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Theorem dvmulf 21376
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 } )
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
dvmulf  |-  ( ph  ->  ( S  _D  ( F  oF  x.  G
) )  =  ( ( ( S  _D  F )  oF  x.  G )  oF  +  ( ( S  _D  G )  oF  x.  F
) ) )

Proof of Theorem dvmulf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvaddf.f . . . . 5  |-  ( ph  ->  F : X --> CC )
21adantr 462 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  F : X --> CC )
3 dvaddf.df . . . . . 6  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
4 dvbsss 21336 . . . . . 6  |-  dom  ( S  _D  F )  C_  S
53, 4syl6eqssr 3404 . . . . 5  |-  ( ph  ->  X  C_  S )
65adantr 462 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  X  C_  S )
7 dvaddf.g . . . . 5  |-  ( ph  ->  G : X --> CC )
87adantr 462 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  G : X --> CC )
9 dvaddf.s . . . . 5  |-  ( ph  ->  S  e.  { RR ,  CC } )
109adantr 462 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  S  e.  { RR ,  CC } )
113eleq2d 2508 . . . . 5  |-  ( ph  ->  ( x  e.  dom  ( S  _D  F
)  <->  x  e.  X
) )
1211biimpar 482 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  F ) )
13 dvaddf.dg . . . . . 6  |-  ( ph  ->  dom  ( S  _D  G )  =  X )
1413eleq2d 2508 . . . . 5  |-  ( ph  ->  ( x  e.  dom  ( S  _D  G
)  <->  x  e.  X
) )
1514biimpar 482 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  G ) )
162, 6, 8, 6, 10, 12, 15dvmul 21374 . . 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
) ) ) )
1716mpteq2dva 4375 . 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 ) ) ) ) )
18 dvfg 21340 . . . . 5  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  ( F  oF  x.  G ) ) : dom  ( S  _D  ( F  oF  x.  G )
) --> CC )
199, 18syl 16 . . . 4  |-  ( ph  ->  ( S  _D  ( F  oF  x.  G
) ) : dom  ( S  _D  ( F  oF  x.  G
) ) --> CC )
20 recnprss 21338 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
219, 20syl 16 . . . . . . 7  |-  ( ph  ->  S  C_  CC )
22 mulcl 9362 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
2322adantl 463 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
249, 5ssexd 4436 . . . . . . . 8  |-  ( ph  ->  X  e.  _V )
25 inidm 3556 . . . . . . . 8  |-  ( X  i^i  X )  =  X
2623, 1, 7, 24, 24, 25off 6333 . . . . . . 7  |-  ( ph  ->  ( F  oF  x.  G ) : X --> CC )
2721, 26, 5dvbss 21335 . . . . . 6  |-  ( ph  ->  dom  ( S  _D  ( F  oF  x.  G ) )  C_  X )
2821adantr 462 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  S  C_  CC )
29 fvex 5698 . . . . . . . . . . 11  |-  ( ( S  _D  F ) `
 x )  e. 
_V
3029a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  _V )
31 fvex 5698 . . . . . . . . . . 11  |-  ( ( S  _D  G ) `
 x )  e. 
_V
3231a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  G
) `  x )  e.  _V )
33 dvfg 21340 . . . . . . . . . . . . . 14  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  F ) : dom  ( S  _D  F
) --> CC )
349, 33syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S  _D  F
) : dom  ( S  _D  F ) --> CC )
3534adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  ( S  _D  F ) : dom  ( S  _D  F ) --> CC )
36 ffun 5558 . . . . . . . . . . . 12  |-  ( ( S  _D  F ) : dom  ( S  _D  F ) --> CC 
->  Fun  ( S  _D  F ) )
37 funfvbrb 5813 . . . . . . . . . . . 12  |-  ( Fun  ( S  _D  F
)  ->  ( x  e.  dom  ( S  _D  F )  <->  x ( S  _D  F ) ( ( S  _D  F
) `  x )
) )
3835, 36, 373syl 20 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  dom  ( S  _D  F )  <->  x ( S  _D  F ) ( ( S  _D  F
) `  x )
) )
3912, 38mpbid 210 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  F
) ( ( S  _D  F ) `  x ) )
40 dvfg 21340 . . . . . . . . . . . . . 14  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  G ) : dom  ( S  _D  G
) --> CC )
419, 40syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S  _D  G
) : dom  ( S  _D  G ) --> CC )
4241adantr 462 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  ( S  _D  G ) : dom  ( S  _D  G ) --> CC )
