Metamath Proof Explorer

Theorem dvmptidg

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

Ref Expression
Hypotheses dvmptidg.s 
|- ( ph -> S e. { RR , CC } )
dvmptidg.a 
|- ( ph -> A e. ( ( TopOpen ` CCfld ) |`t S ) )
Assertion dvmptidg 
|- ( ph -> ( S _D ( x e. A |-> x ) ) = ( x e. A |-> 1 ) )

Proof

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