Metamath Proof Explorer

Theorem zrhunitpreima

Description: The preimage by ZRHom of the units of a division ring is ( ZZ \ { 0 } ) . (Contributed by Thierry Arnoux, 22-Oct-2017)

Ref Expression
Hypotheses zrhker.0 
|- B = ( Base ` R )
zrhker.1 
|- L = ( ZRHom ` R )
zrhker.2 
|- .0. = ( 0g ` R )
Assertion zrhunitpreima 
|- ( ( R e. DivRing /\ ( chr ` R ) = 0 ) -> ( `' L " ( Unit ` R ) ) = ( ZZ \ { 0 } ) )

Proof

Step Hyp Ref Expression
1 zrhker.0 
 |-  B = ( Base ` R )
2 zrhker.1 
 |-  L = ( ZRHom ` R )
3 zrhker.2 
 |-  .0. = ( 0g ` R )
4 eqid 
 |-  ( Unit ` R ) = ( Unit ` R )
5 eqid 
 |-  ( 0g ` R ) = ( 0g ` R )
6 1 4 5 isdrng 
 |-  ( R e. DivRing <-> ( R e. Ring /\ ( Unit ` R ) = ( B \ { ( 0g ` R ) } ) ) )
7 6 simprbi 
 |-  ( R e. DivRing -> ( Unit ` R ) = ( B \ { ( 0g ` R ) } ) )
8 7 imaeq2d 
 |-  ( R e. DivRing -> ( `' L " ( Unit ` R ) ) = ( `' L " ( B \ { ( 0g ` R ) } ) ) )
9 8 adantr 
 |-  ( ( R e. DivRing /\ ( chr ` R ) = 0 ) -> ( `' L " ( Unit ` R ) ) = ( `' L " ( B \ { ( 0g ` R ) } ) ) )
10 drngring 
 |-  ( R e. DivRing -> R e. Ring )
11 2 zrhrhm 
 |-  ( R e. Ring -> L e. ( ZZring RingHom R ) )
12 zringbas 
 |-  ZZ = ( Base ` ZZring )
13 12 1 rhmf 
 |-  ( L e. ( ZZring RingHom R ) -> L : ZZ --> B )
14 ffun 
 |-  ( L : ZZ --> B -> Fun L )
15 11 13 14 3syl 
 |-  ( R e. Ring -> Fun L )
16 difpreima 
 |-  ( Fun L -> ( `' L " ( B \ { ( 0g ` R ) } ) ) = ( ( `' L " B ) \ ( `' L " { ( 0g ` R ) } ) ) )
17 10 15 16 3syl 
 |-  ( R e. DivRing -> ( `' L " ( B \ { ( 0g ` R ) } ) ) = ( ( `' L " B ) \ ( `' L " { ( 0g ` R ) } ) ) )
18 17 adantr 
 |-  ( ( R e. DivRing /\ ( chr ` R ) = 0 ) -> ( `' L " ( B \ { ( 0g ` R ) } ) ) = ( ( `' L " B ) \ ( `' L " { ( 0g ` R ) } ) ) )
19 fimacnv 
 |-  ( L : ZZ --> B -> ( `' L " B ) = ZZ )
20 10 11 13 19 4syl 
 |-  ( R e. DivRing -> ( `' L " B ) = ZZ )
21 20 adantr 
 |-  ( ( R e. DivRing /\ ( chr ` R ) = 0 ) -> ( `' L " B ) = ZZ )
22 1 2 5 zrhker 
 |-  ( R e. Ring -> ( ( chr ` R ) = 0 <-> ( `' L " { ( 0g ` R ) } ) = { 0 } ) )
23 22 biimpa 
 |-  ( ( R e. Ring /\ ( chr ` R ) = 0 ) -> ( `' L " { ( 0g ` R ) } ) = { 0 } )
24 10 23 sylan 
 |-  ( ( R e. DivRing /\ ( chr ` R ) = 0 ) -> ( `' L " { ( 0g ` R ) } ) = { 0 } )
25 21 24 difeq12d 
 |-  ( ( R e. DivRing /\ ( chr ` R ) = 0 ) -> ( ( `' L " B ) \ ( `' L " { ( 0g ` R ) } ) ) = ( ZZ \ { 0 } ) )
26 9 18 25 3eqtrd 
 |-  ( ( R e. DivRing /\ ( chr ` R ) = 0 ) -> ( `' L " ( Unit ` R ) ) = ( ZZ \ { 0 } ) )