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Theorem dvimulf 13837
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 2247 . . . . 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 2247 . . . . 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 13835 . . 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 4090 . 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 7926 . . . . . . 7  |-  CC  e.  _V
1817a1i 9 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
19 mulcl 7929 . . . . . . . 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 4140 . . . . . . 7  |-  ( ph  ->  X  e.  _V )
22 inidm 3344 . . . . . . 7  |-  ( X  i^i  X )  =  X
2320, 1, 5, 21, 21, 22off 6089 . . . . . 6  |-  ( ph  ->  ( F  oF  x.  G ) : X --> CC )
24 elpm2r 6660 . . . . . 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 13824 . . . . 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 13823 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
297, 28syl 14 . . . . . . 7  |-  ( ph  ->  S  C_  CC )
3029, 23, 3dvbss 13821 . . . . . 6  |-  ( ph  ->  dom  ( S  _D  ( F  oF  x.  G ) )  C_  X )
31 reldvg 13815 . . . . . . . . 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 6660 . . . . . . . . . . . . 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 13824 . . . . . . . . . . . 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 5364 . . . . . . . . . . 11  |-  ( ( S  _D  F ) : dom  ( S  _D  F ) --> CC 
->  Fun  ( S  _D  F ) )
40 funfvbrb 5625 . . . . . . . . . . 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 6660 . . . . . . . . . . . . 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 13824 . . . . . . . . . . . 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 5364 . . . . . . . . . . 11  |-  ( ( S  _D  G ) : dom  ( S  _D  G ) --> CC 
->  Fun  ( S  _D  G ) )
49 funfvbrb 5625 . . . . . . . . . . 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 2177 . . . . . . . 8  |-  ( MetOpen `  ( abs  o.  -  )
)  =  ( MetOpen `  ( abs  o.  -  )
)
542, 4, 6, 34, 43, 52, 53dvmulxxbr 13833 . . . . . . 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 4858 . . . . . . 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 3174 . . . . 5  |-  ( ph  ->  dom  ( S  _D  ( F  oF  x.  G ) )  =  X )
5857feq2d 5349 . . . 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 5565 . 2  |-  ( ph  ->  ( S  _D  ( F  oF  x.  G
) )  =  ( x  e.  X  |->  ( ( S  _D  ( F  oF  x.  G
) ) `  x
) ) )
619feq2d 5349 . . . . . 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 5647 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  CC )
645ffvelcdmda 5647 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  CC )
6563, 64mulcld 7968 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  ( G `
 x ) )  e.  CC )
6612feq2d 5349 . . . . . 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 5647 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  G
) `  x )  e.  CC )
691ffvelcdmda 5647 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  CC )
7068, 69mulcld 7968 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  G ) `  x
)  x.  ( F `
 x ) )  e.  CC )
7162feqmptd 5565 . . . 4  |-  ( ph  ->  ( S  _D  F
)  =  ( x  e.  X  |->  ( ( S  _D  F ) `
 x ) ) )
725feqmptd 5565 . . . 4  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
7321, 63, 64, 71, 72offval2 6092 . . 3  |-  ( ph  ->  ( ( S  _D  F )  oF  x.  G )  =  ( x  e.  X  |->  ( ( ( S  _D  F ) `  x )  x.  ( G `  x )
) ) )
7467feqmptd 5565 . . . 4  |-  ( ph  ->  ( S  _D  G
)  =  ( x  e.  X  |->  ( ( S  _D  G ) `
 x ) ) )
751feqmptd 5565 . . . 4  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
7621, 68, 69, 74, 75offval2 6092 . . 3  |-  ( ph  ->  ( ( S  _D  G )  oF  x.  F )  =  ( x  e.  X  |->  ( ( ( S  _D  G ) `  x )  x.  ( F `  x )
) ) )
7721, 65, 70, 73, 76offval2 6092 . 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 2220 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 2148   _Vcvv 2737    C_ wss 3129   {cpr 3592   class class class wbr 4000    |-> cmpt 4061   dom cdm 4623    o. ccom 4627   Rel wrel 4628   Fun wfun 5206   -->wf 5208   ` cfv 5212  (class class class)co 5869    oFcof 6075    ^pm cpm 6643   CCcc 7800   RRcr 7801    + caddc 7805    x. cmul 7807    - cmin 8118   abscabs 10990   MetOpencmopn 13152    _D cdv 13791
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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922  ax-addf 7924  ax-mulf 7925
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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-of 6077  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-map 6644  df-pm 6645  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-xneg 9759  df-xadd 9760  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-rest 12638  df-topgen 12657  df-psmet 13154  df-xmet 13155  df-met 13156  df-bl 13157  df-mopn 13158  df-top 13163  df-topon 13176  df-bases 13208  df-ntr 13263  df-cn 13355  df-cnp 13356  df-tx 13420  df-cncf 13725  df-limced 13792  df-dvap 13793
This theorem is referenced by:  dvexp  13842  dvmptmulx  13849
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