Theorem dvmulxx 15728

Description: The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 15726. (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 8293	. . . . . 6 |- CC e. _V
3	2	a1i 9	. . . . 5 |- ( ph -> CC e. _V )
4		mulcl 8296	. . . . . . 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 4268	. . . . . 6 |- ( ph -> X e. _V )
10		inidm 3440	. . . . . 6 |- ( X i^i X ) = X
11	5, 6, 7, 9, 9, 10	off 6305	. . . . 5 |- ( ph -> ( F oF x. G ) : X --> CC )
12		elpm2r 6930	. . . . 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 1279	. . . 4 |- ( ph -> ( F oF x. G ) e. ( CC ^pm S ) )
14		dvfgg 15712	. . . 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 415	. . 3 |- ( ph -> ( S _D ( F oF x. G ) ) : dom ( S _D ( F oF x. G ) ) --> CC )
16	15	ffund 5532	. 2 |- ( ph -> Fun ( S _D ( F oF x. G ) ) )
17		recnprss 15711	. . . 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 6930	. . . . . . 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 1279	. . . . . 6 |- ( ph -> F e. ( CC ^pm S ) )
22		dvfgg 15712	. . . . . 6 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> ( S _D F ) : dom ( S _D F ) --> CC )
23	1, 21, 22	syl2anc 415	. . . . 5 |- ( ph -> ( S _D F ) : dom ( S _D F ) --> CC )
24		ffun 5531	. . . . 5 |- ( ( S _D F ) : dom ( S _D F ) --> CC -> Fun ( S _D F ) )
25		funfvbrb 5813	. . . . 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 6930	. . . . . . 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 1279	. . . . . 6 |- ( ph -> G e. ( CC ^pm S ) )
31		dvfgg 15712	. . . . . 6 |- ( ( S e. { RR , CC } /\ G e. ( CC ^pm S ) ) -> ( S _D G ) : dom ( S _D G ) --> CC )
32	1, 30, 31	syl2anc 415	. . . . 5 |- ( ph -> ( S _D G ) : dom ( S _D G ) --> CC )
33		ffun 5531	. . . . 5 |- ( ( S _D G ) : dom ( S _D G ) --> CC -> Fun ( S _D G ) )
34		funfvbrb 5813	. . . . 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 2238	. . 3 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
38	6, 8, 7, 18, 27, 36, 37	dvmulxxbr 15726	. 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 5733	. 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 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   {cpr 3706   class class class wbr 4125   dom cdm 4769   o. ccom 4773   Fun wfun 5366   -->wf 5368   ` cfv 5372  (class class class)co 6075   oFcof 6290   ^pm cpm 6913   CCcc 8167   RRcr 8168   + caddc 8172   x. cmul 8174   - cmin 8487   abscabs 11741   MetOpencmopn 14850   _D cdv 15679

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289  ax-addf 8291  ax-mulf 8292

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-of 6292  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-map 6914  df-pm 6915  df-sup 7314  df-inf 7315  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-xneg 10153  df-xadd 10154  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-rest 13572  df-topgen 13591  df-psmet 14852  df-xmet 14853  df-met 14854  df-bl 14855  df-mopn 14856  df-top 15022  df-topon 15035  df-bases 15067  df-ntr 15120  df-cn 15212  df-cnp 15213  df-tx 15277  df-cncf 15595  df-limced 15680  df-dvap 15681

This theorem is referenced by:  dvimulf  15730