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Theorem dviaddf 12721
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 7709 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
21adantl 273 . . 3  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  +  y )  e.  CC )
3 dvaddf.s . . . . 5  |-  ( ph  ->  S  e.  { RR ,  CC } )
4 cnex 7708 . . . . . . 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 6526 . . . . . 6  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : X --> CC  /\  X  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
95, 3, 6, 7, 8syl22anc 1200 . . . . 5  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
10 dvfgg 12709 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  _D  F ) : dom  ( S  _D  F ) --> CC )
113, 9, 10syl2anc 406 . . . 4  |-  ( ph  ->  ( S  _D  F
) : dom  ( S  _D  F ) --> CC )
12 dvaddf.df . . . . 5  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
1312feq2d 5228 . . . 4  |-  ( ph  ->  ( ( S  _D  F ) : dom  ( S  _D  F
) --> CC  <->  ( S  _D  F ) : X --> CC ) )
1411, 13mpbid 146 . . 3  |-  ( ph  ->  ( S  _D  F
) : X --> CC )
15 dvaddf.g . . . . . 6  |-  ( ph  ->  G : X --> CC )
16 elpm2r 6526 . . . . . 6  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( G : X --> CC  /\  X  C_  S ) )  ->  G  e.  ( CC  ^pm  S )
)
175, 3, 15, 7, 16syl22anc 1200 . . . . 5  |-  ( ph  ->  G  e.  ( CC 
^pm  S ) )
18 dvfgg 12709 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  G  e.  ( CC  ^pm  S
) )  ->  ( S  _D  G ) : dom  ( S  _D  G ) --> CC )
193, 17, 18syl2anc 406 . . . 4  |-  ( ph  ->  ( S  _D  G
) : dom  ( S  _D  G ) --> CC )
20 dvaddf.dg . . . . 5  |-  ( ph  ->  dom  ( S  _D  G )  =  X )
2120feq2d 5228 . . . 4  |-  ( ph  ->  ( ( S  _D  G ) : dom  ( S  _D  G
) --> CC  <->  ( S  _D  G ) : X --> CC ) )
2219, 21mpbid 146 . . 3  |-  ( ph  ->  ( S  _D  G
) : X --> CC )
233, 7ssexd 4036 . . 3  |-  ( ph  ->  X  e.  _V )
24 inidm 3253 . . 3  |-  ( X  i^i  X )  =  X
252, 6, 15, 23, 23, 24off 5960 . . . . . 6  |-  ( ph  ->  ( F  oF  +  G ) : X --> CC )
26 elpm2r 6526 . . . . . 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 1200 . . . . 5  |-  ( ph  ->  ( F  oF  +  G )  e.  ( CC  ^pm  S
) )
28 dvfgg 12709 . . . . 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 406 . . . 4  |-  ( ph  ->  ( S  _D  ( F  oF  +  G
) ) : dom  ( S  _D  ( F  oF  +  G
) ) --> CC )
30 recnprss 12708 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
313, 30syl 14 . . . . . . 7  |-  ( ph  ->  S  C_  CC )
3231, 25, 7dvbss 12706 . . . . . 6  |-  ( ph  ->  dom  ( S  _D  ( F  oF  +  G ) )  C_  X )
33 reldvg 12700 . . . . . . . . 9  |-  ( ( S  C_  CC  /\  ( F  oF  +  G
)  e.  ( CC 
^pm  S ) )  ->  Rel  ( S  _D  ( F  oF  +  G ) ) )
3431, 27, 33syl2anc 406 . . . . . . . 8  |-  ( ph  ->  Rel  ( S  _D  ( F  oF  +  G ) ) )
3534adantr 272 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  Rel  ( S  _D  ( F  oF  +  G
) ) )
366adantr 272 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  F : X --> CC )
377adantr 272 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  X  C_  S )
3815adantr 272 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  G : X --> CC )
3931adantr 272 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  S  C_  CC )
4012eleq2d 2185 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  dom  ( S  _D  F
)  <->  x  e.  X
) )
4140biimpar 293 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  F ) )
42 ffun 5243 . . . . . . . . . . 11  |-  ( ( S  _D  F ) : dom  ( S  _D  F ) --> CC 
->  Fun  ( S  _D  F ) )
