Metamath Proof Explorer


Theorem cycsubmel

Description: Characterization of an element of the set of nonnegative integer powers of an element A . Although this theorem holds for any class G , the definition of F is only meaningful if G is a monoid or at least a unital magma. (Contributed by AV, 28-Dec-2023)

Ref Expression
Hypotheses cycsubm.b B = Base G
cycsubm.t · ˙ = G
cycsubm.f F = x 0 x · ˙ A
cycsubm.c C = ran F
Assertion cycsubmel X C i 0 X = i · ˙ A

Proof

Step Hyp Ref Expression
1 cycsubm.b B = Base G
2 cycsubm.t · ˙ = G
3 cycsubm.f F = x 0 x · ˙ A
4 cycsubm.c C = ran F
5 4 eleq2i X C X ran F
6 ovex x · ˙ A V
7 6 3 fnmpti F Fn 0
8 fvelrnb F Fn 0 X ran F i 0 F i = X
9 7 8 ax-mp X ran F i 0 F i = X
10 oveq1 x = i x · ˙ A = i · ˙ A
11 ovex i · ˙ A V
12 10 3 11 fvmpt i 0 F i = i · ˙ A
13 12 eqeq1d i 0 F i = X i · ˙ A = X
14 eqcom i · ˙ A = X X = i · ˙ A
15 13 14 bitrdi i 0 F i = X X = i · ˙ A
16 15 rexbiia i 0 F i = X i 0 X = i · ˙ A
17 5 9 16 3bitri X C i 0 X = i · ˙ A