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Theorem dvmulf 19292
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 451 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  F : X --> CC )
3 dvaddf.df . . . . . 6  |-  ( ph  ->  dom  ( S  _D  F )  =  X )
4 dvbsss 19252 . . . . . . 7  |-  dom  ( S  _D  F )  C_  S
54a1i 10 . . . . . 6  |-  ( ph  ->  dom  ( S  _D  F )  C_  S
)
63, 5eqsstr3d 3213 . . . . 5  |-  ( ph  ->  X  C_  S )
76adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  X  C_  S )
8 dvaddf.g . . . . 5  |-  ( ph  ->  G : X --> CC )
98adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  G : X --> CC )
10 dvaddf.s . . . . 5  |-  ( ph  ->  S  e.  { RR ,  CC } )
1110adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  S  e.  { RR ,  CC } )
123eleq2d 2350 . . . . 5  |-  ( ph  ->  ( x  e.  dom  ( S  _D  F
)  <->  x  e.  X
) )
1312biimpar 471 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  F ) )
14 dvaddf.dg . . . . . 6  |-  ( ph  ->  dom  ( S  _D  G )  =  X )
1514eleq2d 2350 . . . . 5  |-  ( ph  ->  ( x  e.  dom  ( S  _D  G
)  <->  x  e.  X
) )
1615biimpar 471 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  G ) )
172, 7, 9, 7, 11, 13, 16dvmul 19290 . . 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
) ) ) )
1817mpteq2dva 4106 . 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 ) ) ) ) )
19 dvfg 19256 . . . . 5  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  ( F  o F  x.  G ) ) : dom  ( S  _D  ( F  o F  x.  G )
) --> CC )
2010, 19syl 15 . . . 4  |-  ( ph  ->  ( S  _D  ( F  o F  x.  G
) ) : dom  ( S  _D  ( F  o F  x.  G
) ) --> CC )
21 recnprss 19254 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
2210, 21syl 15 . . . . . . 7  |-  ( ph  ->  S  C_  CC )
23 mulcl 8821 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
2423adantl 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  y  e.  CC ) )  -> 
( x  x.  y
)  e.  CC )
256, 22sstrd 3189 . . . . . . . . 9  |-  ( ph  ->  X  C_  CC )
26 cnex 8818 . . . . . . . . 9  |-  CC  e.  _V
27 ssexg 4160 . . . . . . . . 9  |-  ( ( X  C_  CC  /\  CC  e.  _V )  ->  X  e.  _V )
2825, 26, 27sylancl 643 . . . . . . . 8  |-  ( ph  ->  X  e.  _V )
29 inidm 3378 . . . . . . . 8  |-  ( X  i^i  X )  =  X
3024, 1, 8, 28, 28, 29off 6093 . . . . . . 7  |-  ( ph  ->  ( F  o F  x.  G ) : X --> CC )
3122, 30, 6dvbss 19251 . . . . . 6  |-  ( ph  ->  dom  ( S  _D  ( F  o F  x.  G ) )  C_  X )
3222adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  S  C_  CC )
33 fvex 5539 . . . . . . . . . . 11  |-  ( ( S  _D  F ) `
 x )  e. 
_V
3433a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  _V )
35 fvex 5539 . . . . . . . . . . 11  |-  ( ( S  _D  G ) `
 x )  e. 
_V
3635a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  G
) `  x )  e.  _V )
37 dvfg 19256 . . . . . . . . . . . . . 14  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  F ) : dom  ( S  _D  F
) --> CC )
3810, 37syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S  _D  F
) : dom  ( S  _D  F ) --> CC )
3938adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  ( S  _D  F ) : dom  ( S  _D  F ) --> CC )
40 ffun 5391 . . . . . . . . . . . 12  |-  ( ( S  _D  F ) : dom  ( S  _D  F ) --> CC 
->  Fun  ( S  _D  F ) )
41 funfvbrb 5638 . . . . . . . . . . . 12  |-  ( Fun  ( S  _D  F
)  ->  ( x  e.  dom  ( S  _D  F )  <->  x ( S  _D  F ) ( ( S  _D  F
) `  x )
) )
4239, 40, 413syl 18 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  dom  ( S  _D  F )  <->  x ( S  _D  F ) ( ( S  _D  F
) `  x )
) )
4313, 42mpbid 201 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  F
) ( ( S  _D  F ) `  x ) )
44 dvfg 19256 . . . . . . . . . . . . . 14  |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  G ) : dom  ( S  _D  G
) --> CC )
4510, 44syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( S  _D  G
) : dom  ( S  _D  G ) --> CC )
