Metamath Proof Explorer


Theorem mgpplusg

Description: Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014)

Ref Expression
Hypotheses mgpval.1 M = mulGrp R
mgpval.2 · ˙ = R
Assertion mgpplusg · ˙ = + M

Proof

Step Hyp Ref Expression
1 mgpval.1 M = mulGrp R
2 mgpval.2 · ˙ = R
3 2 fvexi · ˙ V
4 plusgid + 𝑔 = Slot + ndx
5 4 setsid R V · ˙ V · ˙ = + R sSet + ndx · ˙
6 3 5 mpan2 R V · ˙ = + R sSet + ndx · ˙
7 1 2 mgpval M = R sSet + ndx · ˙
8 7 fveq2i + M = + R sSet + ndx · ˙
9 6 8 syl6eqr R V · ˙ = + M
10 4 str0 = +
11 fvprc ¬ R V R =
12 2 11 syl5eq ¬ R V · ˙ =
13 fvprc ¬ R V mulGrp R =
14 1 13 syl5eq ¬ R V M =
15 14 fveq2d ¬ R V + M = +
16 10 12 15 3eqtr4a ¬ R V · ˙ = + M
17 9 16 pm2.61i · ˙ = + M