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Theorem dvmulf 21545
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 465 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  F : X --> CC )
3 dvaddf.df . . . . . 6  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
4 dvbsss 21505 . . . . . 6  |-  dom  ( S  _D  F )  C_  S
53, 4syl6eqssr 3510 . . . . 5  |-  ( ph  ->  X  C_  S )
65adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  X  C_  S )
7 dvaddf.g . . . . 5  |-  ( ph  ->  G : X --> CC )
87adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  G : X --> CC )
9 dvaddf.s . . . . 5  |-  ( ph  ->  S  e.  { RR ,  CC } )
109adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  S  e.  { RR ,  CC } )
113eleq2d 2522 . . . . 5  |-  ( ph  ->  ( x  e.  dom  ( S  _D  F
)  <->  x  e.  X
) )
1211biimpar 485 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  F ) )
13 dvaddf.dg . . . . . 6  |-  ( ph  ->  dom  ( S  _D  G )  =  X )
1413eleq2d 2522 . . . . 5  |-  ( ph  ->  ( x  e.  dom  ( S  _D  G
)  <->  x  e.  X
) )
1514biimpar 485 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  G ) )
162, 6, 8, 6, 10, 12, 15dvmul 21543 . . 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 4481 . 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 21509 . . . . 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 21507 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
219, 20syl 16 . . . . . . 7  |-  ( ph  ->  S  C_  CC )
22 mulcl 9472 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
2322adantl 466 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
249, 5ssexd 4542 . . . . . . . 8  |-  ( ph  ->  X  e.  _V )
25 inidm 3662 . . . . . . . 8  |-  ( X  i^i  X )  =  X
2623, 1, 7, 24, 24, 25off 6439 . . . . . . 7  |-  ( ph  ->  ( F  oF  x.  G ) : X --> CC )
2721, 26, 5dvbss 21504 . . . . . 6  |-  ( ph  ->  dom  ( S  _D  ( F  oF  x.  G ) )  C_  X )
2821adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  S  C_  CC )
29 fvex 5804 . . . . . . . . . . 11  |-  ( ( S  _D  F ) `
 x )  e. 
_V
3029a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  _V )
31 fvex 5804 . . . . . . . . . . 11  |-  ( ( S  _D  G ) `
 x )  e. 
_V
3231a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  G
) `  x )  e.  _V )
33 dvfg 21509 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  ( S  _D  F ) : dom  ( S  _D  F ) --> CC )
36 ffun 5664 . . . . . . . . . . . 12  |-  ( ( S  _D  F ) : dom  ( S  _D  F ) --> CC 
->  Fun  ( S  _D  F ) )
37 funfvbrb 5920 . . . . . . . . . . . 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 21509 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  ( S  _D  G ) : dom  ( S  _D  G ) --> CC )
43 ffun 5664 . . . . . . . . . . . 12  |-  ( ( S  _D  G ) : dom  ( S  _D  G ) --> CC 
->  Fun  ( S  _D  G ) )
44 funfvbrb 5920 . . . . . . . . . . . 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 2452 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
482, 6, 8, 6, 28, 30, 32, 39, 46, 47dvmulbr 21541 . . . . . . . . 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 21473 . . . . . . . . . 10  |-  Rel  ( S  _D  ( F  oF  x.  G )
)
5049releldmi 5179 . . . . . . . . 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 3465 . . . . . 6  |-  ( ph  ->  X  C_  dom  ( S  _D  ( F  oF  x.  G )
) )
5427, 53eqssd 3476 . . . . 5  |-  ( ph  ->  dom  ( S  _D  ( F  oF  x.  G ) )  =  X )
5554feq2d 5650 . . . 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 5848 . 2  |-  ( ph  ->  ( S  _D  ( F  oF  x.  G
) )  =  ( x  e.  X  |->  ( ( S  _D  ( F  oF  x.  G
) ) `  x
) ) )
58 ovex 6220 . . . 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 6220 . . . 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 5804 . . . . 5  |-  ( G `
 x )  e. 
