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Metamath Proof Explorer

Theorem rngidpropd

Description: The ring unity depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014)

Ref Expression
Hypotheses rngidpropd.1 
|- ( ph -> B = ( Base ` K ) )
rngidpropd.2 
|- ( ph -> B = ( Base ` L ) )
rngidpropd.3 
|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
Assertion rngidpropd 
|- ( ph -> ( 1r ` K ) = ( 1r ` L ) )

Proof

Step Hyp Ref Expression
1 rngidpropd.1 
 |-  ( ph -> B = ( Base ` K ) )
2 rngidpropd.2 
 |-  ( ph -> B = ( Base ` L ) )
3 rngidpropd.3 
 |-  ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )
4 eqid 
 |-  ( mulGrp ` K ) = ( mulGrp ` K )
5 eqid 
 |-  ( Base ` K ) = ( Base ` K )
6 4 5 mgpbas 
 |-  ( Base ` K ) = ( Base ` ( mulGrp ` K ) )
7 1 6 eqtrdi 
 |-  ( ph -> B = ( Base ` ( mulGrp ` K ) ) )
8 eqid 
 |-  ( mulGrp ` L ) = ( mulGrp ` L )
9 eqid 
 |-  ( Base ` L ) = ( Base ` L )
10 8 9 mgpbas 
 |-  ( Base ` L ) = ( Base ` ( mulGrp ` L ) )
11 2 10 eqtrdi 
 |-  ( ph -> B = ( Base ` ( mulGrp ` L ) ) )
12 eqid 
 |-  ( .r ` K ) = ( .r ` K )
13 4 12 mgpplusg 
 |-  ( .r ` K ) = ( +g ` ( mulGrp ` K ) )
14 13 oveqi 
 |-  ( x ( .r ` K ) y ) = ( x ( +g ` ( mulGrp ` K ) ) y )
15 eqid 
 |-  ( .r ` L ) = ( .r ` L )
16 8 15 mgpplusg 
 |-  ( .r ` L ) = ( +g ` ( mulGrp ` L ) )
17 16 oveqi 
 |-  ( x ( .r ` L ) y ) = ( x ( +g ` ( mulGrp ` L ) ) y )
18 3 14 17 3eqtr3g 
 |-  ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` ( mulGrp ` K ) ) y ) = ( x ( +g ` ( mulGrp ` L ) ) y ) )
19 7 11 18 grpidpropd 
 |-  ( ph -> ( 0g ` ( mulGrp ` K ) ) = ( 0g ` ( mulGrp ` L ) ) )
20 eqid 
 |-  ( 1r ` K ) = ( 1r ` K )
21 4 20 ringidval 
 |-  ( 1r ` K ) = ( 0g ` ( mulGrp ` K ) )
22 eqid 
 |-  ( 1r ` L ) = ( 1r ` L )
23 8 22 ringidval 
 |-  ( 1r ` L ) = ( 0g ` ( mulGrp ` L ) )
24 19 21 23 3eqtr4g 
 |-  ( ph -> ( 1r ` K ) = ( 1r ` L ) )