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Theorem ofsubge0 9761
Description: Function analog of subge0 9303. (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 956 . . . . 5  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  F : A --> RR )
2 ffvelrn 5679 . . . . 5  |-  ( ( F : A --> RR  /\  x  e.  A )  ->  ( F `  x
)  e.  RR )
31, 2sylan 457 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( F `  x )  e.  RR )
4 simp3 957 . . . . 5  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  G : A --> RR )
5 ffvelrn 5679 . . . . 5  |-  ( ( G : A --> RR  /\  x  e.  A )  ->  ( G `  x
)  e.  RR )
64, 5sylan 457 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( G `  x )  e.  RR )
73, 6subge0d 9378 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( 0  <_  ( ( F `
 x )  -  ( G `  x ) )  <->  ( G `  x )  <_  ( F `  x )
) )
87ralbidva 2572 . 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 ) ) )
9 0cn 8847 . . . 4  |-  0  e.  CC
10 fnconstg 5445 . . . 4  |-  ( 0  e.  CC  ->  ( A  X.  { 0 } )  Fn  A )
119, 10mp1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( A  X.  { 0 } )  Fn  A )
12 ffn 5405 . . . . 5  |-  ( F : A --> RR  ->  F  Fn  A )
131, 12syl 15 . . . 4  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  F  Fn  A
)
14 ffn 5405 . . . . 5  |-  ( G : A --> RR  ->  G  Fn  A )
154, 14syl 15 . . . 4  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  G  Fn  A
)
16 simp1 955 . . . 4  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  A  e.  V
)
17 inidm 3391 . . . 4  |-  ( A  i^i  A )  =  A
1813, 15, 16, 16, 17offn 6105 . . 3  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( F  o F  -  G )  Fn  A )
19 c0ex 8848 . . . . 5  |-  0  e.  _V
2019fvconst2 5745 . . . 4  |-  ( x  e.  A  ->  (
( A  X.  {
0 } ) `  x )  =  0 )
2120adantl 452 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( ( A  X.  { 0 } ) `  x )  =  0 )
22 eqidd 2297 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( F `  x )  =  ( F `  x ) )
23 eqidd 2297 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( G `  x )  =  ( G `  x ) )
2413, 15, 16, 16, 17, 22, 23ofval 6103 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( ( F  o F  -  G
) `  x )  =  ( ( F `
 x )  -  ( G `  x ) ) )
2511, 18, 16, 16, 17, 21, 24ofrfval 6102 . 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 ) ) ) )
2615, 13, 16, 16, 17, 23, 22ofrfval 6102 . 2  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( G  o R  <_  F  <->  A. x  e.  A  ( G `  x )  <_  ( F `  x )
) )
278, 25, 263bitr4d 276 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {csn 3653   class class class wbr 4039    X. cxp 4703    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092    o Rcofr 6093   CCcc 8751   RRcr 8752   0cc0 8753    <_ cle 8884    - cmin 9053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056
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