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Theorem dvaddxx 12850
Description: The sum rule for derivatives at a point. For the (more general) relation version, see dvaddxxbr 12848. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.)
Hypotheses
Ref Expression
dvadd.f  |-  ( ph  ->  F : X --> CC )
dvadd.x  |-  ( ph  ->  X  C_  S )
dvaddxx.g  |-  ( ph  ->  G : X --> CC )
dvadd.s  |-  ( ph  ->  S  e.  { RR ,  CC } )
dvadd.df  |-  ( ph  ->  C  e.  dom  ( S  _D  F ) )
dvadd.dg  |-  ( ph  ->  C  e.  dom  ( S  _D  G ) )
Assertion
Ref Expression
dvaddxx  |-  ( ph  ->  ( ( S  _D  ( F  oF  +  G ) ) `  C )  =  ( ( ( S  _D  F ) `  C
)  +  ( ( S  _D  G ) `
 C ) ) )

Proof of Theorem dvaddxx
Dummy variables  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvadd.s . . . 4  |-  ( ph  ->  S  e.  { RR ,  CC } )
2 cnex 7756 . . . . . 6  |-  CC  e.  _V
32a1i 9 . . . . 5  |-  ( ph  ->  CC  e.  _V )
4 addcl 7757 . . . . . . 7  |-  ( ( u  e.  CC  /\  v  e.  CC )  ->  ( u  +  v )  e.  CC )
54adantl 275 . . . . . 6  |-  ( (
ph  /\  ( u  e.  CC  /\  v  e.  CC ) )  -> 
( u  +  v )  e.  CC )
6 dvadd.f . . . . . 6  |-  ( ph  ->  F : X --> CC )
7 dvaddxx.g . . . . . 6  |-  ( ph  ->  G : X --> CC )
8 dvadd.x . . . . . . 7  |-  ( ph  ->  X  C_  S )
91, 8ssexd 4068 . . . . . 6  |-  ( ph  ->  X  e.  _V )
10 inidm 3285 . . . . . 6  |-  ( X  i^i  X )  =  X
115, 6, 7, 9, 9, 10off 5994 . . . . 5  |-  ( ph  ->  ( F  oF  +  G ) : X --> CC )
12 elpm2r 6560 . . . . 5  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( ( F  oF  +  G ) : X --> CC  /\  X  C_  S ) )  -> 
( F  oF  +  G )  e.  ( CC  ^pm  S
) )
133, 1, 11, 8, 12syl22anc 1217 . . . 4  |-  ( ph  ->  ( F  oF  +  G )  e.  ( CC  ^pm  S
) )
14 dvfgg 12840 . . . 4  |-  ( ( S  e.  { RR ,  CC }  /\  ( F  oF  +  G
)  e.  ( CC 
^pm  S ) )  ->  ( S  _D  ( F  oF  +  G ) ) : dom  ( S  _D  ( F  oF  +  G ) ) --> CC )
151, 13, 14syl2anc 408 . . 3  |-  ( ph  ->  ( S  _D  ( F  oF  +  G
) ) : dom  ( S  _D  ( F  oF  +  G
) ) --> CC )
1615ffund 5276 . 2  |-  ( ph  ->  Fun  ( S  _D  ( F  oF  +  G ) ) )
17 recnprss 12839 . . . 4  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
181, 17syl 14 . . 3  |-  ( ph  ->  S  C_  CC )
19 dvadd.df . . . 4  |-  ( ph  ->  C  e.  dom  ( S  _D  F ) )
20 elpm2r 6560 . . . . . . 7  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : X --> CC  /\  X  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
213, 1, 6, 8, 20syl22anc 1217 . . . . . 6  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
22 dvfgg 12840 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  _D  F ) : dom  ( S  _D  F ) --> CC )
231, 21, 22syl2anc 408 . . . . 5  |-  ( ph  ->  ( S  _D  F
) : dom  ( S  _D  F ) --> CC )
24 ffun 5275 . . . . 5  |-  ( ( S  _D  F ) : dom  ( S  _D  F ) --> CC 
->  Fun  ( S  _D  F ) )
25 funfvbrb 5533 . . . . 5  |-  ( Fun  ( S  _D  F
)  ->  ( C  e.  dom  ( S  _D  F )  <->  C ( S  _D  F ) ( ( S  _D  F
