Theorem dvmulxx 14940

Description: The product rule for derivatives at a point. For the (more general) relation version, see dvmulxxbr 14938. (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 8003	. . . . . 6 |- CC e. _V
3	2	a1i 9	. . . . 5 |- ( ph -> CC e. _V )
4		mulcl 8006	. . . . . . 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 4173	. . . . . 6 |- ( ph -> X e. _V )
10		inidm 3372	. . . . . 6 |- ( X i^i X ) = X
11	5, 6, 7, 9, 9, 10	off 6148	. . . . 5 |- ( ph -> ( F oF x. G ) : X --> CC )
12		elpm2r 6725	. . . . 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 14924	. . . 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 5411	. 2 |- ( ph -> Fun ( S _D ( F oF x. G ) ) )
17		recnprss 14923	. . . 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 6725	. . . . . . 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 14924	. . . . . 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 5410	. . . . 5 |- ( ( S _D F ) : dom ( S _D F ) --> CC -> Fun ( S _D F ) )
25		funfvbrb 5675	. . . . 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 6725	. . . . . . 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 14924	. . . . . 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 5410	. . . . 5 |- ( ( S _D G ) : dom ( S _D G ) --> CC -> Fun ( S _D G ) )
34		funfvbrb 5675	. . . . 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 2196	. . 3 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
38	6, 8, 7, 18, 27, 36, 37	dvmulxxbr 14938	. 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 5599	. 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 2167   _Vcvv 2763   C_ wss 3157   {cpr 3623   class class class wbr 4033   dom cdm 4663   o. ccom 4667   Fun wfun 5252   -->wf 5254   ` cfv 5258  (class class class)co 5922   oFcof 6133   ^pm cpm 6708   CCcc 7877   RRcr 7878   + caddc 7882   x. cmul 7884   - cmin 8197   abscabs 11162   MetOpencmopn 14097   _D cdv 14891

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999  ax-addf 8001  ax-mulf 8002

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-of 6135  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-map 6709  df-pm 6710  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-xneg 9847  df-xadd 9848  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-rest 12912  df-topgen 12931  df-psmet 14099  df-xmet 14100  df-met 14101  df-bl 14102  df-mopn 14103  df-top 14234  df-topon 14247  df-bases 14279  df-ntr 14332  df-cn 14424  df-cnp 14425  df-tx 14489  df-cncf 14807  df-limced 14892  df-dvap 14893

This theorem is referenced by:  dvimulf  14942