Theorem dvmulf 23706

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 } )
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
dvmulf	|- ( ph -> ( S _D ( F oF x. G ) ) = ( ( ( S _D F ) oF x. G ) oF + ( ( S _D G ) oF x. F ) ) )

Proof of Theorem dvmulf

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 481	. . . 4 |- ( ( ph /\ x e. X ) -> F : X --> CC )
3		dvaddf.df	. . . . . 6 |- ( ph -> dom ( S _D F ) = X )
4		dvbsss 23666	. . . . . 6 |- dom ( S _D F ) C_ S
5	3, 4	syl6eqssr 3656	. . . . 5 |- ( ph -> X C_ S )
6	5	adantr 481	. . . 4 |- ( ( ph /\ x e. X ) -> X C_ S )
7		dvaddf.g	. . . . 5 |- ( ph -> G : X --> CC )
8	7	adantr 481	. . . 4 |- ( ( ph /\ x e. X ) -> G : X --> CC )
9		dvaddf.s	. . . . 5 |- ( ph -> S e. { RR , CC } )
10	9	adantr 481	. . . 4 |- ( ( ph /\ x e. X ) -> S e. { RR , CC } )
11	3	eleq2d 2687	. . . . 5 |- ( ph -> ( x e. dom ( S _D F ) <-> x e. X ) )
12	11	biimpar 502	. . . 4 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D F ) )
13		dvaddf.dg	. . . . . 6 |- ( ph -> dom ( S _D G ) = X )
14	13	eleq2d 2687	. . . . 5 |- ( ph -> ( x e. dom ( S _D G ) <-> x e. X ) )
15	14	biimpar 502	. . . 4 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D G ) )
16	2, 6, 8, 6, 10, 12, 15	dvmul 23704	. . 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 ) ) ) )
17	16	mpteq2dva 4744	. 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 ) ) ) ) )
18		dvfg 23670	. . . . 5 |- ( S e. { RR , CC } -> ( S _D ( F oF x. G ) ) : dom ( S _D ( F oF x. G ) ) --> CC )
19	9, 18	syl 17	. . . 4 |- ( ph -> ( S _D ( F oF x. G ) ) : dom ( S _D ( F oF x. G ) ) --> CC )
20		recnprss 23668	. . . . . . . 8 |- ( S e. { RR , CC } -> S C_ CC )
21	9, 20	syl 17	. . . . . . 7 |- ( ph -> S C_ CC )
22		mulcl 10020	. . . . . . . . 9 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
23	22	adantl 482	. . . . . . . 8 |- ( ( ph /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
24	9, 5	ssexd 4805	. . . . . . . 8 |- ( ph -> X e. _V )
25		inidm 3822	. . . . . . . 8 |- ( X i^i X ) = X
26	23, 1, 7, 24, 24, 25	off 6912	. . . . . . 7 |- ( ph -> ( F oF x. G ) : X --> CC )
27	21, 26, 5	dvbss 23665	. . . . . 6 |- ( ph -> dom ( S _D ( F oF x. G ) ) C_ X )
28	21	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. X ) -> S C_ CC )
29		fvexd 6203	. . . . . . . . . 10 |- ( ( ph /\ x e. X ) -> ( ( S _D F ) ` x ) e. _V )
30		fvexd 6203	. . . . . . . . . 10 |- ( ( ph /\ x e. X ) -> ( ( S _D G ) ` x ) e. _V )
31		dvfg 23670	. . . . . . . . . . . . . 14 |- ( S e. { RR , CC } -> ( S _D F ) : dom ( S _D F ) --> CC )
32	9, 31	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( S _D F ) : dom ( S _D F ) --> CC )
33	32	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ x e. X ) -> ( S _D F ) : dom ( S _D F ) --> CC )
34		ffun 6048	. . . . . . . . . . . 12 |- ( ( S _D F ) : dom ( S _D F ) --> CC -> Fun ( S _D F ) )
35		funfvbrb 6330	. . . . . . . . . . . 12 |- ( Fun ( S _D F ) -> ( x e. dom ( S _D F ) <-> x ( S _D F ) ( ( S _D F ) ` x ) ) )
36	33, 34, 35	3syl 18	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> ( x e. dom ( S _D F ) <-> x ( S _D F ) ( ( S _D F ) ` x ) ) )
37	12, 36	mpbid 222	. . . . . . . . . 10 |- ( ( ph /\ x e. X ) -> x ( S _D F ) ( ( S _D F ) ` x ) )
38		dvfg 23670	. . . . . . . . . . . . . 14 |- ( S e. { RR , CC } -> ( S _D G ) : dom ( S _D G ) --> CC )
39	9, 38	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( S _D G ) : dom ( S _D G ) --> CC )