) `  C )
) )
2623, 24, 253syl 17 . . . 4  |-  ( ph  ->  ( C  e.  dom  ( S  _D  F
)  <->  C ( S  _D  F ) ( ( S  _D  F ) `
 C ) ) )
2719, 26mpbid 146 . . 3  |-  ( ph  ->  C ( S  _D  F ) ( ( S  _D  F ) `
 C ) )
28 dvadd.dg . . . 4  |-  ( ph  ->  C  e.  dom  ( S  _D  G ) )
29 elpm2r 6560 . . . . . . 7  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( G : X --> CC  /\  X  C_  S ) )  ->  G  e.  ( CC  ^pm  S )
)
303, 1, 7, 8, 29syl22anc 1217 . . . . . 6  |-  ( ph  ->  G  e.  ( CC 
^pm  S ) )
31 dvfgg 12840 . . . . . 6  |-  ( ( S  e.  { RR ,  CC }  /\  G  e.  ( CC  ^pm  S
) )  ->  ( S  _D  G ) : dom  ( S  _D  G ) --> CC )
321, 30, 31syl2anc 408 . . . . 5  |-  ( ph  ->  ( S  _D  G
) : dom  ( S  _D  G ) --> CC )
33 ffun 5275 . . . . 5  |-  ( ( S  _D  G ) : dom  ( S  _D  G ) --> CC 
->  Fun  ( S  _D  G ) )
34 funfvbrb 5533 . . . . 5  |-  ( Fun  ( S  _D  G
)  ->  ( C  e.  dom  ( S  _D  G )  <->  C ( S  _D  G ) ( ( S  _D  G
) `  C )
) )
3532, 33, 343syl 17 . . . 4  |-  ( ph  ->  ( C  e.  dom  ( S  _D  G
)  <->  C ( S  _D  G ) ( ( S  _D  G ) `
 C ) ) )
3628, 35mpbid 146 . . 3  |-  ( ph  ->  C ( S  _D  G ) ( ( S  _D  G ) `
 C ) )
37 eqid 2139 . . 3  |-  ( MetOpen `  ( abs  o.  -  )
)  =  ( MetOpen `  ( abs  o.  -  )
)
386, 8, 7, 18, 27, 36, 37dvaddxxbr 12848 . 2  |-  ( ph  ->  C ( S  _D  ( F  oF  +  G ) ) ( ( ( S  _D  F ) `  C
)  +  ( ( S  _D  G ) `
 C ) ) )
39 funbrfv 5460 . 2  |-  ( Fun  ( S  _D  ( F  oF  +  G
) )  ->  ( C ( S  _D  ( F  oF  +  G ) ) ( ( ( S  _D  F ) `  C
)  +  ( ( S  _D  G ) `
 C ) )  ->  ( ( S  _D  ( F  oF  +  G )
) `  C )  =  ( ( ( S  _D  F ) `
 C )  +  ( ( S  _D  G ) `  C
) ) ) )
4016, 38, 39sylc 62 1  |-  ( ph  ->  ( ( S  _D  ( F  oF  +  G ) ) `  C )  =  ( ( ( S  _D  F ) `  C
)  +  ( ( S  _D  G ) `
 C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   _Vcvv 2686    C_ wss 3071   {cpr 3528   class class class wbr 3929   dom cdm 4539    o. ccom 4543   Fun wfun 5117   -->wf 5119   ` cfv 5123  (class class class)co 5774    oFcof 5980    ^pm cpm 6543   CCcc 7630   RRcr 7631    + caddc 7635    - cmin 7945   abscabs 10781   MetOpencmopn 12168    _D cdv 12807
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752  ax-addf 7754
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-of 5982  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-map 6544  df-pm 6545  df-sup 6871  df-inf 6872  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-xneg 9571  df-xadd 9572  df-seqfrec 10231  df-exp 10305  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-rest 12136  df-topgen 12155  df-psmet 12170  df-xmet 12171  df-met 12172  df-bl 12173  df-mopn 12174  df-top 12179  df-topon 12192  df-bases 12224  df-ntr 12279  df-cn 12371  df-cnp 12372  df-tx 12436  df-limced 12808  df-dvap 12809
This theorem is referenced by:  dviaddf  12852
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