Theorem dvimulf 15436

Description: The product rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

Hypotheses

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

Assertion

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

Proof of Theorem dvimulf

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvaddf.f	. . . . 5 |- ( ph -> F : X --> CC )
2	1	adantr 276	. . . 4 |- ( ( ph /\ x e. X ) -> F : X --> CC )
3		dviaddf.x	. . . . 5 |- ( ph -> X C_ S )
4	3	adantr 276	. . . 4 |- ( ( ph /\ x e. X ) -> X C_ S )
5		dvaddf.g	. . . . 5 |- ( ph -> G : X --> CC )
6	5	adantr 276	. . . 4 |- ( ( ph /\ x e. X ) -> G : X --> CC )
7		dvaddf.s	. . . . 5 |- ( ph -> S e. { RR , CC } )
8	7	adantr 276	. . . 4 |- ( ( ph /\ x e. X ) -> S e. { RR , CC } )
9		dvaddf.df	. . . . . 6 |- ( ph -> dom ( S _D F ) = X )
10	9	eleq2d 2301	. . . . 5 |- ( ph -> ( x e. dom ( S _D F ) <-> x e. X ) )
11	10	biimpar 297	. . . 4 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D F ) )
12		dvaddf.dg	. . . . . 6 |- ( ph -> dom ( S _D G ) = X )
13	12	eleq2d 2301	. . . . 5 |- ( ph -> ( x e. dom ( S _D G ) <-> x e. X ) )
14	13	biimpar 297	. . . 4 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D G ) )
15	2, 4, 6, 8, 11, 14	dvmulxx 15434	. . 3 |- ( ( ph /\ x e. X ) -> ( ( S _D ( F oF x. G ) ) ` x ) = ( ( ( ( S _D F ) ` x ) x. ( G ` x ) ) + ( ( ( S _D G ) ` x ) x. ( F ` x ) ) ) )
16	15	mpteq2dva 4179	. 2 |- ( ph -> ( x e. X |-> ( ( S _D ( F oF x. G ) ) ` x ) ) = ( x e. X |-> ( ( ( ( S _D F ) ` x ) x. ( G ` x ) ) + ( ( ( S _D G ) ` x ) x. ( F ` x ) ) ) ) )
17		cnex 8156	. . . . . . 7 |- CC e. _V
18	17	a1i 9	. . . . . 6 |- ( ph -> CC e. _V )
19		mulcl 8159	. . . . . . . 8 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
20	19	adantl 277	. . . . . . 7 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
21	7, 3	ssexd 4229	. . . . . . 7 |- ( ph -> X e. _V )
22		inidm 3416	. . . . . . 7 |- ( X i^i X ) = X
23	20, 1, 5, 21, 21, 22	off 6248	. . . . . 6 |- ( ph -> ( F oF x. G ) : X --> CC )
24		elpm2r 6835	. . . . . 6 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( ( F oF x. G ) : X --> CC /\ X C_ S ) ) -> ( F oF x. G ) e. ( CC ^pm S ) )
25	18, 7, 23, 3, 24	syl22anc 1274	. . . . 5 |- ( ph -> ( F oF x. G ) e. ( CC ^pm S ) )
26		dvfgg 15418	. . . . 5 |- ( ( 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 )
27	7, 25, 26	syl2anc 411	. . . 4 |- ( ph -> ( S _D ( F oF x. G ) ) : dom ( S _D ( F oF x. G ) ) --> CC )
28		recnprss 15417	. . . . . . . 8 |- ( S e. { RR , CC } -> S C_ CC )
29	7, 28	syl 14	. . . . . . 7 |- ( ph -> S C_ CC )
30	29, 23, 3	dvbss 15415	. . . . . 6 |- ( ph -> dom ( S _D ( F oF x. G ) ) C_ X )
31		reldvg 15409	. . . . . . . . 9 |- ( ( S C_ CC /\ ( F oF x. G ) e. ( CC ^pm S ) ) -> Rel ( S _D ( F oF x. G ) ) )
32	29, 25, 31	syl2anc 411	. . . . . . . 8 |- ( ph -> Rel ( S _D ( F oF x. G ) ) )
33	32	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. X ) -> Rel ( S _D ( F oF x. G ) ) )
34	29	adantr 276	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> S C_ CC )
