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Theorem dvimulf 13739
Description: The product 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
dvimulf  |-  ( ph  ->  ( S  _D  ( F  oF  x.  G
) )  =  ( ( ( S  _D  F )  oF  x.  G )  oF  +  ( ( S  _D  G )  oF  x.  F
) ) )

Proof of Theorem dvimulf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvaddf.f . . . . 5  |-  ( ph  ->  F : X --> CC )
21adantr 276 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  F : X --> CC )
3 dviaddf.x . . . . 5  |-  ( ph  ->  X  C_  S )
43adantr 276 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  X  C_  S )
5 dvaddf.g . . . . 5  |-  ( ph  ->  G : X --> CC )
65adantr 276 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  G : X --> CC )
7 dvaddf.s . . . . 5  |-  ( ph  ->  S  e.  { RR ,  CC } )
87adantr 276 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  S  e.  { RR ,  CC } )
9 dvaddf.df . . . . . 6  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
109eleq2d 2245 . . . . 5  |-  ( ph  ->  ( x  e.  dom  ( S  _D  F
)  <->  x  e.  X
) )
1110biimpar 297 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  F ) )
12 dvaddf.dg . . . . . 6  |-  ( ph  ->  dom  ( S  _D  G )  =  X )
1312eleq2d 2245 . . . . 5  |-  ( ph  ->  ( x  e.  dom  ( S  _D  G
)  <->  x  e.  X
) )
1413biimpar 297 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  G ) )
152, 4, 6, 8, 11, 14dvmulxx 13737 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  ( F  oF  x.  G
) ) `  x
)  =  ( ( ( ( S  _D  F ) `  x
)  x.  ( G `
 x ) )  +  ( ( ( S  _D  G ) `
 x )  x.  ( F `  x
) ) ) )
1615mpteq2dva 4088 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( ( S  _D  ( F  oF  x.  G ) ) `  x ) )  =  ( x  e.  X  |->  ( ( ( ( S  _D  F ) `
 x )  x.  ( G `  x
) )  +  ( ( ( S  _D  G ) `  x
)  x.  ( F `
 x ) ) ) ) )
17 cnex 7910 . . . . . . 7  |-  CC  e.  _V
1817a1i 9 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
19 mulcl 7913 . . . . . . . 8  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
2019adantl 277 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
217, 3ssexd 4138 . . . . . . 7  |-  ( ph  ->  X  e.  _V )
22 inidm 3342 . . . . . . 7  |-  ( X  i^i  X )  =  X
2320, 1, 5, 21, 21, 22off 6085 . . . . . 6  |-  ( ph  ->  ( F  oF  x.  G ) : X --> CC )
24 elpm2r 6656 . . . . . 6  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( ( F  oF  x.  G ) : X --> CC  /\  X  C_  S ) )  -> 
( F  oF  x.  G )  e.  ( CC  ^pm  S
) )
2518, 7, 23, 3, 24syl22anc 1239 . . . . 5  |-  ( ph  ->  ( F  oF  x.  G )  e.  ( CC  ^pm  S
) )
26 dvfgg 13726 . . . . 5  |-  ( ( S  e.  { RR ,  CC }  /\  ( F  oF  x.  G
)  e.  ( CC 
^pm  S ) )  ->  ( S  _D  ( F  oF  x.  G ) ) : dom  ( S  _D  ( F  oF  x.  G ) ) --> CC )
277, 25, 26syl2anc 411 . . . 4  |-  ( ph  ->  ( S  _D  ( F  oF  x.  G
) ) : dom  ( S  _D  ( F  oF  x.  G
) ) --> CC )
28 recnprss 13725 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
297, 28syl 14 . . . . . . 7  |-  ( ph  ->  S  C_  CC )
3029, 23, 3dvbss 13723 . . . . . 6  |-  ( ph  ->  dom  ( S  _D  ( F  oF  x.  G ) )  C_  X )
31 reldvg 13717 . . . . . . . . 9  |-  ( ( S  C_  CC  /\  ( F  oF  x.  G
)  e.  ( CC 
^pm  S ) )  ->  Rel  ( S  _D  ( F  oF  x.  G ) ) )
3229, 25, 31syl2anc 411 . . . . . . . 8  |-  ( ph  ->  Rel  ( S  _D  ( F  oF  x.  G ) ) )
3332adantr 276 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  Rel  ( S  _D  ( F  oF  x.  G
