Theorem dvmulxx 14628

Description: The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 14626. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 2-Dec-2023.)

Hypotheses

Ref	Expression
dvadd.f	|- ( ph -> F : X --> CC )
dvadd.x	|- ( ph -> X C_ S )
dvaddxx.g	|- ( ph -> G : X --> CC )
dvadd.s	|- ( ph -> S e. { RR , CC } )
dvadd.df	|- ( ph -> C e. dom ( S _D F ) )
dvadd.dg	|- ( ph -> C e. dom ( S _D G ) )

Assertion

Ref	Expression
dvmulxx	|- ( ph -> ( ( S _D ( F oF x. G ) ) ` C ) = ( ( ( ( S _D F ) ` C ) x. ( G ` C ) ) + ( ( ( S _D G ) ` C ) x. ( F ` C ) ) ) )

Proof of Theorem dvmulxx

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvadd.s	. . . 4 |- ( ph -> S e. { RR , CC } )
2		cnex 7965	. . . . . 6 |- CC e. _V
3	2	a1i 9	. . . . 5 |- ( ph -> CC e. _V )
4		mulcl 7968	. . . . . . 7 |- ( ( u e. CC /\ v e. CC ) -> ( u x. v ) e. CC )
5	4	adantl 277	. . . . . 6 |- ( ( ph /\ ( u e. CC /\ v e. CC ) ) -> ( u x. v ) e. CC )
6		dvadd.f	. . . . . 6 |- ( ph -> F : X --> CC )
7		dvaddxx.g	. . . . . 6 |- ( ph -> G : X --> CC )
8		dvadd.x	. . . . . . 7 |- ( ph -> X C_ S )
9	1, 8	ssexd 4158	. . . . . 6 |- ( ph -> X e. _V )
10		inidm 3359	. . . . . 6 |- ( X i^i X ) = X
11	5, 6, 7, 9, 9, 10	off 6119	. . . . 5 |- ( ph -> ( F oF x. G ) : X --> CC )
12		elpm2r 6692	. . . . 5 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( ( F oF x. G ) : X --> CC /\ X C_ S ) ) -> ( F oF x. G ) e. ( CC ^pm S ) )
13	3, 1, 11, 8, 12	syl22anc 1250	. . . 4 |- ( ph -> ( F oF x. G ) e. ( CC ^pm S ) )
14		dvfgg 14617	. . . 4 |- ( ( S e. { RR , CC } /\ ( F oF x. G ) e. ( CC ^pm S ) ) -> ( S _D ( F oF x. G ) ) : dom ( S _D ( F oF x. G ) ) --> CC )
15	1, 13, 14	syl2anc 411	. . 3 |- ( ph -> ( S _D ( F oF x. G ) ) : dom ( S _D ( F oF x. G ) ) --> CC )
16	15	ffund 5388	. 2 |- ( ph -> Fun ( S _D ( F oF x. G ) ) )
17		recnprss 14616	. . . 4 |- ( S e. { RR , CC } -> S C_ CC )
18	1, 17	syl 14	. . 3 |- ( ph -> S C_ CC )
19		dvadd.df	. . . 4 |- ( ph -> C e. dom ( S _D F ) )
20		elpm2r 6692	. . . . . . 7 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( F : X --> CC /\ X C_ S ) ) -> F e. ( CC ^pm S ) )
21	3, 1, 6, 8, 20	syl22anc 1250	. . . . . 6 |- ( ph -> F e. ( CC ^pm S ) )
22		dvfgg 14617	. . . . . 6 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> ( S _D F ) : dom ( S _D F ) --> CC )
23	1, 21, 22	syl2anc 411	. . . . 5 |- ( ph -> ( S _D F ) : dom ( S _D F ) --> CC )
24		ffun 5387	. . . . 5 |- ( ( S _D F ) : dom ( S _D F ) --> CC -> Fun ( S _D F ) )
25		funfvbrb 5650	. . . . 5 |- ( Fun ( S _D F ) -> ( C e. dom ( S _D F ) <-> C ( S _D F ) ( ( S _D F ) ` C ) ) )
26	23, 24, 25	3syl 17	. . . 4 |- ( ph -> ( C e. dom ( S _D F ) <-> C ( S _D F ) ( ( S _D F ) ` C ) ) )
