Metamath Proof Explorer


Theorem rngplusg

Description: The additive operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013) (Revised by Mario Carneiro, 30-Apr-2015)

Ref Expression
Hypothesis rngfn.r
|- R = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. }
Assertion rngplusg
|- ( .+ e. V -> .+ = ( +g ` R ) )

Proof

Step Hyp Ref Expression
1 rngfn.r
 |-  R = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. }
2 1 rngstr
 |-  R Struct <. 1 , 3 >.
3 plusgid
 |-  +g = Slot ( +g ` ndx )
4 snsstp2
 |-  { <. ( +g ` ndx ) , .+ >. } C_ { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. }
5 4 1 sseqtrri
 |-  { <. ( +g ` ndx ) , .+ >. } C_ R
6 2 3 5 strfv
 |-  ( .+ e. V -> .+ = ( +g ` R ) )