43 funfvbrb 5499 . . . . . . . . . . 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 272 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  dom  ( S  _D  F )  <->  x ( S  _D  F ) ( ( S  _D  F
) `  x )
) )
4641, 45mpbid 146 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  F
) ( ( S  _D  F ) `  x ) )
4720eleq2d 2185 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  dom  ( S  _D  G
)  <->  x  e.  X
) )
4847biimpar 293 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  G ) )
49 ffun 5243 . . . . . . . . . . 11  |-  ( ( S  _D  G ) : dom  ( S  _D  G ) --> CC 
->  Fun  ( S  _D  G ) )
50 funfvbrb 5499 . . . . . . . . . . 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 272 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  dom  ( S  _D  G )  <->  x ( S  _D  G ) ( ( S  _D  G
) `  x )
) )
5348, 52mpbid 146 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  G
) ( ( S  _D  G ) `  x ) )
54 eqid 2115 . . . . . . . 8  |-  ( MetOpen `  ( abs  o.  -  )
)  =  ( MetOpen `  ( abs  o.  -  )
)
5536, 37, 38, 39, 46, 53, 54dvaddxxbr 12717 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  ( F  oF  +  G
) ) ( ( ( S  _D  F
) `  x )  +  ( ( S  _D  G ) `  x ) ) )
56 releldm 4742 . . . . . . 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 406 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  ( F  oF  +  G ) ) )
5832, 57eqelssd 3084 . . . . 5  |-  ( ph  ->  dom  ( S  _D  ( F  oF  +  G ) )  =  X )
5958feq2d 5228 . . . 4  |-  ( ph  ->  ( ( S  _D  ( F  oF  +  G ) ) : dom  ( S  _D  ( F  oF  +  G ) ) --> CC  <->  ( S  _D  ( F  oF  +  G
) ) : X --> CC ) )
6029, 59mpbid 146 . . 3  |-  ( ph  ->  ( S  _D  ( F  oF  +  G
) ) : X --> CC )
61 eqidd 2116 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  =  ( ( S  _D  F ) `  x ) )
62 eqidd 2116 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  G
) `  x )  =  ( ( S  _D  G ) `  x ) )
633adantr 272 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  S  e.  { RR ,  CC } )
6436, 37, 38, 63, 41, 48dvaddxx 12719 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  ( F  oF  +  G
) ) `  x
)  =  ( ( ( S  _D  F
) `  x )  +  ( ( S  _D  G ) `  x ) ) )
6564eqcomd 2121 . . 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 5961 . 2  |-  ( ph  ->  ( ( S  _D  F )  oF  +  ( S  _D  G ) )  =  ( S  _D  ( F  oF  +  G
) ) )
6766eqcomd 2121 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 103    <-> wb 104    = wceq 1314    e. wcel 1463   _Vcvv 2658    C_ wss 3039   {cpr 3496   class class class wbr 3897   dom cdm 4507    o. ccom 4511   Rel wrel 4512   Fun wfun 5085   -->wf 5087   ` cfv 5091  (class class class)co 5740    oFcof 5946    ^pm cpm 6509   CCcc 7582   RRcr 7583    + caddc 7587    - cmin 7897   abscabs 10709   MetOpencmopn 12049    _D cdv 12676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704  ax-addf 7706
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-isom 5100  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-of 5948  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-map 6510  df-pm 6511  df-sup 6837  df-inf 6838  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-2 8736  df-3 8737  df-4 8738  df-n0 8929  df-z 9006  df-uz 9276  df-q 9361  df-rp 9391  df-xneg 9499  df-xadd 9500  df-seqfrec 10159  df-exp 10233  df-cj 10554  df-re 10555  df-im 10556  df-rsqrt 10710  df-abs 10711  df-rest 12017  df-topgen 12036  df-psmet 12051  df-xmet 12052  df-met 12053  df-bl 12054  df-mopn 12055  df-top 12060  df-topon 12073  df-bases 12105  df-ntr 12160  df-cn 12252  df-cnp 12253  df-tx 12317  df-limced 12677  df-dvap 12678
This theorem is referenced by:  dvmptaddx  12733
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