Theorem dvmptc 15470

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 15449	. . . . . 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 4750	. . . . 5 |- ( ( ph /\ S = RR ) -> ( S X. { A } ) = ( RR X. { A } ) )
7	5, 6	oveq12d 6041	. . . 4 |- ( ( ph /\ S = RR ) -> ( S _D ( S X. { A } ) ) = ( RR _D ( RR X. { A } ) ) )
8	5	xpeq1d 4750	. . . 4 |- ( ( ph /\ S = RR ) -> ( S X. { 0 } ) = ( RR X. { 0 } ) )
9	4, 7, 8	3eqtr4d 2273	. . 3 |- ( ( ph /\ S = RR ) -> ( S _D ( S X. { A } ) ) = ( S X. { 0 } ) )
10		dvconst 15447	. . . . . 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 4750	. . . . 5 |- ( ( ph /\ S = CC ) -> ( S X. { A } ) = ( CC X. { A } ) )
15	13, 14	oveq12d 6041	. . . 4 |- ( ( ph /\ S = CC ) -> ( S _D ( S X. { A } ) ) = ( CC _D ( CC X. { A } ) ) )
16	13	xpeq1d 4750	. . . 4 |- ( ( ph /\ S = CC ) -> ( S X. { 0 } ) = ( CC X. { 0 } ) )
17	12, 15, 16	3eqtr4d 2273	. . 3 |- ( ( ph /\ S = CC ) -> ( S _D ( S X. { A } ) ) = ( S X. { 0 } ) )
18		dvmptid.1	. . . 4 |- ( ph -> S e. { RR , CC } )
19		elpri 3693	. . . 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 805	. 2 |- ( ph -> ( S _D ( S X. { A } ) ) = ( S X. { 0 } ) )
22		fconstmpt 4775	. . 3 |- ( S X. { A } ) = ( x e. S |-> A )
23	22	oveq2i 6034	. 2 |- ( S _D ( S X. { A } ) ) = ( S _D ( x e. S |-> A ) )
24		fconstmpt 4775	. 2 |- ( S X. { 0 } ) = ( x e. S |-> 0 )
25	21, 23, 24	3eqtr3g 2286	1 |- ( ph -> ( S _D ( x e. S |-> A ) ) = ( x e. S |-> 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 715   = wceq 1397   e. wcel 2201   {csn 3670   {cpr 3671   |-> cmpt 4151   X. cxp 4725  (class class class)co 6023   CCcc 8035   RRcr 8036   0cc0 8037   _D cdv 15408

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 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-mulrcl 8136  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-precex 8147  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153  ax-pre-mulgt0 8154  ax-pre-mulext 8155  ax-arch 8156  ax-caucvg 8157

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 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-po 4395  df-iso 4396  df-iord 4465  df-on 4467  df-ilim 4468  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-isom 5337  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-frec 6562  df-map 6824  df-pm 6825  df-sup 7188  df-inf 7189  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-reap 8760  df-ap 8767  df-div 8858  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-n0 9408  df-z 9485  df-uz 9761  df-q 9859  df-rp 9894  df-xneg 10012  df-xadd 10013  df-ioo 10132  df-seqfrec 10716  df-exp 10807  df-cj 11425  df-re 11426  df-im 11427  df-rsqrt 11581  df-abs 11582  df-rest 13347  df-topgen 13366  df-psmet 14581  df-xmet 14582  df-met 14583  df-bl 14584  df-mopn 14585  df-top 14751  df-topon 14764  df-bases 14796  df-ntr 14849  df-cn 14941  df-cnp 14942  df-cncf 15324  df-limced 15409  df-dvap 15410

This theorem is referenced by: (None)