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Theorem stoweidlem33 37183
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 2402 . 2  |-  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  =  ( t  e.  T  |->  ( ( F `
 t )  -  ( G `  t ) ) )
5 eqid 2402 . 2  |-  ( t  e.  T  |->  -u 1
)  =  ( t  e.  T  |->  -u 1
)
6 eqid 2402 . 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 37172 1  |-  ( (
ph  /\  F  e.  A  /\  G  e.  A
)  ->  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974   F/wnf 1637    e. wcel 1842   F/_wnfc 2550    |-> cmpt 4453   -->wf 5565   ` cfv 5569  (class class class)co 6278   RRcr 9521   1c1 9523    + caddc 9525    x. cmul 9527    - cmin 9841   -ucneg 9842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-ltxr 9663  df-sub 9843  df-neg 9844
This theorem is referenced by:  stoweidlem40  37190  stoweidlem41  37191
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