Theorem dvmptc 15412

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

Hypotheses

Ref	Expression
dvmptid.1	|- ( ph -> S e. { RR , CC } )
dvmptc.2	|- ( ph -> A e. CC )

Assertion

Ref	Expression
dvmptc	|- ( ph -> ( S _D ( x e. S |-> A ) ) = ( x e. S |-> 0 ) )

Distinct variable groups:   x, A   ph, x   x, S

Proof of Theorem dvmptc

Step	Hyp	Ref	Expression
1		dvmptc.2	. . . . . 6 |- ( ph -> A e. CC )
2		dvconstre 15391	. . . . . 6 |- ( A e. CC -> ( RR _D ( RR X. { A } ) ) = ( RR X. { 0 } ) )
3	1, 2	syl 14	. . . . 5 |- ( ph -> ( RR _D ( RR X. { A } ) ) = ( RR X. { 0 } ) )
4	3	adantr 276	. . . 4 |- ( ( ph /\ S = RR ) -> ( RR _D ( RR X. { A } ) ) = ( RR X. { 0 } ) )
5		simpr 110	. . . . 5 |- ( ( ph /\ S = RR ) -> S = RR )
6	5	xpeq1d 4743	. . . . 5 |- ( ( ph /\ S = RR ) -> ( S X. { A } ) = ( RR X. { A } ) )
7	5, 6	oveq12d 6028	. . . 4 |- ( ( ph /\ S = RR ) -> ( S _D ( S X. { A } ) ) = ( RR _D ( RR X. { A } ) ) )
8	5	xpeq1d 4743	. . . 4 |- ( ( ph /\ S = RR ) -> ( S X. { 0 } ) = ( RR X. { 0 } ) )
9	4, 7, 8	3eqtr4d 2272	. . 3 |- ( ( ph /\ S = RR ) -> ( S _D ( S X. { A } ) ) = ( S X. { 0 } ) )
10		dvconst 15389	. . . . . 6 |- ( A e. CC -> ( CC _D ( CC X. { A } ) ) = ( CC X. { 0 } ) )
11	1, 10	syl 14	. . . . 5 |- ( ph -> ( CC _D ( CC X. { A } ) ) = ( CC X. { 0 } ) )
12	11	adantr 276	. . . 4 |- ( ( ph /\ S = CC ) -> ( CC _D ( CC X. { A } ) ) = ( CC X. { 0 } ) )
13		simpr 110	. . . . 5 |- ( ( ph /\ S = CC ) -> S = CC )
14	13	xpeq1d 4743	. . . . 5 |- ( ( ph /\ S = CC ) -> ( S X. { A } ) = ( CC X. { A } ) )
15	13, 14	oveq12d 6028	. . . 4 |- ( ( ph /\ S = CC ) -> ( S _D ( S X. { A } ) ) = ( CC _D ( CC X. { A } ) ) )
16	13	xpeq1d 4743	. . . 4 |- ( ( ph /\ S = CC ) -> ( S X. { 0 } ) = ( CC X. { 0 } ) )
17	12, 15, 16	3eqtr4d 2272	. . 3 |- ( ( ph /\ S = CC ) -> ( S _D ( S X. { A } ) ) = ( S X. { 0 } ) )
18		dvmptid.1	. . . 4 |- ( ph -> S e. { RR , CC } )
19		elpri 3689	. . . 4 |- ( S e. { RR , CC } -> ( S = RR \/ S = CC ) )
20	18, 19	syl 14	. . 3 |- ( ph -> ( S = RR \/ S = CC ) )
21	9, 17, 20	mpjaodan 803	. 2 |- ( ph -> ( S _D ( S X. { A } ) ) = ( S X. { 0 } ) )
22		fconstmpt 4768	. . 3 |- ( S X. { A } ) = ( x e. S |-> A )
23	22	oveq2i 6021	. 2 |- ( S _D ( S X. { A } ) ) = ( S _D ( x e. S |-> A ) )
24		fconstmpt 4768	. 2 |- ( S X. { 0 } ) = ( x e. S |-> 0 )
25	21, 23, 24	3eqtr3g 2285	1 |- ( ph -> ( S _D ( x e. S |-> A ) ) = ( x e. S |-> 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   {csn 3666   {cpr 3667   |-> cmpt 4145   X. cxp 4718  (class class class)co 6010   CCcc 8013   RRcr 8014   0cc0 8015   _D cdv 15350

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133  ax-arch 8134  ax-caucvg 8135

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-isom 5330  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-map 6810  df-pm 6811  df-sup 7167  df-inf 7168  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-n0 9386  df-z 9463  df-uz 9739  df-q 9832  df-rp 9867  df-xneg 9985  df-xadd 9986  df-ioo 10105  df-seqfrec 10687  df-exp 10778  df-cj 11374  df-re 11375  df-im 11376  df-rsqrt 11530  df-abs 11531  df-rest 13295  df-topgen 13314  df-psmet 14528  df-xmet 14529  df-met 14530  df-bl 14531  df-mopn 14532  df-top 14693  df-topon 14706  df-bases 14738  df-ntr 14791  df-cn 14883  df-cnp 14884  df-cncf 15266  df-limced 15351  df-dvap 15352

This theorem is referenced by: (None)