Metamath Proof Explorer

Theorem dvmptconst

Description: Function-builder for derivative: derivative of a constant. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses dvmptconst.s 
|- ( ph -> S e. { RR , CC } )
dvmptconst.a 
|- ( ph -> A e. ( ( TopOpen ` CCfld ) |`t S ) )
dvmptconst.b 
|- ( ph -> B e. CC )
Assertion dvmptconst 
|- ( ph -> ( S _D ( x e. A |-> B ) ) = ( x e. A |-> 0 ) )

Proof

Step Hyp Ref Expression
1 dvmptconst.s 
 |-  ( ph -> S e. { RR , CC } )
2 dvmptconst.a 
 |-  ( ph -> A e. ( ( TopOpen ` CCfld ) |`t S ) )
3 dvmptconst.b 
 |-  ( ph -> B e. CC )
4 3 adantr 
 |-  ( ( ph /\ x e. S ) -> B e. CC )
5 0red 
 |-  ( ( ph /\ x e. S ) -> 0 e. RR )
6 1 3 dvmptc 
 |-  ( ph -> ( S _D ( x e. S |-> B ) ) = ( x e. S |-> 0 ) )
7 eqid 
 |-  ( TopOpen ` CCfld ) = ( TopOpen ` CCfld )
8 7 cnfldtopon 
 |-  ( TopOpen ` CCfld ) e. ( TopOn ` CC )
9 8 a1i 
 |-  ( ph -> ( TopOpen ` CCfld ) e. ( TopOn ` CC ) )
10 ax-resscn 
 |-  RR C_ CC
11 sseq1 
 |-  ( S = RR -> ( S C_ CC <-> RR C_ CC ) )
12 10 11 mpbiri 
 |-  ( S = RR -> S C_ CC )
13 eqimss 
 |-  ( S = CC -> S C_ CC )
14 12 13 pm3.2i 
 |-  ( ( S = RR -> S C_ CC ) /\ ( S = CC -> S C_ CC ) )
15 elpri 
 |-  ( S e. { RR , CC } -> ( S = RR \/ S = CC ) )
16 1 15 syl 
 |-  ( ph -> ( S = RR \/ S = CC ) )
17 pm3.44 
 |-  ( ( ( S = RR -> S C_ CC ) /\ ( S = CC -> S C_ CC ) ) -> ( ( S = RR \/ S = CC ) -> S C_ CC ) )
18 14 16 17 mpsyl 
 |-  ( ph -> S C_ CC )
19 resttopon 
 |-  ( ( ( TopOpen ` CCfld ) e. ( TopOn ` CC ) /\ S C_ CC ) -> ( ( TopOpen ` CCfld ) |`t S ) e. ( TopOn ` S ) )
20 9 18 19 syl2anc 
 |-  ( ph -> ( ( TopOpen ` CCfld ) |`t S ) e. ( TopOn ` S ) )
21 toponss 
 |-  ( ( ( ( TopOpen ` CCfld ) |`t S ) e. ( TopOn ` S ) /\ A e. ( ( TopOpen ` CCfld ) |`t S ) ) -> A C_ S )
22 20 2 21 syl2anc 
 |-  ( ph -> A C_ S )
23 eqid 
 |-  ( ( TopOpen ` CCfld ) |`t S ) = ( ( TopOpen ` CCfld ) |`t S )
24 1 4 5 6 22 23 7 2 dvmptres 
 |-  ( ph -> ( S _D ( x e. A |-> B ) ) = ( x e. A |-> 0 ) )