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Theorem ofsubge0 9745
Description: Function analog of subge0 9287. (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 5663 . . . . 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 5663 . . . . 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 9362 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( 0  <_  ( ( F `
 x )  -  ( G `  x ) )  <->  ( G `  x )  <_  ( F `  x )
) )
87ralbidva 2559 . 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 8831 . . . 4  |-  0  e.  CC
10 fnconstg 5429 . . . 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 5389 . . . . 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 5389 . . . . 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 3378 . . . 4  |-  ( A  i^i  A )  =  A
1813, 15, 16, 16, 17offn 6089 . . 3  |-  ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  ->  ( F  o F  -  G )  Fn  A )
19 c0ex 8832 . . . . 5  |-  0  e.  _V
2019fvconst2 5729 . . . 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 2284 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( F `  x )  =  ( F `  x ) )
23 eqidd 2284 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> RR  /\  G : A --> RR )  /\  x  e.  A
)  ->  ( G `  x )  =  ( G `  x ) )
2413, 15, 16, 16, 17, 22, 23ofval 6087 . . 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 6086 . 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 6086 . 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 1623    e. wcel 1684   A.wral 2543   {csn 3640   class class class wbr 4023    X. cxp 4687    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076    o Rcofr 6077   CCcc 8735   RRcr 8736   0cc0 8737    <_ cle 8868    - cmin 9037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040
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