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Theorem ofnegsub 10576
Description: Function analog of negsub 9905. (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 999 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  A  e.  V
)
2 simp2 1000 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  F : A --> CC )
3 ffn 5716 . . 3  |-  ( F : A --> CC  ->  F  Fn  A )
42, 3syl 17 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  F  Fn  A
)
5 ax-1cn 9582 . . . . 5  |-  1  e.  CC
65negcli 9925 . . . 4  |-  -u 1  e.  CC
7 fnconstg 5758 . . . 4  |-  ( -u
1  e.  CC  ->  ( A  X.  { -u
1 } )  Fn  A )
86, 7mp1i 13 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( A  X.  { -u 1 } )  Fn  A )
9 simp3 1001 . . . 4  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  G : A --> CC )
10 ffn 5716 . . . 4  |-  ( G : A --> CC  ->  G  Fn  A )
119, 10syl 17 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  G  Fn  A
)
12 inidm 3650 . . 3  |-  ( A  i^i  A )  =  A
138, 11, 1, 1, 12offn 6534 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( A  X.  { -u 1 } )  oF  x.  G )  Fn  A )
144, 11, 1, 1, 12offn 6534 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( F  oF  -  G )  Fn  A )
15 eqidd 2405 . 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 2405 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  =  ( G `  x ) )
181, 16, 11, 17ofc1 6547 . . 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 6011 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  e.  CC )
2019mulm1d 10051 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( -u 1  x.  ( G `  x
) )  =  -u ( G `  x ) )
2118, 20eqtrd 2445 . 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 6011 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  e.  CC )
2322, 19negsubd 9975 . . 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 6532 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( ( F  oF  -  G
) `  x )  =  ( ( F `
 x )  -  ( G `  x ) ) )
2523, 24eqtr4d 2448 . 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 6545 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 976    = wceq 1407    e. wcel 1844   {csn 3974    X. cxp 4823    Fn wfn 5566   -->wf 5567   ` cfv 5571  (class class class)co 6280    oFcof 6521   CCcc 9522   1c1 9525    + caddc 9527    x. cmul 9529    - cmin 9843   -ucneg 9844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-po 4746  df-so 4747  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-of 6523  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-ltxr 9665  df-sub 9845  df-neg 9846
This theorem is referenced by:  i1fsub  22409  itg1sub  22410  plysub  22910  coesub  22948  dgrsub  22963  basellem9  23745  expgrowth  36101
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