_V
6362a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  _V )
643feq2d 5650 . . . . . 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 5848 . . . 4  |-  ( ph  ->  ( S  _D  F
)  =  ( x  e.  X  |->  ( ( S  _D  F ) `
 x ) ) )
677feqmptd 5848 . . . 4  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
6824, 30, 63, 66, 67offval2 6441 . . 3  |-  ( ph  ->  ( ( S  _D  F )  oF  x.  G )  =  ( x  e.  X  |->  ( ( ( S  _D  F ) `  x )  x.  ( G `  x )
) ) )
69 fvex 5804 . . . . 5  |-  ( F `
 x )  e. 
_V
7069a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  _V )
7113feq2d 5650 . . . . . 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 5848 . . . 4  |-  ( ph  ->  ( S  _D  G
)  =  ( x  e.  X  |->  ( ( S  _D  G ) `
 x ) ) )
741feqmptd 5848 . . . 4  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
7524, 32, 70, 73, 74offval2 6441 . . 3  |-  ( ph  ->  ( ( S  _D  G )  oF  x.  F )  =  ( x  e.  X  |->  ( ( ( S  _D  G ) `  x )  x.  ( F `  x )
) ) )
7624, 59, 61, 68, 75offval2 6441 . 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 2503 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 1370    e. wcel 1758   _Vcvv 3072    C_ wss 3431   {cpr 3982   class class class wbr 4395    |-> cmpt 4453   dom cdm 4943   Fun wfun 5515   -->wf 5517   ` cfv 5521  (class class class)co 6195    oFcof 6423   CCcc 9386   RRcr 9387    + caddc 9391    x. cmul 9393   TopOpenctopn 14474  ℂfldccnfld 17938    _D cdv 21466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466  ax-addf 9467  ax-mulf 9468
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-of 6425  df-om 6582  df-1st 6682  df-2nd 6683  df-supp 6796  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-er 7206  df-map 7321  df-pm 7322  df-ixp 7369  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-fsupp 7727  df-fi 7767  df-sup 7797  df-oi 7830  df-card 8215  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-z 10753  df-dec 10862  df-uz 10968  df-q 11060  df-rp 11098  df-xneg 11195  df-xadd 11196  df-xmul 11197  df-icc 11413  df-fz 11550  df-fzo 11661  df-seq 11919  df-exp 11978  df-hash 12216  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-starv 14367  df-sca 14368  df-vsca 14369  df-ip 14370  df-tset 14371  df-ple 14372  df-ds 14374  df-unif 14375  df-hom 14376  df-cco 14377  df-rest 14475  df-topn 14476  df-0g 14494  df-gsum 14495  df-topgen 14496  df-pt 14497  df-prds 14500  df-xrs 14554  df-qtop 14559  df-imas 14560  df-xps 14562  df-mre 14638  df-mrc 14639  df-acs 14641  df-mnd 15529  df-submnd 15579  df-mulg 15662  df-cntz 15949  df-cmn 16395  df-psmet 17929  df-xmet 17930  df-met 17931  df-bl 17932  df-mopn 17933  df-fbas 17934  df-fg 17935  df-cnfld 17939  df-top 18630  df-bases 18632  df-topon 18633  df-topsp 18634  df-cld 18750  df-ntr 18751  df-cls 18752  df-nei 18829  df-lp 18867  df-perf 18868  df-cn 18958  df-cnp 18959  df-haus 19046  df-tx 19262  df-hmeo 19455  df-fil 19546  df-fm 19638  df-flim 19639  df-flf 19640  df-xms 20022  df-ms 20023  df-tms 20024  df-cncf 20581  df-limc 21469  df-dv 21470
This theorem is referenced by:  dvcmulf  21547  dvexp  21555  dvmptmul  21563  expgrowth  29752
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