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Theorem dviaddf 14100
Description: The sum rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)
Hypotheses
Ref Expression
dvaddf.s  |-  ( ph  ->  S  e.  { RR ,  CC } )
dviaddf.x  |-  ( ph  ->  X  C_  S )
dvaddf.f  |-  ( ph  ->  F : X --> CC )
dvaddf.g  |-  ( ph  ->  G : X --> CC )
dvaddf.df  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
dvaddf.dg  |-  ( ph  ->  dom  ( S  _D  G )  =  X )
Assertion
Ref Expression
dviaddf  |-  ( ph  ->  ( S  _D  ( F  oF  +  G
) )  =  ( ( S  _D  F
)  oF  +  ( S  _D  G
) ) )

Proof of Theorem dviaddf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addcl 7935 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
21adantl 277 . . 3  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  +  y )  e.  CC )
3 dvaddf.s . . . . 5  |-  ( ph  ->  S  e.  { RR ,  CC } )
4 cnex 7934 . . . . . . 7  |-  CC  e.  _V
54a1i 9 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
6 dvaddf.f . . . . . 6  |-  ( ph  ->  F : X --> CC )
7 dviaddf.x . . . . . 6  |-  ( ph  ->  X  C_  S )
8 elpm2r 6665 . . . . . 6  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : X --> CC  /\  X  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
95, 3, 6, 7, 8syl22anc 1239 . . . . 5  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
10 dvfgg 14088 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  _D  F ) : dom  ( S  _D  F ) --> CC )
113, 9, 10syl2anc 411 . . . 4  |-  ( ph  ->  ( S  _D  F
) : dom  ( S  _D  F ) --> CC )
12 dvaddf.df . . . . 5  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
1312feq2d 5353 . . . 4  |-  ( ph  ->  ( ( S  _D  F ) : dom  ( S  _D  F
) --> CC  <->  ( S  _D  F ) : X --> CC ) )
1411, 13mpbid 147 . . 3  |-  ( ph  ->  ( S  _D  F
) : X --> CC )
15 dvaddf.g . . . . . 6  |-  ( ph  ->  G : X --> CC )
16 elpm2r 6665 . . . . . 6  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( G : X --> CC  /\  X  C_  S ) )  ->  G  e.  ( CC  ^pm  S )
)
175, 3, 15, 7, 16syl22anc 1239 . . . . 5  |-  ( ph  ->  G  e.  ( CC 
^pm  S ) )
18 dvfgg 14088 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  G  e.  ( CC  ^pm  S
) )  ->  ( S  _D  G ) : dom  ( S  _D  G ) --> CC )
193, 17, 18syl2anc 411 . . . 4  |-  ( ph  ->  ( S  _D  G
) : dom  ( S  _D  G ) --> CC )
20 dvaddf.dg . . . . 5  |-  ( ph  ->  dom  ( S  _D  G )  =  X )
2120feq2d 5353 . . . 4  |-  ( ph  ->  ( ( S  _D  G ) : dom  ( S  _D  G
) --> CC  <->  ( S  _D  G ) : X --> CC ) )
2219, 21mpbid 147 . . 3  |-  ( ph  ->  ( S  _D  G
) : X --> CC )
233, 7ssexd 4143 . . 3  |-  ( ph  ->  X  e.  _V )
24 inidm 3344 . . 3  |-  ( X  i^i  X )  =  X
252, 6, 15, 23, 23, 24off 6094 . . . . . 6  |-  ( ph  ->  ( F  oF  +  G ) : X --> CC )
26 elpm2r 6665 . . . . . 6  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( ( F  oF  +  G ) : X --> CC  /\  X  C_  S ) )  -> 
( F  oF  +  G )  e.  ( CC  ^pm  S
) )
275, 3, 25, 7, 26syl22anc 1239 . . . . 5  |-  ( ph  ->  ( F  oF  +  G )  e.  ( CC  ^pm  S
) )
28 dvfgg 14088 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  ( F  oF  +  G
)  e.  ( CC 
^pm  S ) )  ->  ( S  _D  ( F  oF  +  G ) ) : dom  ( S  _D  ( F  oF  +  G ) ) --> CC )
293, 27, 28syl2anc 411 . . . 4  |-  ( ph  ->  ( S  _D  ( F  oF  +  G
) ) : dom  ( S  _D  ( F  oF  +  G
) ) --> CC )
30 recnprss 14087 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
313, 30syl 14 . . . . . . 7  |-  ( ph  ->  S  C_  CC )
3231, 25, 7dvbss 14085 . . . . . 6  |-  ( ph  ->  dom  ( S  _D  ( F  oF  +  G ) )  C_  X )
33 reldvg 14079 . . . . . . . . 9  |-  ( ( S  C_  CC  /\  ( F  oF  +  G
)  e.  ( CC 
^pm  S ) )  ->  Rel  ( S  _D  ( F  oF  +  G ) ) )
3431, 27, 33syl2anc 411 . . . . . . . 8  |-  ( ph  ->  Rel  ( S  _D  ( F  oF  +  G ) ) )
3534adantr 276 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  Rel  ( S  _D  ( F  oF  +  G
) ) )
366adantr 276 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  F : X --> CC )
377adantr 276 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  X  C_  S )
3815adantr 276 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  G : X --> CC )
3931adantr 276 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  S  C_  CC )
4012eleq2d 2247 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  dom  ( S  _D  F
)  <->  x  e.  X
) )
4140biimpar 297 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  F ) )
