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Theorem ofsubge0 9932
Description: Function analog of subge0 9474. (Contributed by Mario Carneiro, 24-Jul-2014.)
Assertion
Ref Expression
ofsubge0  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( ( A  X.  { 0 } )  o R  <_ 
( F  o F  -  G )  <->  G  o R  <_  F ) )

Proof of Theorem ofsubge0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp2 958 . . . . 5  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  F : A --> RR )
21ffvelrnda 5810 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( F `  x )  e.  RR )
3 simp3 959 . . . . 5  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  G : A --> RR )
43ffvelrnda 5810 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( G `  x )  e.  RR )
52, 4subge0d 9549 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( 0  <_  ( ( F `
 x )  -  ( G `  x ) )  <->  ( G `  x )  <_  ( F `  x )
) )
65ralbidva 2666 . 2  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( A. x  e.  A  0  <_  ( ( F `  x
)  -  ( G `
 x ) )  <->  A. x  e.  A  ( G `  x )  <_  ( F `  x ) ) )
7 0cn 9018 . . . 4  |-  0  e.  CC
8 fnconstg 5572 . . . 4  |-  ( 0  e.  CC  ->  ( A  X.  { 0 } )  Fn  A )
97, 8mp1i 12 . . 3  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( A  X.  { 0 } )  Fn  A )
10 ffn 5532 . . . . 5  |-  ( F : A --> RR  ->  F  Fn  A )
111, 10syl 16 . . . 4  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  F  Fn  A
)
12 ffn 5532 . . . . 5  |-  ( G : A --> RR  ->  G  Fn  A )
133, 12syl 16 . . . 4  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  G  Fn  A
)
14 simp1 957 . . . 4  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  A  e.  V
)
15 inidm 3494 . . . 4  |-  ( A  i^i  A )  =  A
1611, 13, 14, 14, 15offn 6256 . . 3  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( F  o F  -  G )  Fn  A )
17 c0ex 9019 . . . . 5  |-  0  e.  _V
1817fvconst2 5887 . . . 4  |-  ( x  e.  A  ->  (
( A  X.  {
0 } ) `  x )  =  0 )
1918adantl 453 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( ( A  X.  { 0 } ) `  x )  =  0 )
20 eqidd 2389 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( F `  x )  =  ( F `  x ) )
21 eqidd 2389 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( G `  x )  =  ( G `  x ) )
2211, 13, 14, 14, 15, 20, 21ofval 6254 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( ( F  o F  -  G
) `  x )  =  ( ( F `
 x )  -  ( G `  x ) ) )
239, 16, 14, 14, 15, 19, 22ofrfval 6253 . 2  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( ( A  X.  { 0 } )  o R  <_ 
( F  o F  -  G )  <->  A. x  e.  A  0  <_  ( ( F `  x
)  -  ( G `
 x ) ) ) )
2413, 11, 14, 14, 15, 21, 20ofrfval 6253 . 2  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( G  o R  <_  F  <->  A. x  e.  A  ( G `  x )  <_  ( F `  x )
) )
256, 23, 243bitr4d 277 1  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( ( A  X.  { 0 } )  o R  <_ 
( F  o F  -  G )  <->  G  o R  <_  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2650   {csn 3758   class class class wbr 4154    X. cxp 4817    Fn wfn 5390   -->wf 5391   ` cfv 5395  (class class class)co 6021    o Fcof 6243    o Rcofr 6244   CCcc 8922   RRcr 8923   0cc0 8924    <_ cle 9055    - cmin 9224
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-po 4445  df-so 4446  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-ofr 6246  df-riota 6486  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227
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