Theorem dvaddxx 12836

Description: The sum rule for derivatives at a point. For the (more general) relation version, see dvaddxxbr 12834. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-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
dvaddxx	|- ( ph -> ( ( S _D ( F oF + G ) ) ` C ) = ( ( ( S _D F ) ` C ) + ( ( S _D G ) ` C ) ) )

Proof of Theorem dvaddxx

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 7744	. . . . . 6 |- CC e. _V
3	2	a1i 9	. . . . 5 |- ( ph -> CC e. _V )
4		addcl 7745	. . . . . . 7 |- ( ( u e. CC /\ v e. CC ) -> ( u + v ) e. CC )
5	4	adantl 275	. . . . . 6 |- ( ( ph /\ ( u e. CC /\ v e. CC ) ) -> ( u + 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 4068	. . . . . 6 |- ( ph -> X e. _V )
10		inidm 3285	. . . . . 6 |- ( X i^i X ) = X
11	5, 6, 7, 9, 9, 10	off 5994	. . . . 5 |- ( ph -> ( F oF + G ) : X --> CC )
12		elpm2r 6560	. . . . 5 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( ( F oF + G ) : X --> CC /\ X C_ S ) ) -> ( F oF + G ) e. ( CC ^pm S ) )
13	3, 1, 11, 8, 12	syl22anc 1217	. . . 4 |- ( ph -> ( F oF + G ) e. ( CC ^pm S ) )
14		dvfgg 12826	. . . 4 |- ( ( S e. { RR , CC } /\ ( F oF + G ) e. ( CC ^pm S ) ) -> ( S _D ( F oF + G ) ) : dom ( S _D ( F oF + G ) ) --> CC )
15	1, 13, 14	syl2anc 408	. . 3 |- ( ph -> ( S _D ( F oF + G ) ) : dom ( S _D ( F oF + G ) ) --> CC )
16	15	ffund 5276	. 2 |- ( ph -> Fun ( S _D ( F oF + G ) ) )
17		recnprss 12825	. . . 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 6560	. . . . . . 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 1217	. . . . . 6 |- ( ph -> F e. ( CC ^pm S ) )
22		dvfgg 12826	. . . . . 6 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> ( S _D F ) : dom ( S _D F ) --> CC )
23	1, 21, 22	syl2anc 408	. . . . 5 |- ( ph -> ( S _D F ) : dom ( S _D F ) --> CC )
24		ffun 5275	. . . . 5 |- ( ( S _D F ) : dom ( S _D F ) --> CC -> Fun ( S _D F ) )
25		funfvbrb 5533	. . . . 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 146	. . 3 |- ( ph -> C ( S _D F ) ( ( S _D F ) ` C ) )
28		dvadd.dg	. . . 4 |- ( ph -> C e. dom ( S _D G ) )
29		elpm2r 6560	. . . . . . 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 1217	. . . . . 6 |- ( ph -> G e. ( CC ^pm S ) )
31		dvfgg 12826	. . . . . 6 |- ( ( S e. { RR , CC } /\ G e. ( CC ^pm S ) ) -> ( S _D G ) : dom ( S _D G ) --> CC )
32	1, 30, 31	syl2anc 408	. . . . 5 |- ( ph -> ( S _D G ) : dom ( S _D G ) --> CC )
33		ffun 5275	. . . . 5 |- ( ( S _D G ) : dom ( S _D G ) --> CC -> Fun ( S _D G ) )
34		funfvbrb 5533	. . . . 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 146	. . 3 |- ( ph -> C ( S _D G ) ( ( S _D G ) ` C ) )
37		eqid 2139	. . 3 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
38	6, 8, 7, 18, 27, 36, 37	dvaddxxbr 12834	. 2 |- ( ph -> C ( S _D ( F oF + G ) ) ( ( ( S _D F ) ` C ) + ( ( S _D G ) ` C ) ) )
39		funbrfv 5460	. 2 |- ( Fun ( S _D ( F oF + G ) ) -> ( C ( S _D ( F oF + G ) ) ( ( ( S _D F ) ` C ) + ( ( S _D G ) ` C ) ) -> ( ( S _D ( F oF + G ) ) ` C ) = ( ( ( S _D F ) ` C ) + ( ( S _D G ) ` C ) ) ) )
40	16, 38, 39	sylc 62	1 |- ( ph -> ( ( S _D ( F oF + G ) ) ` C ) = ( ( ( S _D F ) ` C ) + ( ( S _D G ) ` C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   _Vcvv 2686   C_ wss 3071   {cpr 3528   class class class wbr 3929   dom cdm 4539   o. ccom 4543   Fun wfun 5117   -->wf 5119   ` cfv 5123  (class class class)co 5774   oFcof 5980   ^pm cpm 6543   CCcc 7618   RRcr 7619   + caddc 7623   - cmin 7933   abscabs 10769   MetOpencmopn 12154   _D cdv 12793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739  ax-caucvg 7740  ax-addf 7742

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-of 5982  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-map 6544  df-pm 6545  df-sup 6871  df-inf 6872  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-3 8780  df-4 8781  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-xneg 9559  df-xadd 9560  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-rest 12122  df-topgen 12141  df-psmet 12156  df-xmet 12157  df-met 12158  df-bl 12159  df-mopn 12160  df-top 12165  df-topon 12178  df-bases 12210  df-ntr 12265  df-cn 12357  df-cnp 12358  df-tx 12422  df-limced 12794  df-dvap 12795

This theorem is referenced by:  dviaddf  12838