Theorem dvmptc 15631

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 15610	. . . . . 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 4774	. . . . 5 |- ( ( ph /\ S = RR ) -> ( S X. { A } ) = ( RR X. { A } ) )
7	5, 6	oveq12d 6070	. . . 4 |- ( ( ph /\ S = RR ) -> ( S _D ( S X. { A } ) ) = ( RR _D ( RR X. { A } ) ) )
8	5	xpeq1d 4774	. . . 4 |- ( ( ph /\ S = RR ) -> ( S X. { 0 } ) = ( RR X. { 0 } ) )
9	4, 7, 8	3eqtr4d 2277	. . 3 |- ( ( ph /\ S = RR ) -> ( S _D ( S X. { A } ) ) = ( S X. { 0 } ) )
10		dvconst 15608	. . . . . 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 4774	. . . . 5 |- ( ( ph /\ S = CC ) -> ( S X. { A } ) = ( CC X. { A } ) )
15	13, 14	oveq12d 6070	. . . 4 |- ( ( ph /\ S = CC ) -> ( S _D ( S X. { A } ) ) = ( CC _D ( CC X. { A } ) ) )
16	13	xpeq1d 4774	. . . 4 |- ( ( ph /\ S = CC ) -> ( S X. { 0 } ) = ( CC X. { 0 } ) )
17	12, 15, 16	3eqtr4d 2277	. . 3 |- ( ( ph /\ S = CC ) -> ( S _D ( S X. { A } ) ) = ( S X. { 0 } ) )
18		dvmptid.1	. . . 4 |- ( ph -> S e. { RR , CC } )
19		elpri 3714	. . . 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 806	. 2 |- ( ph -> ( S _D ( S X. { A } ) ) = ( S X. { 0 } ) )
22		fconstmpt 4799	. . 3 |- ( S X. { A } ) = ( x e. S |-> A )
23	22	oveq2i 6063	. 2 |- ( S _D ( S X. { A } ) ) = ( S _D ( x e. S |-> A ) )
24		fconstmpt 4799	. 2 |- ( S X. { 0 } ) = ( x e. S |-> 0 )
25	21, 23, 24	3eqtr3g 2290	1 |- ( ph -> ( S _D ( x e. S |-> A ) ) = ( x e. S |-> 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2205   {csn 3691   {cpr 3692   |-> cmpt 4173   X. cxp 4749  (class class class)co 6052   CCcc 8130   RRcr 8131   0cc0 8132   _D cdv 15569

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251  ax-caucvg 8252

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-map 6886  df-pm 6887  df-sup 7277  df-inf 7278  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-n0 9502  df-z 9583  df-uz 9860  df-q 9958  df-rp 9993  df-xneg 10111  df-xadd 10112  df-ioo 10231  df-seqfrec 10817  df-exp 10908  df-cj 11535  df-re 11536  df-im 11537  df-rsqrt 11691  df-abs 11692  df-rest 13475  df-topgen 13494  df-psmet 14740  df-xmet 14741  df-met 14742  df-bl 14743  df-mopn 14744  df-top 14912  df-topon 14925  df-bases 14957  df-ntr 15010  df-cn 15102  df-cnp 15103  df-cncf 15485  df-limced 15570  df-dvap 15571

This theorem is referenced by: (None)