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Theorem dvmulf 22976
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 472 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  F : X --> CC )
3 dvaddf.df . . . . . 6  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
4 dvbsss 22936 . . . . . 6  |-  dom  ( S  _D  F )  C_  S
53, 4syl6eqssr 3469 . . . . 5  |-  ( ph  ->  X  C_  S )
65adantr 472 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  X  C_  S )
7 dvaddf.g . . . . 5  |-  ( ph  ->  G : X --> CC )
87adantr 472 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  G : X --> CC )
9 dvaddf.s . . . . 5  |-  ( ph  ->  S  e.  { RR ,  CC } )
109adantr 472 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  S  e.  { RR ,  CC } )
113eleq2d 2534 . . . . 5  |-  ( ph  ->  ( x  e.  dom  ( S  _D  F
)  <->  x  e.  X
) )
1211biimpar 493 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  F ) )
13 dvaddf.dg . . . . . 6  |-  ( ph  ->  dom  ( S  _D  G )  =  X )
1413eleq2d 2534 . . . . 5  |-  ( ph  ->  ( x  e.  dom  ( S  _D  G
)  <->  x  e.  X
) )
1514biimpar 493 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  G ) )
162, 6, 8, 6, 10, 12, 15dvmul 22974 . . 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 4482 . 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 22940 . . . . 5  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  ( F  oF  x.  G ) ) : dom  ( S  _D  ( F  oF  x.  G )
) --> CC )
199, 18syl 17 . . . 4  |-  ( ph  ->  ( S  _D  ( F  oF  x.  G
) ) : dom  ( S  _D  ( F  oF  x.  G
) ) --> CC )
20 recnprss 22938 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
219, 20syl 17 . . . . . . 7  |-  ( ph  ->  S  C_  CC )
22 mulcl 9641 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
2322adantl 473 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
249, 5ssexd 4543 . . . . . . . 8  |-  ( ph  ->  X  e.  _V )
25 inidm 3632 . . . . . . . 8  |-  ( X  i^i  X )  =  X
2623, 1, 7, 24, 24, 25off 6565 . . . . . . 7  |-  ( ph  ->  ( F  oF  x.  G ) : X --> CC )
2721, 26, 5dvbss 22935 . . . . . 6  |-  ( ph  ->  dom  ( S  _D  ( F  oF  x.  G ) )  C_  X )
2821adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  S  C_  CC )
29 fvex 5889 . . . . . . . . . . 11  |-  ( ( S  _D  F ) `
 x )  e. 
_V
3029a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  _V )
31 fvex 5889 . . . . . . . . . . 11  |-  ( ( S  _D  G ) `
 x )  e. 
_V
3231a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  G
) `  x )  e.  _V )
33 dvfg 22940 . . . . . . . . . . . . . 14  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  F ) : dom  ( S  _D  F
) --> CC )
349, 33syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S  _D  F
) : dom  ( S  _D  F ) --> CC )
3534adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  ( S  _D  F ) : dom  ( S  _D  F ) --> CC )
36 ffun 5742 . . . . . . . . . . . 12  |-  ( ( S  _D  F ) : dom  ( S  _D  F ) --> CC 
->  Fun  ( S  _D  F ) )
37 funfvbrb 6010 . . . . . . . . . . . 12  |-  ( Fun  ( S  _D  F
)  ->  ( x  e.  dom  ( S  _D  F )  <->  x ( S  _D  F ) ( ( S  _D  F
) `  x )
) )
3835, 36, 373syl 18 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  dom  ( S  _D  F )  <->  x ( S  _D  F ) ( ( S  _D  F
) `  x )
) )
3912, 38mpbid 215 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  F
) ( ( S  _D  F ) `  x ) )
40 dvfg 22940 . . . . . . . . . . . . . 14  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  G ) : dom  ( S  _D  G
) --> CC )
419, 40syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S  _D  G
) : dom  ( S  _D  G ) --> CC )
