Metamath Proof Explorer

Theorem drngid2

Description: Properties showing that an element I is the identity element of a division ring. (Contributed by Mario Carneiro, 11-Oct-2013)

Ref Expression
Hypotheses drngid2.b 
|- B = ( Base ` R )
drngid2.t 
|- .x. = ( .r ` R )
drngid2.o 
|- .0. = ( 0g ` R )
drngid2.u 
|- .1. = ( 1r ` R )
Assertion drngid2 
|- ( R e. DivRing -> ( ( I e. B /\ I =/= .0. /\ ( I .x. I ) = I ) <-> .1. = I ) )

Proof

Step Hyp Ref Expression
1 drngid2.b 
 |-  B = ( Base ` R )
2 drngid2.t 
 |-  .x. = ( .r ` R )
3 drngid2.o 
 |-  .0. = ( 0g ` R )
4 drngid2.u 
 |-  .1. = ( 1r ` R )
5 df-3an 
 |-  ( ( I e. B /\ I =/= .0. /\ ( I .x. I ) = I ) <-> ( ( I e. B /\ I =/= .0. ) /\ ( I .x. I ) = I ) )
6 eldifsn 
 |-  ( I e. ( B \ { .0. } ) <-> ( I e. B /\ I =/= .0. ) )
7 6 anbi1i 
 |-  ( ( I e. ( B \ { .0. } ) /\ ( I .x. I ) = I ) <-> ( ( I e. B /\ I =/= .0. ) /\ ( I .x. I ) = I ) )
8 5 7 bitr4i 
 |-  ( ( I e. B /\ I =/= .0. /\ ( I .x. I ) = I ) <-> ( I e. ( B \ { .0. } ) /\ ( I .x. I ) = I ) )
9 eqid 
 |-  ( ( mulGrp ` R ) |`s ( B \ { .0. } ) ) = ( ( mulGrp ` R ) |`s ( B \ { .0. } ) )
10 1 3 9 drngmgp 
 |-  ( R e. DivRing -> ( ( mulGrp ` R ) |`s ( B \ { .0. } ) ) e. Grp )
11 difss 
 |-  ( B \ { .0. } ) C_ B
12 eqid 
 |-  ( mulGrp ` R ) = ( mulGrp ` R )
13 12 1 mgpbas 
 |-  B = ( Base ` ( mulGrp ` R ) )
14 9 13 ressbas2 
 |-  ( ( B \ { .0. } ) C_ B -> ( B \ { .0. } ) = ( Base ` ( ( mulGrp ` R ) |`s ( B \ { .0. } ) ) ) )
15 11 14 ax-mp 
 |-  ( B \ { .0. } ) = ( Base ` ( ( mulGrp ` R ) |`s ( B \ { .0. } ) ) )
16 1 fvexi 
 |-  B e. _V
17 difexg 
 |-  ( B e. _V -> ( B \ { .0. } ) e. _V )
18 12 2 mgpplusg 
 |-  .x. = ( +g ` ( mulGrp ` R ) )
19 9 18 ressplusg 
 |-  ( ( B \ { .0. } ) e. _V -> .x. = ( +g ` ( ( mulGrp ` R ) |`s ( B \ { .0. } ) ) ) )
20 16 17 19 mp2b 
 |-  .x. = ( +g ` ( ( mulGrp ` R ) |`s ( B \ { .0. } ) ) )
21 eqid 
 |-  ( 0g ` ( ( mulGrp ` R ) |`s ( B \ { .0. } ) ) ) = ( 0g ` ( ( mulGrp ` R ) |`s ( B \ { .0. } ) ) )
22 15 20 21 isgrpid2 
 |-  ( ( ( mulGrp ` R ) |`s ( B \ { .0. } ) ) e. Grp -> ( ( I e. ( B \ { .0. } ) /\ ( I .x. I ) = I ) <-> ( 0g ` ( ( mulGrp ` R ) |`s ( B \ { .0. } ) ) ) = I ) )
23 10 22 syl 
 |-  ( R e. DivRing -> ( ( I e. ( B \ { .0. } ) /\ ( I .x. I ) = I ) <-> ( 0g ` ( ( mulGrp ` R ) |`s ( B \ { .0. } ) ) ) = I ) )
24 8 23 syl5bb 
 |-  ( R e. DivRing -> ( ( I e. B /\ I =/= .0. /\ ( I .x. I ) = I ) <-> ( 0g ` ( ( mulGrp ` R ) |`s ( B \ { .0. } ) ) ) = I ) )
25 1 3 4 9 drngid 
 |-  ( R e. DivRing -> .1. = ( 0g ` ( ( mulGrp ` R ) |`s ( B \ { .0. } ) ) ) )
26 25 eqeq1d 
 |-  ( R e. DivRing -> ( .1. = I <-> ( 0g ` ( ( mulGrp ` R ) |`s ( B \ { .0. } ) ) ) = I ) )
27 24 26 bitr4d 
 |-  ( R e. DivRing -> ( ( I e. B /\ I =/= .0. /\ ( I .x. I ) = I ) <-> .1. = I ) )