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Theorem dvmulf 19308
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 19268 . . . . . . 7  |-  dom  ( S  _D  F )  C_  S
54a1i 10 . . . . . 6  |-  ( ph  ->  dom  ( S  _D  F )  C_  S
)
63, 5eqsstr3d 3226 . . . . 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 2363 . . . . 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 2363 . . . . 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 19306 . . 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 4122 . 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 19272 . . . . 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 19270 . . . . . . . 8  |-  ( S  e.  { RR ,  CC }  ->  S  C_  CC )
2210, 21syl 15 . . . . . . 7  |-  ( ph  ->  S  C_  CC )
23 mulcl 8837 . . . . . . . . 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 3202 . . . . . . . . 9  |-  ( ph  ->  X  C_  CC )
26 cnex 8834 . . . . . . . . 9  |-  CC  e.  _V
27 ssexg 4176 . . . . . . . . 9  |-  ( ( X  C_  CC  /\  CC  e.  _V )  ->  X  e.  _V )
2825, 26, 27sylancl 643 . . . . . . . 8  |-  ( ph  ->  X  e.  _V )
29 inidm 3391 . . . . . . . 8  |-  ( X  i^i  X )  =  X
3024, 1, 8, 28, 28, 29off 6109 . . . . . . 7  |-  ( ph  ->  ( F  o F  x.  G ) : X --> CC )
3122, 30, 6dvbss 19267 . . . . . 6  |-  ( ph  ->  dom  ( S  _D  ( F  o F  x.  G ) )  C_  X )
3222adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  S  C_  CC )
33 fvex 5555 . . . . . . . . . . 11  |-  ( ( S  _D  F ) `
 x )  e. 
_V
3433a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  F
) `  x )  e.  _V )
35 fvex 5555 . . . . . . . . . . 11  |-  ( ( S  _D  G ) `
 x )  e. 
_V
3635a1i 10 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  X )  ->  (
( S  _D  G
) `  x )  e.  _V )
37 dvfg 19272 . . . . . . . . . . . . . 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 5407 . . . . . . . . . . . 12  |-  ( ( S  _D  F ) : dom  ( S  _D  F ) --> CC 
->  Fun  ( S  _D  F ) )
41 funfvbrb 5654 . . . . . . . . . . . 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 19272 . . . . . . . . . . . . . 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 5407 . . . . . . . . . . . 12  |-  ( ( S  _D  G ) : dom  ( S  _D  G ) --> CC 
->  Fun  ( S  _D  G ) )
48 funfvbrb 5654 . . . . . . . . . . . 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 2296 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
522, 7, 9, 7, 32, 34, 36, 43, 50, 51dvmulbr 19304 . . . . . . . . 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 19236 . . . . . . . . . 10  |-  Rel  ( S  _D  ( F  o F  x.  G )
)
5453releldmi 4931 . . . . . . . . 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 3198 . . . . . 6  |-  ( ph  ->  X  C_  dom  ( S  _D  ( F  o F  x.  G )
) )
5831, 57eqssd 3209 . . . . 5  |-  ( ph  ->  dom  ( S  _D  ( F  o F  x.  G ) )  =  X )
5958feq2d 5396 . . . 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 5591 . 2  |-  ( ph  ->  ( S  _D  ( F  o F  x.  G
) )  =  ( x  e.  X  |->  ( ( S  _D  ( F  o F  x.  G
) ) `  x
) ) )
62 ovex 5899 . . . 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 5899 . . . 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 5555 . . . . 5  |-  ( G `
 x )  e. 
_V
6766a1i 10 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( G `  x )  e.  _V )
683feq2d 5396 . . . . . 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 5591 . . . 4  |-  ( ph  ->  ( S  _D  F
)  =  ( x  e.  X  |->  ( ( S  _D  F ) `
 x ) ) )
718feqmptd 5591 . . . 4  |-  ( ph  ->  G  =  ( x  e.  X  |->  ( G `
 x ) ) )
7228, 34, 67, 70, 71offval2 6111 . . 3  |-  ( ph  ->  ( ( S  _D  F )  o F  x.  G )  =  ( x  e.  X  |->  ( ( ( S  _D  F ) `  x )  x.  ( G `  x )
) ) )
73 fvex 5555 . . . . 5  |-  ( F `
 x )  e. 
_V
7473a1i 10 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( F `  x )  e.  _V )
7514feq2d 5396 . . . . . 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 5591 . . . 4  |-  ( ph  ->  ( S  _D  G
)  =  ( x  e.  X  |->  ( ( S  _D  G ) `
 x ) ) )
781feqmptd 5591 . . . 4  |-  ( ph  ->  F  =  ( x  e.  X  |->  ( F `
 x ) ) )
7928, 36, 74, 77, 78offval2 6111 . . 3  |-  ( ph  ->  ( ( S  _D  G )  o F  x.  F )  =  ( x  e.  X  |->  ( ( ( S  _D  G ) `  x )  x.  ( F `  x )
) ) )
8028, 63, 65, 72, 79offval2 6111 . 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 2338 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 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   {cpr 3654   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   RRcr 8752    + caddc 8756    x. cmul 8758   TopOpenctopn 13342  ℂfldccnfld 16393    _D cdv 19229
This theorem is referenced by:  dvcmulf  19310  dvexp  19318  dvmptmul  19326  expgrowth  27655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233
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