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Theorem ofsubge0 10536
Description: Function analog of subge0 10066. (Contributed by Mario Carneiro, 24-Jul-2014.)
Assertion
Ref Expression
ofsubge0  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( ( A  X.  { 0 } )  oR  <_ 
( F  oF  -  G )  <->  G  oR  <_  F ) )

Proof of Theorem ofsubge0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp2 997 . . . . 5  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  F : A --> RR )
21ffvelrnda 6022 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( F `  x )  e.  RR )
3 simp3 998 . . . . 5  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  G : A --> RR )
43ffvelrnda 6022 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( G `  x )  e.  RR )
52, 4subge0d 10143 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( 0  <_  ( ( F `
 x )  -  ( G `  x ) )  <->  ( G `  x )  <_  ( F `  x )
) )
65ralbidva 2900 . 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 9589 . . . 4  |-  0  e.  CC
8 fnconstg 5773 . . . 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 5731 . . . . 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 5731 . . . . 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 996 . . . 4  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  A  e.  V
)
15 inidm 3707 . . . 4  |-  ( A  i^i  A )  =  A
1611, 13, 14, 14, 15offn 6536 . . 3  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( F  oF  -  G )  Fn  A )
17 c0ex 9591 . . . . 5  |-  0  e.  _V
1817fvconst2 6117 . . . 4  |-  ( x  e.  A  ->  (
( A  X.  {
0 } ) `  x )  =  0 )
1918adantl 466 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( ( A  X.  { 0 } ) `  x )  =  0 )
20 eqidd 2468 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( F `  x )  =  ( F `  x ) )
21 eqidd 2468 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( G `  x )  =  ( G `  x ) )
2211, 13, 14, 14, 15, 20, 21ofval 6534 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( ( F  oF  -  G
) `  x )  =  ( ( F `
 x )  -  ( G `  x ) ) )
239, 16, 14, 14, 15, 19, 22ofrfval 6533 . 2  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( ( A  X.  { 0 } )  oR  <_ 
( F  oF  -  G )  <->  A. x  e.  A  0  <_  ( ( F `  x
)  -  ( G `
 x ) ) ) )
2413, 11, 14, 14, 15, 21, 20ofrfval 6533 . 2  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( G  oR  <_  F  <->  A. x  e.  A  ( G `  x )  <_  ( F `  x )
) )
256, 23, 243bitr4d 285 1  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( ( A  X.  { 0 } )  oR  <_ 
( F  oF  -  G )  <->  G  oR  <_  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   {csn 4027   class class class wbr 4447    X. cxp 4997    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6285    oFcof 6523    oRcofr 6524   CCcc 9491   RRcr 9492   0cc0 9493    <_ cle 9630    - cmin 9806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-of 6525  df-ofr 6526  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809
This theorem is referenced by: (None)
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