35		elpm2r 6835	. . . . . . . . . . . . 13 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( F : X --> CC /\ X C_ S ) ) -> F e. ( CC ^pm S ) )
36	18, 7, 1, 3, 35	syl22anc 1274	. . . . . . . . . . . 12 |- ( ph -> F e. ( CC ^pm S ) )
37		dvfgg 15418	. . . . . . . . . . . 12 |- ( ( S e. { RR , CC } /\ F e. ( CC ^pm S ) ) -> ( S _D F ) : dom ( S _D F ) --> CC )
38	7, 36, 37	syl2anc 411	. . . . . . . . . . 11 |- ( ph -> ( S _D F ) : dom ( S _D F ) --> CC )
39		ffun 5485	. . . . . . . . . . 11 |- ( ( S _D F ) : dom ( S _D F ) --> CC -> Fun ( S _D F ) )
40		funfvbrb 5760	. . . . . . . . . . 11 |- ( Fun ( S _D F ) -> ( x e. dom ( S _D F ) <-> x ( S _D F ) ( ( S _D F ) ` x ) ) )
41	38, 39, 40	3syl 17	. . . . . . . . . 10 |- ( ph -> ( x e. dom ( S _D F ) <-> x ( S _D F ) ( ( S _D F ) ` x ) ) )
42	41	adantr 276	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> ( x e. dom ( S _D F ) <-> x ( S _D F ) ( ( S _D F ) ` x ) ) )
43	11, 42	mpbid 147	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> x ( S _D F ) ( ( S _D F ) ` x ) )
44		elpm2r 6835	. . . . . . . . . . . . 13 |- ( ( ( CC e. _V /\ S e. { RR , CC } ) /\ ( G : X --> CC /\ X C_ S ) ) -> G e. ( CC ^pm S ) )
45	18, 7, 5, 3, 44	syl22anc 1274	. . . . . . . . . . . 12 |- ( ph -> G e. ( CC ^pm S ) )
46		dvfgg 15418	. . . . . . . . . . . 12 |- ( ( S e. { RR , CC } /\ G e. ( CC ^pm S ) ) -> ( S _D G ) : dom ( S _D G ) --> CC )
47	7, 45, 46	syl2anc 411	. . . . . . . . . . 11 |- ( ph -> ( S _D G ) : dom ( S _D G ) --> CC )
48		ffun 5485	. . . . . . . . . . 11 |- ( ( S _D G ) : dom ( S _D G ) --> CC -> Fun ( S _D G ) )
49		funfvbrb 5760	. . . . . . . . . . 11 |- ( Fun ( S _D G ) -> ( x e. dom ( S _D G ) <-> x ( S _D G ) ( ( S _D G ) ` x ) ) )
50	47, 48, 49	3syl 17	. . . . . . . . . 10 |- ( ph -> ( x e. dom ( S _D G ) <-> x ( S _D G ) ( ( S _D G ) ` x ) ) )
51	50	adantr 276	. . . . . . . . 9 |- ( ( ph /\ x e. X ) -> ( x e. dom ( S _D G ) <-> x ( S _D G ) ( ( S _D G ) ` x ) ) )
52	14, 51	mpbid 147	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> x ( S _D G ) ( ( S _D G ) ` x ) )
53		eqid 2231	. . . . . . . 8 |- ( MetOpen ` ( abs o. - ) ) = ( MetOpen ` ( abs o. - ) )
54	2, 4, 6, 34, 43, 52, 53	dvmulxxbr 15432	. . . . . . 7 |- ( ( ph /\ x e. X ) -> x ( S _D ( F oF x. G ) ) ( ( ( ( S _D F ) ` x ) x. ( G ` x ) ) + ( ( ( S _D G ) ` x ) x. ( F ` x ) ) ) )
55		releldm 4967	. . . . . . 7 |- ( ( Rel ( S _D ( F oF x. G ) ) /\ x ( S _D ( F oF x. G ) ) ( ( ( ( S _D F ) ` x ) x. ( G ` x ) ) + ( ( ( S _D G ) ` x ) x. ( F ` x ) ) ) ) -> x e. dom ( S _D ( F oF x. G ) ) )
56	33, 54, 55	syl2anc 411	. . . . . 6 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D ( F oF x. G ) ) )
57	30, 56	eqelssd 3246	. . . . 5 |- ( ph -> dom ( S _D ( F oF x. G ) ) = X )
58	57	feq2d 5470	. . . 4 |- ( ph -> ( ( S _D ( F oF x. G ) ) : dom ( S _D ( F oF x. G ) ) --> CC <-> ( S _D ( F oF x. G ) ) : X --> CC ) )
59	27, 58	mpbid 147	. . 3 |- ( ph -> ( S _D ( F oF x. G ) ) : X --> CC )
