Theorem dvmptmulx 15463

Description: Function-builder for derivative, product rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.)

Hypotheses

Ref	Expression
dvmptadd.s	|- ( ph -> S e. { RR , CC } )
dvmptadd.a	|- ( ( ph /\ x e. X ) -> A e. CC )
dvmptadd.b	|- ( ( ph /\ x e. X ) -> B e. V )
dvmptadd.da	|- ( ph -> ( S _D ( x e. X |-> A ) ) = ( x e. X |-> B ) )
dvmptclx.ss	|- ( ph -> X C_ S )
dvmptadd.c	|- ( ( ph /\ x e. X ) -> C e. CC )
dvmptadd.d	|- ( ( ph /\ x e. X ) -> D e. W )
dvmptadd.dc	|- ( ph -> ( S _D ( x e. X |-> C ) ) = ( x e. X |-> D ) )

Assertion

Ref	Expression
dvmptmulx	|- ( ph -> ( S _D ( x e. X |-> ( A x. C ) ) ) = ( x e. X |-> ( ( B x. C ) + ( D x. A ) ) ) )

Distinct variable groups:   ph, x   x, S   x, V   x, W   x, X

Allowed substitution hints:   A( x)   B( x)   C( x)   D( x)

Proof of Theorem dvmptmulx

Step	Hyp	Ref	Expression
1		dvmptadd.s	. . 3 |- ( ph -> S e. { RR , CC } )
2		dvmptclx.ss	. . 3 |- ( ph -> X C_ S )
3		dvmptadd.a	. . . 4 |- ( ( ph /\ x e. X ) -> A e. CC )
4	3	fmpttd 5802	. . 3 |- ( ph -> ( x e. X |-> A ) : X --> CC )
5		dvmptadd.c	. . . 4 |- ( ( ph /\ x e. X ) -> C e. CC )
6	5	fmpttd 5802	. . 3 |- ( ph -> ( x e. X |-> C ) : X --> CC )
7		dvmptadd.da	. . . . 5 |- ( ph -> ( S _D ( x e. X |-> A ) ) = ( x e. X |-> B ) )
8	7	dmeqd 4933	. . . 4 |- ( ph -> dom ( S _D ( x e. X |-> A ) ) = dom ( x e. X |-> B ) )
9		dvmptadd.b	. . . . . 6 |- ( ( ph /\ x e. X ) -> B e. V )
10	9	ralrimiva 2605	. . . . 5 |- ( ph -> A. x e. X B e. V )
11		dmmptg 5234	. . . . 5 |- ( A. x e. X B e. V -> dom ( x e. X |-> B ) = X )
12	10, 11	syl 14	. . . 4 |- ( ph -> dom ( x e. X |-> B ) = X )
13	8, 12	eqtrd 2264	. . 3 |- ( ph -> dom ( S _D ( x e. X |-> A ) ) = X )
14		dvmptadd.dc	. . . . 5 |- ( ph -> ( S _D ( x e. X |-> C ) ) = ( x e. X |-> D ) )
15	14	dmeqd 4933	. . . 4 |- ( ph -> dom ( S _D ( x e. X |-> C ) ) = dom ( x e. X |-> D ) )
16		dvmptadd.d	. . . . . 6 |- ( ( ph /\ x e. X ) -> D e. W )
17	16	ralrimiva 2605	. . . . 5 |- ( ph -> A. x e. X D e. W )
18		dmmptg 5234	. . . . 5 |- ( A. x e. X D e. W -> dom ( x e. X |-> D ) = X )
19	17, 18	syl 14	. . . 4 |- ( ph -> dom ( x e. X |-> D ) = X )
20	15, 19	eqtrd 2264	. . 3 |- ( ph -> dom ( S _D ( x e. X |-> C ) ) = X )
21	1, 2, 4, 6, 13, 20	dvimulf 15449	. 2 |- ( ph -> ( S _D ( ( x e. X |-> A ) oF x. ( x e. X |-> C ) ) ) = ( ( ( S _D ( x e. X |-> A ) ) oF x. ( x e. X |-> C ) ) oF + ( ( S _D ( x e. X |-> C ) ) oF x. ( x e. X |-> A ) ) ) )
22	1, 2	ssexd 4229	. . . 4 |- ( ph -> X e. _V )
23		eqidd 2232	. . . 4 |- ( ph -> ( x e. X |-> A ) = ( x e. X |-> A ) )
24		eqidd 2232	. . . 4 |- ( ph -> ( x e. X |-> C ) = ( x e. X |-> C ) )
25	22, 3, 5, 23, 24	offval2 6251	. . 3 |- ( ph -> ( ( x e. X |-> A ) oF x. ( x e. X |-> C ) ) = ( x e. X |-> ( A x. C ) ) )
26	25	oveq2d 6034	. 2 |- ( ph -> ( S _D ( ( x e. X |-> A ) oF x. ( x e. X |-> C ) ) ) = ( S _D ( x e. X |-> ( A x. C ) ) ) )
27	1, 3, 9, 7, 2	dvmptclx 15461	. . . 4 |- ( ( ph /\ x e. X ) -> B e. CC )
28	27, 5	mulcld 8200	. . 3 |- ( ( ph /\ x e. X ) -> ( B x. C ) e. CC )
29	1, 5, 16, 14, 2	dvmptclx 15461	. . . 4 |- ( ( ph /\ x e. X ) -> D e. CC )
30	29, 3	mulcld 8200	. . 3 |- ( ( ph /\ x e. X ) -> ( D x. A ) e. CC )
31	22, 9, 5, 7, 24	offval2 6251	. . 3 |- ( ph -> ( ( S _D ( x e. X |-> A ) ) oF x. ( x e. X |-> C ) ) = ( x e. X |-> ( B x. C ) ) )
32	22, 16, 3, 14, 23	offval2 6251	. . 3 |- ( ph -> ( ( S _D ( x e. X |-> C ) ) oF x. ( x e. X |-> A ) ) = ( x e. X |-> ( D x. A ) ) )
33	22, 28, 30, 31, 32	offval2 6251	. 2 |- ( ph -> ( ( ( S _D ( x e. X |-> A ) ) oF x. ( x e. X |-> C ) ) oF + ( ( S _D ( x e. X |-> C ) ) oF x. ( x e. X |-> A ) ) ) = ( x e. X |-> ( ( B x. C ) + ( D x. A ) ) ) )
34	21, 26, 33	3eqtr3d 2272	1 |- ( ph -> ( S _D ( x e. X |-> ( A x. C ) ) ) = ( x e. X |-> ( ( B x. C ) + ( D x. A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   C_ wss 3200   {cpr 3670   |-> cmpt 4150   dom cdm 4725  (class class class)co 6018   oFcof 6233   CCcc 8030   RRcr 8031   + caddc 8035   x. cmul 8037   _D cdv 15398

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 11420  df-re 11421  df-im 11422  df-rsqrt 11576  df-abs 11577  df-rest 13342  df-topgen 13361  df-psmet 14576  df-xmet 14577  df-met 14578  df-bl 14579  df-mopn 14580  df-top 14741  df-topon 14754  df-bases 14786  df-ntr 14839  df-cn 14931  df-cnp 14932  df-tx 14996  df-cncf 15314  df-limced 15399  df-dvap 15400

This theorem is referenced by:  dvmptcmulcn  15464