42 ffun 5368 . . . . . . . . . . 11  |-  ( ( S  _D  F ) : dom  ( S  _D  F ) --> CC 
->  Fun  ( S  _D  F ) )
43 funfvbrb 5629 . . . . . . . . . . 11  |-  ( Fun  ( S  _D  F
)  ->  ( x  e.  dom  ( S  _D  F )  <->  x ( S  _D  F ) ( ( S  _D  F
) `  x )
) )
4411, 42, 433syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  dom  ( S  _D  F
)  <->  x ( S  _D  F ) ( ( S  _D  F
) `  x )
) )
4544adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  dom  ( S  _D  F )  <->  x ( S  _D  F ) ( ( S  _D  F
) `  x )
) )
4641, 45mpbid 147 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  F
) ( ( S  _D  F ) `  x ) )
4720eleq2d 2247 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  dom  ( S  _D  G
)  <->  x  e.  X
) )
4847biimpar 297 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  G ) )
49 ffun 5368 . . . . . . . . . . 11  |-  ( ( S  _D  G ) : dom  ( S  _D  G ) --> CC 
->  Fun  ( S  _D  G ) )
50 funfvbrb 5629 . . . . . . . . . . 11  |-  ( Fun  ( S  _D  G
)  ->  ( x  e.  dom  ( S  _D  G )  <->  x ( S  _D  G ) ( ( S  _D  G
) `  x )
) )
5119, 49, 503syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  dom  ( S  _D  G
)  <->  x ( S  _D  G ) ( ( S  _D  G
) `  x )
) )
5251adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  dom  ( S  _D  G )  <->  x ( S  _D  G ) ( ( S  _D  G
) `  x )
) )
5348, 52mpbid 147 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  G
) ( ( S  _D  G ) `  x ) )
54 eqid 2177 . . . . . . . 8  |-  ( MetOpen `  ( abs  o.  -  )
)  =  ( MetOpen `  ( abs  o.  -  )
)
5536, 37, 38, 39, 46, 53, 54dvaddxxbr 14096 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  ( F  oF  +  G
) ) ( ( ( S  _D  F
) `  x )  +  ( ( S  _D  G ) `  x ) ) )
56 releldm 4862 . . . . . . 7  |-  ( ( Rel  ( S  _D  ( F  oF  +  G ) )  /\  x ( S  _D  ( F  oF  +  G ) ) ( ( ( S  _D  F ) `  x
)  +  ( ( S  _D  G ) `
 x ) ) )  ->  x  e.  dom  ( S  _D  ( F  oF  +  G
) ) )
5735, 55, 56syl2anc 411 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  ( F  oF  +  G ) ) )
5832, 57eqelssd 3174 . . . . 5  |-  ( ph  ->  dom  ( S  _D  ( F  oF  +  G ) )  =  X )
5958feq2d 5353 . . . 4  |-  ( ph  ->  ( ( S  _D  ( F  oF  +  G ) ) : dom  ( S  _D  ( F  oF  +  G ) ) --> CC  <->  ( S  _D  ( F  oF  +  G
) ) : X --> CC ) )
6029, 59mpbid 147 . . 3  |-  ( ph  ->  ( S  _D  ( F  oF  +  G
) ) : X --> CC )
61 eqidd 2178 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  =  ( ( S  _D  F ) `  x ) )
62 eqidd 2178 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  G
) `  x )  =  ( ( S  _D  G ) `  x ) )
633adantr 276 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  S  e.  { RR ,  CC } )
6436, 37, 38, 63, 41, 48dvaddxx 14098 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  ( F  oF  +  G
) ) `  x
)  =  ( ( ( S  _D  F
) `  x )  +  ( ( S  _D  G ) `  x ) ) )
6564eqcomd 2183 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  +  ( ( S  _D  G ) `
 x ) )  =  ( ( S  _D  ( F  oF  +  G )
) `  x )
)
662, 14, 22, 23, 23, 24, 60, 61, 62, 65offeq 6095 . 2  |-  ( ph  ->  ( ( S  _D  F )  oF  +  ( S  _D  G ) )  =  ( S  _D  ( F  oF  +  G
) ) )
6766eqcomd 2183 1  |-  ( ph  ->  ( S  _D  ( F  oF  +  G
) )  =  ( ( S  _D  F
)  oF  +  ( S  _D  G
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   _Vcvv 2737    C_ wss 3129   {cpr 3593   class class class wbr 4003   dom cdm 4626    o. ccom 4630   Rel wrel 4631   Fun wfun 5210   -->wf 5212   ` cfv 5216  (class class class)co 5874    oFcof 6080    ^pm cpm 6648   CCcc 7808   RRcr 7809    + caddc 7813    - cmin 8126   abscabs 11001   MetOpencmopn 13376    _D cdv 14055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930  ax-addf 7932
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-of 6082  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-map 6649  df-pm 6650  df-sup 6982  df-inf 6983  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-q 9618  df-rp 9652  df-xneg 9770  df-xadd 9771  df-seqfrec 10443  df-exp 10517  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003  df-rest 12684  df-topgen 12703  df-psmet 13378  df-xmet 13379  df-met 13380  df-bl 13381  df-mopn 13382  df-top 13429  df-topon 13442  df-bases 13474  df-ntr 13527  df-cn 13619  df-cnp 13620  df-tx 13684  df-limced 14056  df-dvap 14057
This theorem is referenced by:  dvmptaddx  14112
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