27	19, 26	mpbid 147	. . 3 |- ( ph -> C ( S _D F ) ( ( S _D F ) ` C ) )
28		dvadd.dg	. . . 4 |- ( ph -> C e. dom ( S _D G ) )
29		elpm2r 6692	. . . . . . 7 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( G : X --> CC /\ X C_ S ) ) -> G e. ( CC ^pm S ) )
30	3, 1, 7, 8, 29	syl22anc 1250	. . . . . 6 |- ( ph -> G e. ( CC ^pm S ) )
31		dvfgg 14617	. . . . . 6 |- ( ( S e. { RR , CC } /\ G e. ( CC ^pm S ) ) -> ( S _D G ) : dom ( S _D G ) --> CC )
32	1, 30, 31	syl2anc 411	. . . . 5 |- ( ph -> ( S _D G ) : dom ( S _D G ) --> CC )
33		ffun 5387	. . . . 5 |- ( ( S _D G ) : dom ( S _D G ) --> CC -> Fun ( S _D G ) )
34		funfvbrb 5650	. . . . 5 |- ( Fun ( S _D G ) -> ( C e. dom ( S _D G ) <-> C ( S _D G ) ( ( S _D G ) ` C ) ) )
35	32, 33, 34	3syl 17	. . . 4 |- ( ph -> ( C e. dom ( S _D G ) <-> C ( S _D G ) ( ( S _D G ) ` C ) ) )
36	28, 35	mpbid 147	. . 3 |- ( ph -> C ( S _D G ) ( ( S _D G ) ` C ) )
37		eqid 2189	. . 3 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
38	6, 8, 7, 18, 27, 36, 37	dvmulxxbr 14626	. 2 |- ( ph -> C ( S _D ( F oF x. G ) ) ( ( ( ( S _D F ) ` C ) x. ( G ` C ) ) + ( ( ( S _D G ) ` C ) x. ( F ` C ) ) ) )
39		funbrfv 5575	. 2 |- ( Fun ( S _D ( F oF x. G ) ) -> ( C ( S _D ( F oF x. G ) ) ( ( ( ( S _D F ) ` C ) x. ( G ` C ) ) + ( ( ( S _D G ) ` C ) x. ( F ` C ) ) ) -> ( ( S _D ( F oF x. G ) ) ` C ) = ( ( ( ( S _D F ) ` C ) x. ( G ` C ) ) + ( ( ( S _D G ) ` C ) x. ( F ` C ) ) ) ) )
40	16, 38, 39	sylc 62	1 |- ( ph -> ( ( S _D ( F oF x. G ) ) ` C ) = ( ( ( ( S _D F ) ` C ) x. ( G ` C ) ) + ( ( ( S _D G ) ` C ) x. ( F ` C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2160   _Vcvv 2752   C_ wss 3144   {cpr 3608   class class class wbr 4018   dom cdm 4644   o. ccom 4648   Fun wfun 5229   -->wf 5231   ` cfv 5235  (class class class)co 5896   oFcof 6104   ^pm cpm 6675   CCcc 7839   RRcr 7840   + caddc 7844   x. cmul 7846   - cmin 8158   abscabs 11038   MetOpencmopn 13854   _D cdv 14584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959  ax-arch 7960  ax-caucvg 7961  ax-addf 7963  ax-mulf 7964

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-of 6106  df-1st 6165  df-2nd 6166  df-recs 6330  df-frec 6416  df-map 6676  df-pm 6677  df-sup 7013  df-inf 7014  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-n0 9207  df-z 9284  df-uz 9559  df-q 9650  df-rp 9684  df-xneg 9802  df-xadd 9803  df-seqfrec 10477  df-exp 10551  df-cj 10883  df-re 10884  df-im 10885  df-rsqrt 11039  df-abs 11040  df-rest 12746  df-topgen 12765  df-psmet 13856  df-xmet 13857  df-met 13858  df-bl 13859  df-mopn 13860  df-top 13958  df-topon 13971  df-bases 14003  df-ntr 14056  df-cn 14148  df-cnp 14149  df-tx 14213  df-cncf 14518  df-limced 14585  df-dvap 14586

This theorem is referenced by:  dvimulf  14630