Metamath Proof Explorer


Theorem erngmul-rN

Description: Ring addition operation. (Contributed by NM, 10-Jun-2013) (New usage is discouraged.)

Ref Expression
Hypotheses erngset.h-r H = LHyp K
erngset.t-r T = LTrn K W
erngset.e-r E = TEndo K W
erngset.d-r D = EDRing R K W
erng.m-r · ˙ = D
Assertion erngmul-rN K X W H U E V E U · ˙ V = V U

Proof

Step Hyp Ref Expression
1 erngset.h-r H = LHyp K
2 erngset.t-r T = LTrn K W
3 erngset.e-r E = TEndo K W
4 erngset.d-r D = EDRing R K W
5 erng.m-r · ˙ = D
6 1 2 3 4 5 erngfmul-rN K X W H · ˙ = s E , t E t s
7 6 adantr K X W H U E V E · ˙ = s E , t E t s
8 7 oveqd K X W H U E V E U · ˙ V = U s E , t E t s V
9 coexg V E U E V U V
10 9 ancoms U E V E V U V
11 coeq2 s = U t s = t U
12 coeq1 t = V t U = V U
13 eqid s E , t E t s = s E , t E t s
14 11 12 13 ovmpog U E V E V U V U s E , t E t s V = V U
15 10 14 mpd3an3 U E V E U s E , t E t s V = V U
16 15 adantl K X W H U E V E U s E , t E t s V = V U
17 8 16 eqtrd K X W H U E V E U · ˙ V = V U