60	59	feqmptd 5699	. 2 |- ( ph -> ( S _D ( F oF x. G ) ) = ( x e. X |-> ( ( S _D ( F oF x. G ) ) ` x ) ) )
61	9	feq2d 5470	. . . . . 6 |- ( ph -> ( ( S _D F ) : dom ( S _D F ) --> CC <-> ( S _D F ) : X --> CC ) )
62	38, 61	mpbid 147	. . . . 5 |- ( ph -> ( S _D F ) : X --> CC )
63	62	ffvelcdmda 5782	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) e. CC )
64	5	ffvelcdmda 5782	. . . 4 |- ( ( ph /\ x e. X ) -> ( G ` x ) e. CC )
65	63, 64	mulcld 8200	. . 3 |- ( ( ph /\ x e. X ) -> ( ( ( S _D F ) ` x ) x. ( G ` x ) ) e. CC )
66	12	feq2d 5470	. . . . . 6 |- ( ph -> ( ( S _D G ) : dom ( S _D G ) --> CC <-> ( S _D G ) : X --> CC ) )
67	47, 66	mpbid 147	. . . . 5 |- ( ph -> ( S _D G ) : X --> CC )
68	67	ffvelcdmda 5782	. . . 4 |- ( ( ph /\ x e. X ) -> ( ( S _D G ) ` x ) e. CC )
69	1	ffvelcdmda 5782	. . . 4 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. CC )
70	68, 69	mulcld 8200	. . 3 |- ( ( ph /\ x e. X ) -> ( ( ( S _D G ) ` x ) x. ( F ` x ) ) e. CC )
71	62	feqmptd 5699	. . . 4 |- ( ph -> ( S _D F ) = ( x e. X |-> ( ( S _D F ) ` x ) ) )
72	5	feqmptd 5699	. . . 4 |- ( ph -> G = ( x e. X |-> ( G ` x ) ) )
73	21, 63, 64, 71, 72	offval2 6251	. . 3 |- ( ph -> ( ( S _D F ) oF x. G ) = ( x e. X |-> ( ( ( S _D F ) ` x ) x. ( G ` x ) ) ) )
74	67	feqmptd 5699	. . . 4 |- ( ph -> ( S _D G ) = ( x e. X |-> ( ( S _D G ) ` x ) ) )
75	1	feqmptd 5699	. . . 4 |- ( ph -> F = ( x e. X |-> ( F ` x ) ) )
76	21, 68, 69, 74, 75	offval2 6251	. . 3 |- ( ph -> ( ( S _D G ) oF x. F ) = ( x e. X |-> ( ( ( S _D G ) ` x ) x. ( F ` x ) ) ) )
77	21, 65, 70, 73, 76	offval2 6251	. 2 |- ( ph -> ( ( ( S _D F ) oF x. G ) oF + ( ( S _D G ) oF x. F ) ) = ( x e. X |-> ( ( ( ( S _D F ) ` x ) x. ( G ` x ) ) + ( ( ( S _D G ) ` x ) x. ( F ` x ) ) ) ) )
78	16, 60, 77	3eqtr4d 2274	1 |- ( ph -> ( S _D ( F oF x. G ) ) = ( ( ( S _D F ) oF x. G ) oF + ( ( S _D G ) oF x. F ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   _Vcvv 2802   C_ wss 3200   {cpr 3670   class class class wbr 4088   |-> cmpt 4150   dom cdm 4725   o. ccom 4729   Rel wrel 4730   Fun wfun 5320   -->wf 5322   ` cfv 5326  (class class class)co 6018   oFcof 6233   ^pm cpm 6818   CCcc 8030   RRcr 8031   + caddc 8035   x. cmul 8037   - cmin 8350   abscabs 11562   MetOpencmopn 14561   _D cdv 15385

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152  ax-addf 8154  ax-mulf 8155

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-of 6235  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-map 6819  df-pm 6820  df-sup 7183  df-inf 7184  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-xneg 10007  df-xadd 10008  df-seqfrec 10711  df-exp 10802  df-cj 11407  df-re 11408  df-im 11409  df-rsqrt 11563  df-abs 11564  df-rest 13329  df-topgen 13348  df-psmet 14563  df-xmet 14564  df-met 14565  df-bl 14566  df-mopn 14567  df-top 14728  df-topon 14741  df-bases 14773  df-ntr 14826  df-cn 14918  df-cnp 14919  df-tx 14983  df-cncf 15301  df-limced 15386  df-dvap 15387

This theorem is referenced by:  dvexp  15441  dvmptmulx  15450