) ) )
3429adantr 276 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  S  C_  CC )
35 elpm2r 6656 . . . . . . . . . . . . 13  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( F : X --> CC  /\  X  C_  S ) )  ->  F  e.  ( CC  ^pm  S )
)
3618, 7, 1, 3, 35syl22anc 1239 . . . . . . . . . . . 12  |-  ( ph  ->  F  e.  ( CC 
^pm  S ) )
37 dvfgg 13726 . . . . . . . . . . . 12  |-  ( ( S  e.  { RR ,  CC }  /\  F  e.  ( CC  ^pm  S
) )  ->  ( S  _D  F ) : dom  ( S  _D  F ) --> CC )
387, 36, 37syl2anc 411 . . . . . . . . . . 11  |-  ( ph  ->  ( S  _D  F
) : dom  ( S  _D  F ) --> CC )
39 ffun 5360 . . . . . . . . . . 11  |-  ( ( S  _D  F ) : dom  ( S  _D  F ) --> CC 
->  Fun  ( S  _D  F ) )
40 funfvbrb 5621 . . . . . . . . . . 11  |-  ( Fun  ( S  _D  F
)  ->  ( x  e.  dom  ( S  _D  F )  <->  x ( S  _D  F ) ( ( S  _D  F
) `  x )
) )
4138, 39, 403syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  dom  ( S  _D  F
)  <->  x ( S  _D  F ) ( ( S  _D  F
) `  x )
) )
4241adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  dom  ( S  _D  F )  <->  x ( S  _D  F ) ( ( S  _D  F
) `  x )
) )
4311, 42mpbid 147 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  F
) ( ( S  _D  F ) `  x ) )
44 elpm2r 6656 . . . . . . . . . . . . 13  |-  ( ( ( CC  e.  _V  /\  S  e.  { RR ,  CC } )  /\  ( G : X --> CC  /\  X  C_  S ) )  ->  G  e.  ( CC  ^pm  S )
)
4518, 7, 5, 3, 44syl22anc 1239 . . . . . . . . . . . 12  |-  ( ph  ->  G  e.  ( CC 
^pm  S ) )
46 dvfgg 13726 . . . . . . . . . . . 12  |-  ( ( S  e.  { RR ,  CC }  /\  G  e.  ( CC  ^pm  S
) )  ->  ( S  _D  G ) : dom  ( S  _D  G ) --> CC )
477, 45, 46syl2anc 411 . . . . . . . . . . 11  |-  ( ph  ->  ( S  _D  G
) : dom  ( S  _D  G ) --> CC )
48 ffun 5360 . . . . . . . . . . 11  |-  ( ( S  _D  G ) : dom  ( S  _D  G ) --> CC 
->  Fun  ( S  _D  G ) )
49 funfvbrb 5621 . . . . . . . . . . 11  |-  ( Fun  ( S  _D  G
)  ->  ( x  e.  dom  ( S  _D  G )  <->  x ( S  _D  G ) ( ( S  _D  G
) `  x )
) )
5047, 48, 493syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  dom  ( S  _D  G
)  <->  x ( S  _D  G ) ( ( S  _D  G
) `  x )
) )
5150adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  dom  ( S  _D  G )  <->  x ( S  _D  G ) ( ( S  _D  G
) `  x )
) )
5214, 51mpbid 147 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  G
) ( ( S  _D  G ) `  x ) )
53 eqid 2175 . . . . . . . 8  |-  ( MetOpen `  ( abs  o.  -  )
)  =  ( MetOpen `  ( abs  o.  -  )
)
542, 4, 6, 34, 43, 52, 53dvmulxxbr 13735 . . . . . . 7  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  ( F  oF  x.  G
) ) ( ( ( ( S  _D  F ) `  x
)  x.  ( G `
 x ) )  +  ( ( ( S  _D  G ) `
 x )  x.  ( F `  x
) ) ) )
55 releldm 4855 . . . . . . 7  |-  ( ( Rel  ( S  _D  ( F  oF  x.  G ) )  /\  x ( S  _D  ( F  oF  x.  G ) ) ( ( ( ( S  _D  F ) `  x )  x.  ( G `  x )
)  +  ( ( ( S  _D  G
) `  x )  x.  ( F `  x
) ) ) )  ->  x  e.  dom  ( S  _D  ( F  oF  x.  G
) ) )
5633, 54, 55syl2anc 411 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  ( F  oF  x.  G ) ) )
5730, 56eqelssd 3172 . . . . 5  |-  ( ph  ->  dom  ( S  _D  ( F  oF  x.  G ) )  =  X )
5857feq2d 5345 . . . 4  |-  ( ph  ->  ( ( S  _D  ( F  oF  x.  G ) ) : dom  ( S  _D  ( F  oF  x.  G ) ) --> CC  <->  ( S  _D  ( F  oF  x.  G
) ) : X --> CC ) )
5927, 58mpbid 147 . . 3  |-  ( ph  ->  ( S  _D  ( F  oF  x.  G
) ) : X --> CC )
