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Theorem ofnegsub 10523
Description: Function analog of negsub 9856. (Contributed by Mario Carneiro, 24-Jul-2014.)
Assertion
Ref Expression
ofnegsub  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( F  oF  +  ( ( A  X.  { -u 1 } )  oF  x.  G ) )  =  ( F  oF  -  G )
)

Proof of Theorem ofnegsub
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp1 991 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  A  e.  V
)
2 simp2 992 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  F : A --> CC )
3 ffn 5722 . . 3  |-  ( F : A --> CC  ->  F  Fn  A )
42, 3syl 16 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  F  Fn  A
)
5 ax-1cn 9539 . . . . 5  |-  1  e.  CC
65negcli 9876 . . . 4  |-  -u 1  e.  CC
7 fnconstg 5764 . . . 4  |-  ( -u
1  e.  CC  ->  ( A  X.  { -u
1 } )  Fn  A )
86, 7mp1i 12 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( A  X.  { -u 1 } )  Fn  A )
9 simp3 993 . . . 4  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  G : A --> CC )
10 ffn 5722 . . . 4  |-  ( G : A --> CC  ->  G  Fn  A )
119, 10syl 16 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  G  Fn  A
)
12 inidm 3700 . . 3  |-  ( A  i^i  A )  =  A
138, 11, 1, 1, 12offn 6526 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( A  X.  { -u 1 } )  oF  x.  G )  Fn  A )
144, 11, 1, 1, 12offn 6526 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( F  oF  -  G )  Fn  A )
15 eqidd 2461 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  =  ( F `  x ) )
166a1i 11 . . . 4  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  -u 1  e.  CC )
17 eqidd 2461 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  =  ( G `  x ) )
181, 16, 11, 17ofc1 6538 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( A  X.  { -u 1 } )  oF  x.  G ) `
 x )  =  ( -u 1  x.  ( G `  x
) ) )
199ffvelrnda 6012 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  e.  CC )
2019mulm1d 9997 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( -u 1  x.  ( G `  x
) )  =  -u ( G `  x ) )
2118, 20eqtrd 2501 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( A  X.  { -u 1 } )  oF  x.  G ) `
 x )  = 
-u ( G `  x ) )
222ffvelrnda 6012 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  e.  CC )
2322, 19negsubd 9925 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( ( F `  x )  +  -u ( G `  x ) )  =  ( ( F `  x )  -  ( G `  x )
) )
244, 11, 1, 1, 12, 15, 17ofval 6524 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( ( F  oF  -  G
) `  x )  =  ( ( F `
 x )  -  ( G `  x ) ) )
2523, 24eqtr4d 2504 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( ( F `  x )  +  -u ( G `  x ) )  =  ( ( F  oF  -  G ) `  x ) )
261, 4, 13, 14, 15, 21, 25offveq 6536 1  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( F  oF  +  ( ( A  X.  { -u 1 } )  oF  x.  G ) )  =  ( F  oF  -  G )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   {csn 4020    X. cxp 4990    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275    oFcof 6513   CCcc 9479   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9794   -ucneg 9795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9796  df-neg 9797
This theorem is referenced by:  i1fsub  21843  itg1sub  21844  plysub  22344  coesub  22381  dgrsub  22396  basellem9  23083  expgrowth  30795
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