43 ffun 5558 . . . . . . . . . . . 12  |-  ( ( S  _D  G ) : dom  ( S  _D  G ) --> CC 
->  Fun  ( S  _D  G ) )
44 funfvbrb 5813 . . . . . . . . . . . 12  |-  ( Fun  ( S  _D  G
)  ->  ( x  e.  dom  ( S  _D  G )  <->  x ( S  _D  G ) ( ( S  _D  G
) `  x )
) )
4542, 43, 443syl 20 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  dom  ( S  _D  G )  <->  x ( S  _D  G ) ( ( S  _D  G
) `  x )
) )
4615, 45mpbid 210 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  G
) ( ( S  _D  G ) `  x ) )
47 eqid 2441 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
482, 6, 8, 6, 28, 30, 32, 39, 46, 47dvmulbr 21372 . . . . . . . . 9  |-  ( (
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
) ) ) )
49 reldv 21304 . . . . . . . . . 10  |-  Rel  ( S  _D  ( F  oF  x.  G )
)
5049releldmi 5072 . . . . . . . . 9  |-  ( 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 )
) )
5148, 50syl 16 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  ( F  oF  x.  G ) ) )
5251ex 434 . . . . . . 7  |-  ( ph  ->  ( x  e.  X  ->  x  e.  dom  ( S  _D  ( F  oF  x.  G )
) ) )
5352ssrdv 3359 . . . . . 6  |-  ( ph  ->  X  C_  dom  ( S  _D  ( F  oF  x.  G )
) )
5427, 53eqssd 3370 . . . . 5  |-  ( ph  ->  dom  ( S  _D  ( F  oF  x.  G ) )  =  X )
5554feq2d 5544 . . . 4  |-  ( ph  ->  ( ( S  _D  ( F  oF  x.  G ) ) : dom  ( S  _D  ( F  oF  x.  G ) ) --> CC  <->  ( S  _D  ( F  oF  x.  G
) ) : X --> CC ) )
5619, 55mpbid 210 . . 3  |-  ( ph  ->  ( S  _D  ( F  oF  x.  G
) ) : X --> CC )
5756feqmptd 5741 . 2  |-  ( ph  ->  ( S  _D  ( F  oF  x.  G
) )  =  ( x  e.  X  |->  ( ( S  _D  ( F  oF  x.  G
) ) `  x
) ) )
58 ovex 6115 . . . 4  |-  ( ( ( S  _D  F
) `  x )  x.  ( G `  x
) )  e.  _V
5958a1i 11 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  ( G `
 x ) )  e.  _V )
60 ovex 6115 . . . 4  |-  ( ( ( S  _D  G
) `  x )  x.  ( F `  x
) )  e.  _V
6160a1i 11 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  G ) `  x
)  x.  ( F `
 x ) )  e.  _V )
62 fvex 5698 . . . . 5  |-  ( G `
 x )  e. 
_V
6362a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  _V )
643feq2d 5544 . . . . . 6  |-  ( ph  ->  ( ( S  _D  F ) : dom  ( S  _D  F
) --> CC  <->  ( S  _D  F ) : X --> CC ) )
6534, 64mpbid 210 . . . . 5  |-  ( ph  ->  ( S  _D  F
) : X --> CC )
6665feqmptd 5741 . . . 4  |-  ( ph  ->  ( S  _D  F
)  =  ( x  e.  X  |->  ( ( S  _D  F ) `
 x ) ) )
677feqmptd 5741 . . . 4  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
6824, 30, 63, 66, 67offval2 6335 . . 3  |-  ( ph  ->  ( ( S  _D  F )  oF  x.  G )  =  ( x  e.  X  |->  ( ( ( S  _D  F ) `  x )  x.  ( G `  x )
) ) )
69 fvex 5698 . . . . 5  |-  ( F `
 x )  e. 
_V
7069a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  _V )
7113feq2d 5544 . . . . . 6  |-  ( ph  ->  ( ( S  _D  G ) : dom  ( S  _D  G
) --> CC  <->  ( S  _D  G ) : X --> CC ) )
7241, 71mpbid 210 . . . . 5  |-  ( ph  ->  ( S  _D  G
) : X --> CC )
7372feqmptd 5741 . . . 4  |-  ( ph  ->  ( S  _D  G
)  =  ( x  e.  X  |->  ( ( S  _D  G ) `
 x ) ) )
741feqmptd 5741 . . . 4  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
7524, 32, 70, 73, 74offval2 6335 . . 3  |-  ( ph  ->  ( ( S  _D  G )  oF  x.  F )  =  ( x  e.  X  |->  ( ( ( S  _D  G ) `  x )  x.  ( F `  x )
) ) )
7624, 59, 61, 68, 75offval2 6335 . 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 ) ) ) ) )
7717, 57, 763eqtr4d 2483 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970    C_ wss 3325   {cpr 3876   class class class wbr 4289    e. cmpt 4347   dom cdm 4836   Fun wfun 5409   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317   CCcc 9276   RRcr 9277    + caddc 9281    x. cmul 9283   TopOpenctopn 14356  ℂfldccnfld 17777    _D cdv 21297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-icc 11303  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301
This theorem is referenced by:  dvcmulf  21378  dvexp  21386  dvmptmul  21394  expgrowth  29534
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