6059feqmptd 5561 . 2  |-  ( ph  ->  ( S  _D  ( F  oF  x.  G
) )  =  ( x  e.  X  |->  ( ( S  _D  ( F  oF  x.  G
) ) `  x
) ) )
619feq2d 5345 . . . . . 6  |-  ( ph  ->  ( ( S  _D  F ) : dom  ( S  _D  F
) --> CC  <->  ( S  _D  F ) : X --> CC ) )
6238, 61mpbid 147 . . . . 5  |-  ( ph  ->  ( S  _D  F
) : X --> CC )
6362ffvelcdmda 5643 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  CC )
645ffvelcdmda 5643 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  CC )
6563, 64mulcld 7952 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  ( G `
 x ) )  e.  CC )
6612feq2d 5345 . . . . . 6  |-  ( ph  ->  ( ( S  _D  G ) : dom  ( S  _D  G
) --> CC  <->  ( S  _D  G ) : X --> CC ) )
6747, 66mpbid 147 . . . . 5  |-  ( ph  ->  ( S  _D  G
) : X --> CC )
6867ffvelcdmda 5643 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  G
) `  x )  e.  CC )
691ffvelcdmda 5643 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  CC )
7068, 69mulcld 7952 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  G ) `  x
)  x.  ( F `
 x ) )  e.  CC )
7162feqmptd 5561 . . . 4  |-  ( ph  ->  ( S  _D  F
)  =  ( x  e.  X  |->  ( ( S  _D  F ) `
 x ) ) )
725feqmptd 5561 . . . 4  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
7321, 63, 64, 71, 72offval2 6088 . . 3  |-  ( ph  ->  ( ( S  _D  F )  oF  x.  G )  =  ( x  e.  X  |->  ( ( ( S  _D  F ) `  x )  x.  ( G `  x )
) ) )
7467feqmptd 5561 . . . 4  |-  ( ph  ->  ( S  _D  G
)  =  ( x  e.  X  |->  ( ( S  _D  G ) `
 x ) ) )
751feqmptd 5561 . . . 4  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
7621, 68, 69, 74, 75offval2 6088 . . 3  |-  ( ph  ->  ( ( S  _D  G )  oF  x.  F )  =  ( x  e.  X  |->  ( ( ( S  _D  G ) `  x )  x.  ( F `  x )
) ) )
7721, 65, 70, 73, 76offval2 6088 . 2  |-  ( ph  ->  ( ( ( S  _D  F )  oF  x.  G )  oF  +  ( ( S  _D  G
)  oF  x.  F ) )  =  ( x  e.  X  |->  ( ( ( ( S  _D  F ) `
 x )  x.  ( G `  x
) )  +  ( ( ( S  _D  G ) `  x
)  x.  ( F `
 x ) ) ) ) )
7816, 60, 773eqtr4d 2218 1  |-  ( ph  ->  ( S  _D  ( F  oF  x.  G
) )  =  ( ( ( S  _D  F )  oF  x.  G )  oF  +  ( ( S  _D  G )  oF  x.  F
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2146   _Vcvv 2735    C_ wss 3127   {cpr 3590   class class class wbr 3998    |-> cmpt 4059   dom cdm 4620    o. ccom 4624   Rel wrel 4625   Fun wfun 5202   -->wf 5204   ` cfv 5208  (class class class)co 5865    oFcof 6071    ^pm cpm 6639   CCcc 7784   RRcr 7785    + caddc 7789    x. cmul 7791    - cmin 8102   abscabs 10972   MetOpencmopn 13054    _D cdv 13693
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906  ax-addf 7908  ax-mulf 7909
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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-isom 5217  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-of 6073  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-map 6640  df-pm 6641  df-sup 6973  df-inf 6974  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-xneg 9741  df-xadd 9742  df-seqfrec 10414  df-exp 10488  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-rest 12610  df-topgen 12629  df-psmet 13056  df-xmet 13057  df-met 13058  df-bl 13059  df-mopn 13060  df-top 13065  df-topon 13078  df-bases 13110  df-ntr 13165  df-cn 13257  df-cnp 13258  df-tx 13322  df-cncf 13627  df-limced 13694  df-dvap 13695
This theorem is referenced by:  dvexp  13744  dvmptmulx  13751
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