40	39	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ x e. X ) -> ( S _D G ) : dom ( S _D G ) --> CC )
41		ffun 6048	. . . . . . . . . . . 12 |- ( ( S _D G ) : dom ( S _D G ) --> CC -> Fun ( S _D G ) )
42		funfvbrb 6330	. . . . . . . . . . . 12 |- ( Fun ( S _D G ) -> ( x e. dom ( S _D G ) <-> x ( S _D G ) ( ( S _D G ) ` x ) ) )
43	40, 41, 42	3syl 18	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> ( x e. dom ( S _D G ) <-> x ( S _D G ) ( ( S _D G ) ` x ) ) )
44	15, 43	mpbid 222	. . . . . . . . . 10 |- ( ( ph /\ x e. X ) -> x ( S _D G ) ( ( S _D G ) ` x ) )
45		eqid 2622	. . . . . . . . . 10 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
46	2, 6, 8, 6, 28, 29, 30, 37, 44, 45	dvmulbr 23702	. . . . . . . . 9 |- ( ( 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 ) ) ) )
47		reldv 23634	. . . . . . . . . 10 |- Rel ( S _D ( F oF x. G ) )
48	47	releldmi 5362	. . . . . . . . 9 |- ( 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 ) ) )
49	46, 48	syl 17	. . . . . . . 8 |- ( ( ph /\ x e. X ) -> x e. dom ( S _D ( F oF x. G ) ) )
50	49	ex 450	. . . . . . 7 |- ( ph -> ( x e. X -> x e. dom ( S _D ( F oF x. G ) ) ) )
51	50	ssrdv 3609	. . . . . 6 |- ( ph -> X C_ dom ( S _D ( F oF x. G ) ) )
52	27, 51	eqssd 3620	. . . . 5 |- ( ph -> dom ( S _D ( F oF x. G ) ) = X )
53	52	feq2d 6031	. . . 4 |- ( ph -> ( ( S _D ( F oF x. G ) ) : dom ( S _D ( F oF x. G ) ) --> CC <-> ( S _D ( F oF x. G ) ) : X --> CC ) )
54	19, 53	mpbid 222	. . 3 |- ( ph -> ( S _D ( F oF x. G ) ) : X --> CC )
55	54	feqmptd 6249	. 2 |- ( ph -> ( S _D ( F oF x. G ) ) = ( x e. X |-> ( ( S _D ( F oF x. G ) ) ` x ) ) )
56		ovexd 6680	. . 3 |- ( ( ph /\ x e. X ) -> ( ( ( S _D F ) ` x ) x. ( G ` x ) ) e. _V )
57		ovexd 6680	. . 3 |- ( ( ph /\ x e. X ) -> ( ( ( S _D G ) ` x ) x. ( F ` x ) ) e. _V )
58		fvexd 6203	. . . 4 |- ( ( ph /\ x e. X ) -> ( G ` x ) e. _V )
59	3	feq2d 6031	. . . . . 6 |- ( ph -> ( ( S _D F ) : dom ( S _D F ) --> CC <-> ( S _D F ) : X --> CC ) )
60	32, 59	mpbid 222	. . . . 5 |- ( ph -> ( S _D F ) : X --> CC )
61	60	feqmptd 6249	. . . 4 |- ( ph -> ( S _D F ) = ( x e. X |-> ( ( S _D F ) ` x ) ) )
62	7	feqmptd 6249	. . . 4 |- ( ph -> G = ( x e. X |-> ( G ` x ) ) )
63	24, 29, 58, 61, 62	offval2 6914	. . 3 |- ( ph -> ( ( S _D F ) oF x. G ) = ( x e. X |-> ( ( ( S _D F ) ` x ) x. ( G ` x ) ) ) )
64		fvexd 6203	. . . 4 |- ( ( ph /\ x e. X ) -> ( F ` x ) e. _V )
65	13	feq2d 6031	. . . . . 6 |- ( ph -> ( ( S _D G ) : dom ( S _D G ) --> CC <-> ( S _D G ) : X --> CC ) )
66	39, 65	mpbid 222	. . . . 5 |- ( ph -> ( S _D G ) : X --> CC )
67	66	feqmptd 6249	. . . 4 |- ( ph -> ( S _D G ) = ( x e. X |-> ( ( S _D G ) ` x ) ) )
68	1	feqmptd 6249	. . . 4 |- ( ph -> F = ( x e. X |-> ( F ` x ) ) )
69	24, 30, 64, 67, 68	offval2 6914	. . 3 |- ( ph -> ( ( S _D G ) oF x. F ) = ( x e. X |-> ( ( ( S _D G ) ` x ) x. ( F ` x ) ) ) )
70	24, 56, 57, 63, 69	offval2 6914	. 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 ) ) ) ) )
71	17, 55, 70	3eqtr4d 2666	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   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   + caddc 9939   x. cmul 9941   TopOpenctopn 16082  ℂ_fldccnfld 19746   _D cdv 23627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  dvcmulf  23708  dvexp  23716  dvmptmul  23724  expgrowth  38534  binomcxplemnotnn0  38555  dvmulcncf  40140