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Metamath Proof Explorer

Theorem cndrngOLD

Description: Obsolete version of cndrng as of 30-Apr-2025. (Contributed by Stefan O'Rear, 27-Nov-2014) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion cndrngOLD 
|- CCfld e. DivRing

Proof

Step Hyp Ref Expression
1 cnfldbas 
 |-  CC = ( Base ` CCfld )
2 1 a1i 
 |-  ( T. -> CC = ( Base ` CCfld ) )
3 cnfldmul 
 |-  x. = ( .r ` CCfld )
4 3 a1i 
 |-  ( T. -> x. = ( .r ` CCfld ) )
5 cnfld0 
 |-  0 = ( 0g ` CCfld )
6 5 a1i 
 |-  ( T. -> 0 = ( 0g ` CCfld ) )
7 cnfld1 
 |-  1 = ( 1r ` CCfld )
8 7 a1i 
 |-  ( T. -> 1 = ( 1r ` CCfld ) )
9 cnring 
 |-  CCfld e. Ring
10 9 a1i 
 |-  ( T. -> CCfld e. Ring )
11 mulne0 
 |-  ( ( ( x e. CC /\ x =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( x x. y ) =/= 0 )
12 11 3adant1 
 |-  ( ( T. /\ ( x e. CC /\ x =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( x x. y ) =/= 0 )
13 ax-1ne0 
 |-  1 =/= 0
14 13 a1i 
 |-  ( T. -> 1 =/= 0 )
15 reccl 
 |-  ( ( x e. CC /\ x =/= 0 ) -> ( 1 / x ) e. CC )
16 15 adantl 
 |-  ( ( T. /\ ( x e. CC /\ x =/= 0 ) ) -> ( 1 / x ) e. CC )
17 recid2 
 |-  ( ( x e. CC /\ x =/= 0 ) -> ( ( 1 / x ) x. x ) = 1 )
18 17 adantl 
 |-  ( ( T. /\ ( x e. CC /\ x =/= 0 ) ) -> ( ( 1 / x ) x. x ) = 1 )
19 2 4 6 8 10 12 14 16 18 isdrngd 
 |-  ( T. -> CCfld e. DivRing )
20 19 mptru 
 |-  CCfld e. DivRing