4241adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  ( S  _D  G ) : dom  ( S  _D  G ) --> CC )
43 ffun 5742 . . . . . . . . . . . 12  |-  ( ( S  _D  G ) : dom  ( S  _D  G ) --> CC 
->  Fun  ( S  _D  G ) )
44 funfvbrb 6010 . . . . . . . . . . . 12  |-  ( Fun  ( S  _D  G
)  ->  ( x  e.  dom  ( S  _D  G )  <->  x ( S  _D  G ) ( ( S  _D  G
) `  x )
) )
4542, 43, 443syl 18 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  dom  ( S  _D  G )  <->  x ( S  _D  G ) ( ( S  _D  G
) `  x )
) )
4615, 45mpbid 215 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  G
) ( ( S  _D  G ) `  x ) )
47 eqid 2471 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
482, 6, 8, 6, 28, 30, 32, 39, 46, 47dvmulbr 22972 . . . . . . . . 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 22904 . . . . . . . . . 10  |-  Rel  ( S  _D  ( F  oF  x.  G )
)
5049releldmi 5077 . . . . . . . . 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 17 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  ( F  oF  x.  G ) ) )
5251ex 441 . . . . . . 7  |-  ( ph  ->  ( x  e.  X  ->  x  e.  dom  ( S  _D  ( F  oF  x.  G )
) ) )
5352ssrdv 3424 . . . . . 6  |-  ( ph  ->  X  C_  dom  ( S  _D  ( F  oF  x.  G )
) )
5427, 53eqssd 3435 . . . . 5  |-  ( ph  ->  dom  ( S  _D  ( F  oF  x.  G ) )  =  X )
5554feq2d 5725 . . . 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 215 . . 3  |-  ( ph  ->  ( S  _D  ( F  oF  x.  G
) ) : X --> CC )
5756feqmptd 5932 . 2  |-  ( ph  ->  ( S  _D  ( F  oF  x.  G
) )  =  ( x  e.  X  |->  ( ( S  _D  ( F  oF  x.  G
) ) `  x
) ) )
58 ovex 6336 . . . 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 6336 . . . 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 5889 . . . . 5  |-  ( G `
 x )  e. 
_V
6362a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  _V )
643feq2d 5725 . . . . . 6  |-  ( ph  ->  ( ( S  _D  F ) : dom  ( S  _D  F
) --> CC  <->  ( S  _D  F ) : X --> CC ) )
6534, 64mpbid 215 . . . . 5  |-  ( ph  ->  ( S  _D  F
) : X --> CC )
6665feqmptd 5932 . . . 4  |-  ( ph  ->  ( S  _D  F
)  =  ( x  e.  X  |->  ( ( S  _D  F ) `
 x ) ) )
677feqmptd 5932 . . . 4  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
6824, 30, 63, 66, 67offval2 6567 . . 3  |-  ( ph  ->  ( ( S  _D  F )  oF  x.  G )  =  ( x  e.  X  |->  ( ( ( S  _D  F ) `  x )  x.  ( G `  x )
) ) )
69 fvex 5889 . . . . 5  |-  ( F `
 x )  e. 
_V
7069a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  _V )
7113feq2d 5725 . . . . . 6  |-  ( ph  ->  ( ( S  _D  G ) : dom  ( S  _D  G
) --> CC  <->  ( S  _D  G ) : X --> CC ) )
7241, 71mpbid 215 . . . . 5  |-  ( ph  ->  ( S  _D  G
) : X --> CC )
7372feqmptd 5932 . . . 4  |-  ( ph  ->  ( S  _D  G
)  =  ( x  e.  X  |->  ( ( S  _D  G ) `
 x ) ) )
741feqmptd 5932 . . . 4  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
7524, 32, 70, 73, 74offval2 6567 . . 3  |-  ( ph  ->  ( ( S  _D  G )  oF  x.  F )  =  ( x  e.  X  |->  ( ( ( S  _D  G ) `  x )  x.  ( F `  x )
) ) )
7624, 59, 61, 68, 75offval2 6567 . 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 2515 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 189    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031    C_ wss 3390   {cpr 3961   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308    oFcof 6548   CCcc 9555   RRcr 9556    + caddc 9560    x. cmul 9562   TopOpenctopn 15398  ℂfldccnfld 19047    _D cdv 22897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901
This theorem is referenced by:  dvcmulf  22978  dvexp  22986  dvmptmul  22994  expgrowth  36754  binomcxplemnotnn0  36775  dvmulcncf  37894
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