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Theorem ofsubge0 9989
Description: Function analog of subge0 9531. (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 5862 . . . 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 5862 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( G `  x )  e.  RR )
52, 4subge0d 9606 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( 0  <_  ( ( F `
 x )  -  ( G `  x ) )  <->  ( G `  x )  <_  ( F `  x )
) )
65ralbidva 2713 . 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 9074 . . . 4  |-  0  e.  CC
8 fnconstg 5623 . . . 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 5583 . . . . 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 5583 . . . . 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 3542 . . . 4  |-  ( A  i^i  A )  =  A
1611, 13, 14, 14, 15offn 6308 . . 3  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( F  o F  -  G )  Fn  A )
17 c0ex 9075 . . . . 5  |-  0  e.  _V
1817fvconst2 5939 . . . 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 2436 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( F `  x )  =  ( F `  x ) )
21 eqidd 2436 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( G `  x )  =  ( G `  x ) )
2211, 13, 14, 14, 15, 20, 21ofval 6306 . . 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 6305 . 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 6305 . 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 1652    e. wcel 1725   A.wral 2697   {csn 3806   class class class wbr 4204    X. cxp 4868    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295    o Rcofr 6296   CCcc 8978   RRcr 8979   0cc0 8980    <_ cle 9111    - cmin 9281
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-ofr 6298  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284
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