4645adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  X )  ->  ( S  _D  G ) : dom  ( S  _D  G ) --> CC )
47 ffun 5391 . . . . . . . . . . . 12  |-  ( ( S  _D  G ) : dom  ( S  _D  G ) --> CC 
->  Fun  ( S  _D  G ) )
48 funfvbrb 5638 . . . . . . . . . . . 12  |-  ( Fun  ( S  _D  G
)  ->  ( x  e.  dom  ( S  _D  G )  <->  x ( S  _D  G ) ( ( S  _D  G
) `  x )
) )
4946, 47, 483syl 18 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  X )  ->  (
x  e.  dom  ( S  _D  G )  <->  x ( S  _D  G ) ( ( S  _D  G
) `  x )
) )
5016, 49mpbid 201 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  x
( S  _D  G
) ( ( S  _D  G ) `  x ) )
51 eqid 2283 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
522, 7, 9, 7, 32, 34, 36, 43, 50, 51dvmulbr 19288 . . . . . . . . 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
) ) ) )
53 reldv 19220 . . . . . . . . . 10  |-  Rel  ( S  _D  ( F  o F  x.  G )
)
5453releldmi 4915 . . . . . . . . 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 )
) )
5552, 54syl 15 . . . . . . . 8  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  dom  ( S  _D  ( F  o F  x.  G ) ) )
5655ex 423 . . . . . . 7  |-  ( ph  ->  ( x  e.  X  ->  x  e.  dom  ( S  _D  ( F  o F  x.  G )
) ) )
5756ssrdv 3185 . . . . . 6  |-  ( ph  ->  X  C_  dom  ( S  _D  ( F  o F  x.  G )
) )
5831, 57eqssd 3196 . . . . 5  |-  ( ph  ->  dom  ( S  _D  ( F  o F  x.  G ) )  =  X )
5958feq2d 5380 . . . 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 ) )
6020, 59mpbid 201 . . 3  |-  ( ph  ->  ( S  _D  ( F  o F  x.  G
) ) : X --> CC )
6160feqmptd 5575 . 2  |-  ( ph  ->  ( S  _D  ( F  o F  x.  G
) )  =  ( x  e.  X  |->  ( ( S  _D  ( F  o F  x.  G
) ) `  x
) ) )
62 ovex 5883 . . . 4  |-  ( ( ( S  _D  F
) `  x )  x.  ( G `  x
) )  e.  _V
6362a1i 10 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  F ) `  x
)  x.  ( G `
 x ) )  e.  _V )
64 ovex 5883 . . . 4  |-  ( ( ( S  _D  G
) `  x )  x.  ( F `  x
) )  e.  _V
6564a1i 10 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( ( S  _D  G ) `  x
)  x.  ( F `
 x ) )  e.  _V )
66 fvex 5539 . . . . 5  |-  ( G `
 x )  e. 
_V
6766a1i 10 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  _V )
683feq2d 5380 . . . . . 6  |-  ( ph  ->  ( ( S  _D  F ) : dom  ( S  _D  F
) --> CC  <->  ( S  _D  F ) : X --> CC ) )
6938, 68mpbid 201 . . . . 5  |-  ( ph  ->  ( S  _D  F
) : X --> CC )
7069feqmptd 5575 . . . 4  |-  ( ph  ->  ( S  _D  F
)  =  ( x  e.  X  |->  ( ( S  _D  F ) `
 x ) ) )
718feqmptd 5575 . . . 4  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
7228, 34, 67, 70, 71offval2 6095 . . 3  |-  ( ph  ->  ( ( S  _D  F )  o F  x.  G )  =  ( x  e.  X  |->  ( ( ( S  _D  F ) `  x )  x.  ( G `  x )
) ) )
73 fvex 5539 . . . . 5  |-  ( F `
 x )  e. 
_V
7473a1i 10 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  _V )
7514feq2d 5380 . . . . . 6  |-  ( ph  ->  ( ( S  _D  G ) : dom  ( S  _D  G
) --> CC  <->  ( S  _D  G ) : X --> CC ) )
7645, 75mpbid 201 . . . . 5  |-  ( ph  ->  ( S  _D  G
) : X --> CC )
7776feqmptd 5575 . . . 4  |-  ( ph  ->  ( S  _D  G
)  =  ( x  e.  X  |->  ( ( S  _D  G ) `
 x ) ) )
781feqmptd 5575 . . . 4  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
7928, 36, 74, 77, 78offval2 6095 . . 3  |-  ( ph  ->  ( ( S  _D  G )  o F  x.  F )  =  ( x  e.  X  |->  ( ( ( S  _D  G ) `  x )  x.  ( F `  x )
) ) )
8028, 63, 65, 72, 79offval2 6095 . 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 ) ) ) ) )
8118, 61, 803eqtr4d 2325 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {cpr 3641   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   RRcr 8736    + caddc 8740    x. cmul 8742   TopOpenctopn 13326  ℂfldccnfld 16377    _D cdv 19213
This theorem is referenced by:  dvcmulf  19294  dvexp  19302  dvmptmul  19310  expgrowth  27552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217
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