Theorem dvaddxx 15336

Description: The sum rule for derivatives at a point. For the (more general) relation version, see dvaddxxbr 15334. (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 8086	. . . . . 6 |- CC e. _V
3	2	a1i 9	. . . . 5 |- ( ph -> CC e. _V )
4		addcl 8087	. . . . . . 7 |- ( ( u e. CC /\ v e. CC ) -> ( u + v ) e. CC )
5	4	adantl 277	. . . . . 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 4201	. . . . . 6 |- ( ph -> X e. _V )
10		inidm 3391	. . . . . 6 |- ( X i^i X ) = X
11	5, 6, 7, 9, 9, 10	off 6196	. . . . 5 |- ( ph -> ( F oF + G ) : X --> CC )
12		elpm2r 6778	. . . . 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 1251	. . . 4 |- ( ph -> ( F oF + G ) e. ( CC ^pm S ) )
14		dvfgg 15321	. . . 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 411	. . 3 |- ( ph -> ( S _D ( F oF + G ) ) : dom ( S _D ( F oF + G ) ) --> CC )
16	15	ffund 5450	. 2 |- ( ph -> Fun ( S _D ( F oF + G ) ) )
17		recnprss 15320	. . . 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 6778	. . . . . . 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 1251	. . . . . 6 |- ( ph -> F e. ( CC ^pm S ) )
22		dvfgg 15321	. . . . . 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 5449	. . . . 5 |- ( ( S _D F ) : dom ( S _D F ) --> CC -> Fun ( S _D F ) )
25		funfvbrb 5718	. . . . 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 6778	. . . . . . 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 1251	. . . . . 6 |- ( ph -> G e. ( CC ^pm S ) )
31		dvfgg 15321	. . . . . 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 5449	. . . . 5 |- ( ( S _D G ) : dom ( S _D G ) --> CC -> Fun ( S _D G ) )
34		funfvbrb 5718	. . . . 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 2207	. . 3 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
38	6, 8, 7, 18, 27, 36, 37	dvaddxxbr 15334	. 2 |- ( ph -> C ( S _D ( F oF + G ) ) ( ( ( S _D F ) ` C ) + ( ( S _D G ) ` C ) ) )
39		funbrfv 5641	. 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 104   <-> wb 105   = wceq 1373   e. wcel 2178   _Vcvv 2777   C_ wss 3175   {cpr 3645   class class class wbr 4060   dom cdm 4694   o. ccom 4698   Fun wfun 5285   -->wf 5287   ` cfv 5291  (class class class)co 5969   oFcof 6181   ^pm cpm 6761   CCcc 7960   RRcr 7961   + caddc 7965   - cmin 8280   abscabs 11469   MetOpencmopn 14464   _D cdv 15288

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4176  ax-sep 4179  ax-nul 4187  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604  ax-iinf 4655  ax-cnex 8053  ax-resscn 8054  ax-1cn 8055  ax-1re 8056  ax-icn 8057  ax-addcl 8058  ax-addrcl 8059  ax-mulcl 8060  ax-mulrcl 8061  ax-addcom 8062  ax-mulcom 8063  ax-addass 8064  ax-mulass 8065  ax-distr 8066  ax-i2m1 8067  ax-0lt1 8068  ax-1rid 8069  ax-0id 8070  ax-rnegex 8071  ax-precex 8072  ax-cnre 8073  ax-pre-ltirr 8074  ax-pre-ltwlin 8075  ax-pre-lttrn 8076  ax-pre-apti 8077  ax-pre-ltadd 8078  ax-pre-mulgt0 8079  ax-pre-mulext 8080  ax-arch 8081  ax-caucvg 8082  ax-addf 8084

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2779  df-sbc 3007  df-csb 3103  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-nul 3470  df-if 3581  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-int 3901  df-iun 3944  df-br 4061  df-opab 4123  df-mpt 4124  df-tr 4160  df-id 4359  df-po 4362  df-iso 4363  df-iord 4432  df-on 4434  df-ilim 4435  df-suc 4437  df-iom 4658  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-ima 4707  df-iota 5252  df-fun 5293  df-fn 5294  df-f 5295  df-f1 5296  df-fo 5297  df-f1o 5298  df-fv 5299  df-isom 5300  df-riota 5924  df-ov 5972  df-oprab 5973  df-mpo 5974  df-of 6183  df-1st 6251  df-2nd 6252  df-recs 6416  df-frec 6502  df-map 6762  df-pm 6763  df-sup 7114  df-inf 7115  df-pnf 8146  df-mnf 8147  df-xr 8148  df-ltxr 8149  df-le 8150  df-sub 8282  df-neg 8283  df-reap 8685  df-ap 8692  df-div 8783  df-inn 9074  df-2 9132  df-3 9133  df-4 9134  df-n0 9333  df-z 9410  df-uz 9686  df-q 9778  df-rp 9813  df-xneg 9931  df-xadd 9932  df-seqfrec 10632  df-exp 10723  df-cj 11314  df-re 11315  df-im 11316  df-rsqrt 11470  df-abs 11471  df-rest 13234  df-topgen 13253  df-psmet 14466  df-xmet 14467  df-met 14468  df-bl 14469  df-mopn 14470  df-top 14631  df-topon 14644  df-bases 14676  df-ntr 14729  df-cn 14821  df-cnp 14822  df-tx 14886  df-limced 15289  df-dvap 15290

This theorem is referenced by:  dviaddf  15338