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Theorem dvmulf 19831
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  o F  x.  G
) )  =  ( ( ( S  _D  F )  o F  x.  G )  o F  +  ( ( S  _D  G )  o F  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 453 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  F : X --> CC )
3 dvaddf.df . . . . . 6  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
4 dvbsss 19791 . . . . . 6  |-  dom  ( S  _D  F )  C_  S
53, 4syl6eqssr 3401 . . . . 5  |-  ( ph  ->  X  C_  S )
65adantr 453 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  X  C_  S )
7 dvaddf.g . . . . 5  |-  ( ph  ->  G : X --> CC )
87adantr 453 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  G : X --> CC )
9 dvaddf.s . . . . 5  |-  ( ph  ->  S  e.  { RR ,  CC } )
109adantr 453 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  S  e.  { RR ,  CC } )
113eleq2d 2505 . . . . 5  |-  ( ph  ->  ( x  e.  dom  ( S  _D  F
)  <->  x  e.  X
) )
1211biimpar 473 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  F ) )
13 dvaddf.dg . . . . . 6  |-  ( ph  ->  dom  ( S  _D  G )  =  X )
1413eleq2d 2505 . . . . 5  |-  ( ph  ->  ( x  e.  dom  ( S  _D  G
)  <->  x  e.  X
) )
1514biimpar 473 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  G ) )
162, 6, 8, 6, 10, 12, 15dvmul 19829 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  ( F  o F  x.  G
) ) `  x
)  =  ( ( ( ( S  _D  F ) `  x
)  x.  ( G `
 x ) )  +  ( ( ( S  _D  G ) `
 x )  x.  ( F `  x
) ) ) )
1716mpteq2dva 4297 . 2  |-  ( ph  ->  ( x  e.  X  |->  ( ( S  _D  ( F  o F  x.  G ) ) `  x ) )  =  ( x  e.  X  |->  ( ( ( ( S  _D  F ) `
 x )  x.  ( G `  x
) )  +  ( ( ( S  _D  G ) `  x
)  x.  ( F `
 x ) ) ) ) )
18 dvfg 19795 . . . . 5  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  ( F  o F  x.  G ) ) : dom  ( S  _D  ( F  o F  x.  G )
) --> CC )
199, 18syl 16 . . . 4  |-  ( ph  ->  ( S  _D  ( F  o F  x.  G
) ) : dom  ( S  _D  ( F  o F  x.  G
) ) --> CC )
20 recnprss 19793 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
219, 20syl 16 . . . . . . 7  |-  ( ph  ->  S  C_  CC )
22 mulcl 9076 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
2322adantl 454 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
249, 5ssexd 4352 . . . . . . . 8  |-  ( ph  ->  X  e.  _V )
25 inidm 3552 . . . . . . . 8  |-  ( X  i^i  X )  =  X
2623, 1, 7, 24, 24, 25off 6322 . . . . . . 7  |-  ( ph  ->  ( F  o F  x.  G ) : X --> CC )
2721, 26, 5dvbss 19790 . . . . . 6  |-  ( ph  ->  dom  ( S  _D  ( F  o F  x.  G ) )  C_  X )
2821adantr 453 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  S  C_  CC )
29 fvex 5744 . . . . . . . . . . 11  |-  ( ( S  _D  F ) `
 x )  e. 
_V
3029a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  _V )
31 fvex 5744 . . . . . . . . . . 11  |-  ( ( S  _D  G ) `
 x )  e. 
_V
3231a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  G
) `  x )  e.  _V )
33 dvfg 19795 . . . . . . . . . . . . . 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 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  ( S  _D  F ) : dom  ( S  _D  F ) --> CC )
36 ffun 5595 . . . . . . . . . . . 12  |-  ( ( S  _D  F ) : dom  ( S  _D  F ) --> CC 
->  Fun  ( S  _D  F ) )
37 funfvbrb 5845 . . . . . . . . . . . 12  |-  ( Fun  ( S  _D  F
)  ->  ( x  e.  dom  ( S  _D  F )  <->  x ( S  _D  F ) ( ( S  _D  F
) `  x )
) )
3835, 36, 373syl 19 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  dom  ( S  _D  F )  <->  x ( S  _D  F ) ( ( S  _D  F
) `  x )
) )
3912, 38mpbid 203 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  F
) ( ( S  _D  F ) `  x ) )
40 dvfg 19795 . . . . . . . . . . . . . 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 453 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  ( S  _D  G ) : dom  ( S  _D  G ) --> CC )
43 ffun 5595 . . . . . . . . . . . 12  |-  ( ( S  _D  G ) : dom  ( S  _D  G ) --> CC 
->  Fun  ( S  _D  G ) )
44 funfvbrb 5845 . . . . . . . . . . . 12  |-  ( Fun  ( S  _D  G
)  ->  ( x  e.  dom  ( S  _D  G )  <->  x ( S  _D  G ) ( ( S  _D  G
) `  x )
) )
4542, 43, 443syl 19 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  dom  ( S  _D  G )  <->  x ( S  _D  G ) ( ( S  _D  G
