  e.  T  |->  -u 1 ) `  t )  x.  ( G `  t )
) )
7 stoweidlem33.4 . 2  |-  ( (
ph  /\  f  e.  A )  ->  f : T --> RR )
8 stoweidlem33.5 . 2  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
9 stoweidlem33.6 . 2  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
10 stoweidlem33.7 . 2  |-  ( (
ph  /\  x  e.  RR )  ->  ( t  e.  T  |->  x )  e.  A )
111, 2, 3, 4, 5, 6, 7, 8, 9, 10stoweidlem22 27771 1  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   F/wnf 1531    e. wcel 1684   F/_wnfc 2406    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   RRcr 8736   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038
This theorem is referenced by:  stoweidlem40  27789  stoweidlem41  27790
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  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
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-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  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-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040
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