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Metamath Proof Explorer

Theorem dvmptcmul

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

Ref Expression
Hypotheses 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 ) )
dvmptcmul.c 
|- ( ph -> C e. CC )
Assertion dvmptcmul 
|- ( ph -> ( S _D ( x e. X |-> ( C x. A ) ) ) = ( x e. X |-> ( C x. B ) ) )

Proof

Step Hyp Ref Expression
1 dvmptadd.s 
 |-  ( ph -> S e. { RR , CC } )
2 dvmptadd.a 
 |-  ( ( ph /\ x e. X ) -> A e. CC )
3 dvmptadd.b 
 |-  ( ( ph /\ x e. X ) -> B e. V )
4 dvmptadd.da 
 |-  ( ph -> ( S _D ( x e. X |-> A ) ) = ( x e. X |-> B ) )
5 dvmptcmul.c 
 |-  ( ph -> C e. CC )
6 5 adantr 
 |-  ( ( ph /\ x e. X ) -> C e. CC )
7 0cnd 
 |-  ( ( ph /\ x e. X ) -> 0 e. CC )
8 5 adantr 
 |-  ( ( ph /\ x e. S ) -> C e. CC )
9 0cnd 
 |-  ( ( ph /\ x e. S ) -> 0 e. CC )
10 1 5 dvmptc 
 |-  ( ph -> ( S _D ( x e. S |-> C ) ) = ( x e. S |-> 0 ) )
11 4 dmeqd 
 |-  ( ph -> dom ( S _D ( x e. X |-> A ) ) = dom ( x e. X |-> B ) )
12 3 ralrimiva 
 |-  ( ph -> A. x e. X B e. V )
13 dmmptg 
 |-  ( A. x e. X B e. V -> dom ( x e. X |-> B ) = X )
14 12 13 syl 
 |-  ( ph -> dom ( x e. X |-> B ) = X )
15 11 14 eqtrd 
 |-  ( ph -> dom ( S _D ( x e. X |-> A ) ) = X )
16 dvbsss 
 |-  dom ( S _D ( x e. X |-> A ) ) C_ S
17 15 16 eqsstrrdi 
 |-  ( ph -> X C_ S )
18 eqid 
 |-  ( ( TopOpen ` CCfld ) |`t S ) = ( ( TopOpen ` CCfld ) |`t S )
19 eqid 
 |-  ( TopOpen ` CCfld ) = ( TopOpen ` CCfld )
20 19 cnfldtopon 
 |-  ( TopOpen ` CCfld ) e. ( TopOn ` CC )
21 recnprss 
 |-  ( S e. { RR , CC } -> S C_ CC )
22 1 21 syl 
 |-  ( ph -> S C_ CC )
23 resttopon 
 |-  ( ( ( TopOpen ` CCfld ) e. ( TopOn ` CC ) /\ S C_ CC ) -> ( ( TopOpen ` CCfld ) |`t S ) e. ( TopOn ` S ) )
24 20 22 23 sylancr 
 |-  ( ph -> ( ( TopOpen ` CCfld ) |`t S ) e. ( TopOn ` S ) )
25 topontop 
 |-  ( ( ( TopOpen ` CCfld ) |`t S ) e. ( TopOn ` S ) -> ( ( TopOpen ` CCfld ) |`t S ) e. Top )
26 24 25 syl 
 |-  ( ph -> ( ( TopOpen ` CCfld ) |`t S ) e. Top )
27 toponuni 
 |-  ( ( ( TopOpen ` CCfld ) |`t S ) e. ( TopOn ` S ) -> S = U. ( ( TopOpen ` CCfld ) |`t S ) )
28 24 27 syl 
 |-  ( ph -> S = U. ( ( TopOpen ` CCfld ) |`t S ) )
29 17 28 sseqtrd 
 |-  ( ph -> X C_ U. ( ( TopOpen ` CCfld ) |`t S ) )
30 eqid 
 |-  U. ( ( TopOpen ` CCfld ) |`t S ) = U. ( ( TopOpen ` CCfld ) |`t S )
31 30 ntrss2 
 |-  ( ( ( ( TopOpen ` CCfld ) |`t S ) e. Top /\ X C_ U. ( ( TopOpen ` CCfld ) |`t S ) ) -> ( ( int ` ( ( TopOpen ` CCfld ) |`t S ) ) ` X ) C_ X )
32 26 29 31 syl2anc 
 |-  ( ph -> ( ( int ` ( ( TopOpen ` CCfld ) |`t S ) ) ` X ) C_ X )
33 2 fmpttd 
 |-  ( ph -> ( x e. X |-> A ) : X --> CC )
34 22 33 17 18 19 dvbssntr 
 |-  ( ph -> dom ( S _D ( x e. X |-> A ) ) C_ ( ( int ` ( ( TopOpen ` CCfld ) |`t S ) ) ` X ) )
35 15 34 eqsstrrd 
 |-  ( ph -> X C_ ( ( int ` ( ( TopOpen ` CCfld ) |`t S ) ) ` X ) )
36 32 35 eqssd 
 |-  ( ph -> ( ( int ` ( ( TopOpen ` CCfld ) |`t S ) ) ` X ) = X )
37 1 8 9 10 17 18 19 36 dvmptres2 
 |-  ( ph -> ( S _D ( x e. X |-> C ) ) = ( x e. X |-> 0 ) )
38 1 6 7 37 2 3 4 dvmptmul 
 |-  ( ph -> ( S _D ( x e. X |-> ( C x. A ) ) ) = ( x e. X |-> ( ( 0 x. A ) + ( B x. C ) ) ) )
39 2 mul02d 
 |-  ( ( ph /\ x e. X ) -> ( 0 x. A ) = 0 )
40 39 oveq1d 
 |-  ( ( ph /\ x e. X ) -> ( ( 0 x. A ) + ( B x. C ) ) = ( 0 + ( B x. C ) ) )
41 1 2 3 4 dvmptcl 
 |-  ( ( ph /\ x e. X ) -> B e. CC )
42 41 6 mulcld 
 |-  ( ( ph /\ x e. X ) -> ( B x. C ) e. CC )
43 42 addlidd 
 |-  ( ( ph /\ x e. X ) -> ( 0 + ( B x. C ) ) = ( B x. C ) )
44 41 6 mulcomd 
 |-  ( ( ph /\ x e. X ) -> ( B x. C ) = ( C x. B ) )
45 40 43 44 3eqtrd 
 |-  ( ( ph /\ x e. X ) -> ( ( 0 x. A ) + ( B x. C ) ) = ( C x. B ) )
46 45 mpteq2dva 
 |-  ( ph -> ( x e. X |-> ( ( 0 x. A ) + ( B x. C ) ) ) = ( x e. X |-> ( C x. B ) ) )
47 38 46 eqtrd 
 |-  ( ph -> ( S _D ( x e. X |-> ( C x. A ) ) ) = ( x e. X |-> ( C x. B ) ) )