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Theorem dvmulf 19782
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 452 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  F : X --> CC )
3 dvaddf.df . . . . . 6  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
4 dvbsss 19742 . . . . . 6  |-  dom  ( S  _D  F )  C_  S
53, 4syl6eqssr 3359 . . . . 5  |-  ( ph  ->  X  C_  S )
65adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  X  C_  S )
7 dvaddf.g . . . . 5  |-  ( ph  ->  G : X --> CC )
87adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  G : X --> CC )
9 dvaddf.s . . . . 5  |-  ( ph  ->  S  e.  { RR ,  CC } )
109adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  S  e.  { RR ,  CC } )
113eleq2d 2471 . . . . 5  |-  ( ph  ->  ( x  e.  dom  ( S  _D  F
)  <->  x  e.  X
) )
1211biimpar 472 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  F ) )
13 dvaddf.dg . . . . . 6  |-  ( ph  ->  dom  ( S  _D  G )  =  X )
1413eleq2d 2471 . . . . 5  |-  ( ph  ->  ( x  e.  dom  ( S  _D  G
)  <->  x  e.  X
) )
1514biimpar 472 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  G ) )
162, 6, 8, 6, 10, 12, 15dvmul 19780 . . 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 4255 . 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 19746 . . . . 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 19744 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
219, 20syl 16 . . . . . . 7  |-  ( ph  ->  S  C_  CC )
22 mulcl 9030 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
2322adantl 453 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
249, 5ssexd 4310 . . . . . . . 8  |-  ( ph  ->  X  e.  _V )
25 inidm 3510 . . . . . . . 8  |-  ( X  i^i  X )  =  X
2623, 1, 7, 24, 24, 25off 6279 . . . . . . 7  |-  ( ph  ->  ( F  o F  x.  G ) : X --> CC )
2721, 26, 5dvbss 19741 . . . . . 6  |-  ( ph  ->  dom  ( S  _D  ( F  o F  x.  G ) )  C_  X )
2821adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  S  C_  CC )
29 fvex 5701 . . . . . . . . . . 11  |-  ( ( S  _D  F ) `
 x )  e. 
_V
3029a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  _V )
31 fvex 5701 . . . . . . . . . . 11  |-  ( ( S  _D  G ) `
 x )  e. 
_V
3231a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  G
) `  x )  e.  _V )
33 dvfg 19746 . . . . . . . . . . . . . 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 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  ( S  _D  F ) : dom  ( S  _D  F ) --> CC )
36 ffun 5552 . . . . . . . . . . . 12  |-  ( ( S  _D  F ) : dom  ( S  _D  F ) --> CC 
->  Fun  ( S  _D  F ) )
37 funfvbrb 5802 . . . . . . . . . . . 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 202 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  F
) ( ( S  _D  F ) `  x ) )
40 dvfg 19746 . . . . . . . . . . . . . 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 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  ( S  _D  G ) : dom  ( S  _D  G ) --> CC )
43 ffun 5552 . . . . . . . . . . . 12  |-  ( ( S  _D  G ) : dom  ( S  _D  G ) --> CC 
->  Fun  ( S  _D  G ) )
44 funfvbrb 5802 . . . . . . . . . . . 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 202 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  G
) ( ( S  _D  G ) `  x ) )
47 eqid 2404 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
482, 6, 8, 6, 28, 30, 32, 39, 46, 47dvmulbr 19778 . . . . . . . . 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 19710 . . . . . . . . . 10  |-  Rel  ( S  _D  ( F  o F  x.  G )
)
5049releldmi 5065 . . . . . . . . 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 424 . . . . . . 7  |-  ( ph  ->  ( x  e.  X  ->  x  e.  dom  ( S  _D  ( F  o F  x.  G )
) ) )
5352ssrdv 3314 . . . . . 6  |-  ( ph  ->  X  C_  dom  ( S  _D  ( F  o F  x.  G )
) )
5427, 53eqssd 3325 . . . . 5  |-  ( ph  ->  dom  ( S  _D  ( F  o F  x.  G ) )  =  X )
5554feq2d 5540 . . . 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 202 . . 3  |-  ( ph  ->  ( S  _D  ( F  o F  x.  G
) ) : X --> CC )
5756feqmptd 5738 . 2  |-  ( ph  ->  ( S  _D  ( F  o F  x.  G
) )  =  ( x  e.  X  |->  ( ( S  _D  ( F  o F  x.  G
) ) `  x
) ) )
58 ovex 6065 . . . 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 6065 . . . 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 5701 . . . . 5  |-  ( G `
 x )  e. 
_V
6362a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  _V )
643feq2d 5540 . . . . . 6  |-  ( ph  ->  ( ( S  _D  F ) : dom  ( S  _D  F
) --> CC  <->  ( S  _D  F ) : X --> CC ) )
6534, 64mpbid 202 . . . . 5  |-  ( ph  ->  ( S  _D  F
) : X --> CC )
6665feqmptd 5738 . . . 4  |-  ( ph  ->  ( S  _D  F
)  =  ( x  e.  X  |->  ( ( S  _D  F ) `
 x ) ) )
677feqmptd 5738 . . . 4  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
6824, 30, 63, 66, 67offval2 6281 . . 3  |-  ( ph  ->  ( ( S  _D  F )  o F  x.  G )  =  ( x  e.  X  |->  ( ( ( S  _D  F ) `  x )  x.  ( G `  x )
) ) )
69 fvex 5701 . . . . 5  |-  ( F `
 x )  e. 
_V
7069a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  _V )
7113feq2d 5540 . . . . . 6  |-  ( ph  ->  ( ( S  _D  G ) : dom  ( S  _D  G
) --> CC  <->  ( S  _D  G ) : X --> CC ) )
7241, 71mpbid 202 . . . . 5  |-  ( ph  ->  ( S  _D  G
) : X --> CC )
7372feqmptd 5738 . . . 4  |-  ( ph  ->  ( S  _D  G
)  =  ( x  e.  X  |->  ( ( S  _D  G ) `
 x ) ) )
741feqmptd 5738 . . . 4  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
7524, 32, 70, 73, 74offval2 6281 . . 3  |-  ( ph  ->  ( ( S  _D  G )  o F  x.  F )  =  ( x  e.  X  |->  ( ( ( S  _D  G ) `  x )  x.  ( F `  x )
) ) )
7624, 59, 61, 68, 75offval2 6281 . 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 2446 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    C_ wss 3280   {cpr 3775   class class class wbr 4172    e. cmpt 4226   dom cdm 4837   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262   CCcc 8944   RRcr 8945    + caddc 8949    x. cmul 8951   TopOpenctopn 13604  ℂfldccnfld 16658    _D cdv 19703
This theorem is referenced by:  dvcmulf  19784  dvexp  19792  dvmptmul  19800  expgrowth  27420
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707
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