) `  x )
) )
4615, 45mpbid 203 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  G
) ( ( S  _D  G ) `  x ) )
47 eqid 2438 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
482, 6, 8, 6, 28, 30, 32, 39, 46, 47dvmulbr 19827 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  ( F  o F  x.  G
) ) ( ( ( ( S  _D  F ) `  x
)  x.  ( G `
 x ) )  +  ( ( ( S  _D  G ) `
 x )  x.  ( F `  x
) ) ) )
49 reldv 19759 . . . . . . . . . 10  |-  Rel  ( S  _D  ( F  o F  x.  G )
)
5049releldmi 5108 . . . . . . . . 9  |-  ( x ( S  _D  ( F  o F  x.  G
) ) ( ( ( ( S  _D  F ) `  x
)  x.  ( G `
 x ) )  +  ( ( ( S  _D  G ) `
 x )  x.  ( F `  x
) ) )  ->  x  e.  dom  ( S  _D  ( F  o F  x.  G )
) )
5148, 50syl 16 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  ( F  o F  x.  G ) ) )
5251ex 425 . . . . . . 7  |-  ( ph  ->  ( x  e.  X  ->  x  e.  dom  ( S  _D  ( F  o F  x.  G )
) ) )
5352ssrdv 3356 . . . . . 6  |-  ( ph  ->  X  C_  dom  ( S  _D  ( F  o F  x.  G )
) )
5427, 53eqssd 3367 . . . . 5  |-  ( ph  ->  dom  ( S  _D  ( F  o F  x.  G ) )  =  X )
5554feq2d 5583 . . . 4  |-  ( ph  ->  ( ( S  _D  ( F  o F  x.  G ) ) : dom  ( S  _D  ( F  o F  x.  G ) ) --> CC  <->  ( S  _D  ( F  o F  x.  G
) ) : X --> CC ) )
5619, 55mpbid 203 . . 3  |-  ( ph  ->  ( S  _D  ( F  o F  x.  G
) ) : X --> CC )
5756feqmptd 5781 . 2  |-  ( ph  ->  ( S  _D  ( F  o F  x.  G
) )  =  ( x  e.  X  |->  ( ( S  _D  ( F  o F  x.  G
) ) `  x
) ) )
58 ovex 6108 . . . 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 6108 . . . 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 5744 . . . . 5  |-  ( G `
 x )  e. 
_V
6362a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  _V )
643feq2d 5583 . . . . . 6  |-  ( ph  ->  ( ( S  _D  F ) : dom  ( S  _D  F
) --> CC  <->  ( S  _D  F ) : X --> CC ) )
6534, 64mpbid 203 . . . . 5  |-  ( ph  ->  ( S  _D  F
) : X --> CC )
6665feqmptd 5781 . . . 4  |-  ( ph  ->  ( S  _D  F
)  =  ( x  e.  X  |->  ( ( S  _D  F ) `
 x ) ) )
677feqmptd 5781 . . . 4  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
6824, 30, 63, 66, 67offval2 6324 . . 3  |-  ( ph  ->  ( ( S  _D  F )  o F  x.  G )  =  ( x  e.  X  |->  ( ( ( S  _D  F ) `  x )  x.  ( G `  x )
) ) )
69 fvex 5744 . . . . 5  |-  ( F `
 x )  e. 
_V
7069a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  _V )
7113feq2d 5583 . . . . . 6  |-  ( ph  ->  ( ( S  _D  G ) : dom  ( S  _D  G
) --> CC  <->  ( S  _D  G ) : X --> CC ) )
7241, 71mpbid 203 . . . . 5  |-  ( ph  ->  ( S  _D  G
) : X --> CC )
7372feqmptd 5781 . . . 4  |-  ( ph  ->  ( S  _D  G
)  =  ( x  e.  X  |->  ( ( S  _D  G ) `
 x ) ) )
741feqmptd 5781 . . . 4  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
7524, 32, 70, 73, 74offval2 6324 . . 3  |-  ( ph  ->  ( ( S  _D  G )  o F  x.  F )  =  ( x  e.  X  |->  ( ( ( S  _D  G ) `  x )  x.  ( F `  x )
) ) )
7624, 59, 61, 68, 75offval2 6324 . 2  |-  ( ph  ->  ( ( ( S  _D  F )  o F  x.  G )  o F  +  ( ( S  _D  G
)  o F  x.  F ) )  =  ( x  e.  X  |->  ( ( ( ( S  _D  F ) `
 x )  x.  ( G `  x
) )  +  ( ( ( S  _D  G ) `  x
)  x.  ( F `
 x ) ) ) ) )
7717, 57, 763eqtr4d 2480 1  |-  ( ph  ->  ( S  _D  ( F  o F  x.  G
) )  =  ( ( ( S  _D  F )  o F  x.  G )  o F  +  ( ( S  _D  G )  o F  x.  F
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   {cpr 3817   class class class wbr 4214    e. cmpt 4268   dom cdm 4880   Fun wfun 5450   -->wf 5452   ` cfv 5456  (class class class)co 6083    o Fcof 6305   CCcc 8990   RRcr 8991    + caddc 8995    x. cmul 8997   TopOpenctopn 13651  ℂfldccnfld 16705    _D cdv 19752
This theorem is referenced by:  dvcmulf  19833  dvexp  19841  dvmptmul  19849  expgrowth  27531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-icc 10925  df-fz 11046  df-fzo 11138  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-fbas 16701  df-fg 16702  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-lp 17202  df-perf 17203  df-cn 17293  df-cnp 17294  df-haus 17381  df-tx 17596  df-hmeo 17789  df-fil 17880  df-fm 17972  df-flim 17973  df-flf 17974  df-xms 18352  df-ms 18353  df-tms 18354  df-cncf 18910  df-limc 19755  df-dv 19756
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