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Metamath Proof Explorer

Theorem dvmptres2

Description: Function-builder for derivative: restrict a derivative to a subset. (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 ) )
dvmptres2.z 
|- ( ph -> Z C_ X )
dvmptres2.j 
|- J = ( K |`t S )
dvmptres2.k 
|- K = ( TopOpen ` CCfld )
dvmptres2.i 
|- ( ph -> ( ( int ` J ) ` Z ) = Y )
Assertion dvmptres2 
|- ( ph -> ( S _D ( x e. Z |-> A ) ) = ( x e. Y |-> 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 dvmptres2.z 
 |-  ( ph -> Z C_ X )
6 dvmptres2.j 
 |-  J = ( K |`t S )
7 dvmptres2.k 
 |-  K = ( TopOpen ` CCfld )
8 dvmptres2.i 
 |-  ( ph -> ( ( int ` J ) ` Z ) = Y )
9 recnprss 
 |-  ( S e. { RR , CC } -> S C_ CC )
10 1 9 syl 
 |-  ( ph -> S C_ CC )
11 2 fmpttd 
 |-  ( ph -> ( x e. X |-> A ) : X --> CC )
12 4 dmeqd 
 |-  ( ph -> dom ( S _D ( x e. X |-> A ) ) = dom ( x e. X |-> B ) )
13 3 ralrimiva 
 |-  ( ph -> A. x e. X B e. V )
14 dmmptg 
 |-  ( A. x e. X B e. V -> dom ( x e. X |-> B ) = X )
15 13 14 syl 
 |-  ( ph -> dom ( x e. X |-> B ) = X )
16 12 15 eqtrd 
 |-  ( ph -> dom ( S _D ( x e. X |-> A ) ) = X )
17 dvbsss 
 |-  dom ( S _D ( x e. X |-> A ) ) C_ S
18 16 17 eqsstrrdi 
 |-  ( ph -> X C_ S )
19 5 18 sstrd 
 |-  ( ph -> Z C_ S )
20 7 6 dvres 
 |-  ( ( ( S C_ CC /\ ( x e. X |-> A ) : X --> CC ) /\ ( X C_ S /\ Z C_ S ) ) -> ( S _D ( ( x e. X |-> A ) |` Z ) ) = ( ( S _D ( x e. X |-> A ) ) |` ( ( int ` J ) ` Z ) ) )
21 10 11 18 19 20 syl22anc 
 |-  ( ph -> ( S _D ( ( x e. X |-> A ) |` Z ) ) = ( ( S _D ( x e. X |-> A ) ) |` ( ( int ` J ) ` Z ) ) )
22 5 resmptd 
 |-  ( ph -> ( ( x e. X |-> A ) |` Z ) = ( x e. Z |-> A ) )
23 22 oveq2d 
 |-  ( ph -> ( S _D ( ( x e. X |-> A ) |` Z ) ) = ( S _D ( x e. Z |-> A ) ) )
24 4 reseq1d 
 |-  ( ph -> ( ( S _D ( x e. X |-> A ) ) |` ( ( int ` J ) ` Z ) ) = ( ( x e. X |-> B ) |` ( ( int ` J ) ` Z ) ) )
25 8 reseq2d 
 |-  ( ph -> ( ( x e. X |-> B ) |` ( ( int ` J ) ` Z ) ) = ( ( x e. X |-> B ) |` Y ) )
26 7 cnfldtopon 
 |-  K e. ( TopOn ` CC )
27 resttopon 
 |-  ( ( K e. ( TopOn ` CC ) /\ S C_ CC ) -> ( K |`t S ) e. ( TopOn ` S ) )
28 26 10 27 sylancr 
 |-  ( ph -> ( K |`t S ) e. ( TopOn ` S ) )
29 6 28 eqeltrid 
 |-  ( ph -> J e. ( TopOn ` S ) )
30 topontop 
 |-  ( J e. ( TopOn ` S ) -> J e. Top )
31 29 30 syl 
 |-  ( ph -> J e. Top )
32 toponuni 
 |-  ( J e. ( TopOn ` S ) -> S = U. J )
33 29 32 syl 
 |-  ( ph -> S = U. J )
34 19 33 sseqtrd 
 |-  ( ph -> Z C_ U. J )
35 eqid 
 |-  U. J = U. J
36 35 ntrss2 
 |-  ( ( J e. Top /\ Z C_ U. J ) -> ( ( int ` J ) ` Z ) C_ Z )
37 31 34 36 syl2anc 
 |-  ( ph -> ( ( int ` J ) ` Z ) C_ Z )
38 8 37 eqsstrrd 
 |-  ( ph -> Y C_ Z )
39 38 5 sstrd 
 |-  ( ph -> Y C_ X )
40 39 resmptd 
 |-  ( ph -> ( ( x e. X |-> B ) |` Y ) = ( x e. Y |-> B ) )
41 24 25 40 3eqtrd 
 |-  ( ph -> ( ( S _D ( x e. X |-> A ) ) |` ( ( int ` J ) ` Z ) ) = ( x e. Y |-> B ) )
42 21 23 41 3eqtr3d 
 |-  ( ph -> ( S _D ( x e. Z |-> A ) ) = ( x e. Y |-> B ) )