Metamath Proof Explorer


Theorem cyggenod2

Description: In an infinite cyclic group, the generator must have infinite order, but this property no longer characterizes the generators. (Contributed by Mario Carneiro, 21-Apr-2016)

Ref Expression
Hypotheses iscyg.1 B = Base G
iscyg.2 · ˙ = G
iscyg3.e E = x B | ran n n · ˙ x = B
cyggenod.o O = od G
Assertion cyggenod2 G Grp X E O X = if B Fin B 0

Proof

Step Hyp Ref Expression
1 iscyg.1 B = Base G
2 iscyg.2 · ˙ = G
3 iscyg3.e E = x B | ran n n · ˙ x = B
4 cyggenod.o O = od G
5 1 2 3 iscyggen X E X B ran n n · ˙ X = B
6 5 simplbi X E X B
7 eqid n n · ˙ X = n n · ˙ X
8 1 4 2 7 dfod2 G Grp X B O X = if ran n n · ˙ X Fin ran n n · ˙ X 0
9 6 8 sylan2 G Grp X E O X = if ran n n · ˙ X Fin ran n n · ˙ X 0
10 5 simprbi X E ran n n · ˙ X = B
11 10 adantl G Grp X E ran n n · ˙ X = B
12 11 eleq1d G Grp X E ran n n · ˙ X Fin B Fin
13 11 fveq2d G Grp X E ran n n · ˙ X = B
14 12 13 ifbieq1d G Grp X E if ran n n · ˙ X Fin ran n n · ˙ X 0 = if B Fin B 0
15 9 14 eqtrd G Grp X E O X = if B Fin B 0