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Metamath Proof Explorer

Theorem mpocnfldmul

Description: The multiplication operation of the field of complex numbers. Version of cnfldmul using maps-to notation, which does not require ax-mulf . (Contributed by GG, 31-Mar-2025)

Ref Expression
Assertion mpocnfldmul 
|- ( x e. CC , y e. CC |-> ( x x. y ) ) = ( .r ` CCfld )

Proof

Step Hyp Ref Expression
1 mpomulex 
 |-  ( x e. CC , y e. CC |-> ( x x. y ) ) e. _V
2 cnfldstr 
 |-  CCfld Struct <. 1 , ; 1 3 >.
3 mulridx 
 |-  .r = Slot ( .r ` ndx )
4 snsstp3 
 |-  { <. ( .r ` ndx ) , ( x e. CC , y e. CC |-> ( x x. y ) ) >. } C_ { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , ( x e. CC , y e. CC |-> ( x + y ) ) >. , <. ( .r ` ndx ) , ( x e. CC , y e. CC |-> ( x x. y ) ) >. }
5 ssun1 
 |-  { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , ( x e. CC , y e. CC |-> ( x + y ) ) >. , <. ( .r ` ndx ) , ( x e. CC , y e. CC |-> ( x x. y ) ) >. } C_ ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , ( x e. CC , y e. CC |-> ( x + y ) ) >. , <. ( .r ` ndx ) , ( x e. CC , y e. CC |-> ( x x. y ) ) >. } u. { <. ( *r ` ndx ) , * >. } )
6 ssun1 
 |-  ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , ( x e. CC , y e. CC |-> ( x + y ) ) >. , <. ( .r ` ndx ) , ( x e. CC , y e. CC |-> ( x x. y ) ) >. } u. { <. ( *r ` ndx ) , * >. } ) C_ ( ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , ( x e. CC , y e. CC |-> ( x + y ) ) >. , <. ( .r ` ndx ) , ( x e. CC , y e. CC |-> ( x x. y ) ) >. } u. { <. ( *r ` ndx ) , * >. } ) u. ( { <. ( TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , ( metUnif ` ( abs o. - ) ) >. } ) )
7 df-cnfld 
 |-  CCfld = ( ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , ( x e. CC , y e. CC |-> ( x + y ) ) >. , <. ( .r ` ndx ) , ( x e. CC , y e. CC |-> ( x x. y ) ) >. } u. { <. ( *r ` ndx ) , * >. } ) u. ( { <. ( TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , ( metUnif ` ( abs o. - ) ) >. } ) )
8 6 7 sseqtrri 
 |-  ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , ( x e. CC , y e. CC |-> ( x + y ) ) >. , <. ( .r ` ndx ) , ( x e. CC , y e. CC |-> ( x x. y ) ) >. } u. { <. ( *r ` ndx ) , * >. } ) C_ CCfld
9 5 8 sstri 
 |-  { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , ( x e. CC , y e. CC |-> ( x + y ) ) >. , <. ( .r ` ndx ) , ( x e. CC , y e. CC |-> ( x x. y ) ) >. } C_ CCfld
10 4 9 sstri 
 |-  { <. ( .r ` ndx ) , ( x e. CC , y e. CC |-> ( x x. y ) ) >. } C_ CCfld
11 2 3 10 strfv 
 |-  ( ( x e. CC , y e. CC |-> ( x x. y ) ) e. _V -> ( x e. CC , y e. CC |-> ( x x. y ) ) = ( .r ` CCfld ) )
12 1 11 ax-mp 
 |-  ( x e. CC , y e. CC |-> ( x x. y ) ) = ( .r ` CCfld )