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Theorem stoweidlem33 29999
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 2454 . 2  |-  ( t  e.  T  |->  ( ( F `  t )  -  ( G `  t ) ) )  =  ( t  e.  T  |->  ( ( F `
 t )  -  ( G `  t ) ) )
5 eqid 2454 . 2  |-  ( t  e.  T  |->  -u 1
)  =  ( t  e.  T  |->  -u 1
)
6 eqid 2454 . 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 29988 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 369    /\ w3a 965   F/wnf 1590    e. wcel 1758   F/_wnfc 2602    |-> cmpt 4461   -->wf 5525   ` cfv 5529  (class class class)co 6203   RRcr 9396   1c1 9398    + caddc 9400    x. cmul 9402    - cmin 9710   -ucneg 9711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-ltxr 9538  df-sub 9712  df-neg 9713
This theorem is referenced by:  stoweidlem40  30006  stoweidlem41  30007
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