Theorem oppgcntr

Description: The center of a group is the same as the center of the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016)

	Ref	Expression
Hypotheses	oppggic.o	$$⊢O={opp}_{𝑔}\left(G\right)$$
	oppgcntr.z	$$⊢Z=\mathrm{Cntr}\left(G\right)$$
Assertion	oppgcntr	$$⊢Z=\mathrm{Cntr}\left(O\right)$$

Proof

Step	Hyp	Ref	Expression
1		oppggic.o	$$⊢O={opp}_{𝑔}\left(G\right)$$
2		oppgcntr.z	$$⊢Z=\mathrm{Cntr}\left(G\right)$$
3		eqid	$$⊢\mathrm{Cntz}\left(G\right)=\mathrm{Cntz}\left(G\right)$$
4	1 3	oppgcntz	$$⊢\mathrm{Cntz}\left(G\right)\left({\mathrm{Base}}_G\right)=\mathrm{Cntz}\left(O\right)\left({\mathrm{Base}}_G\right)$$
5		eqid	$$⊢{\mathrm{Base}}_G={\mathrm{Base}}_G$$
6	5 3	cntrval	$$⊢\mathrm{Cntz}\left(G\right)\left({\mathrm{Base}}_G\right)=\mathrm{Cntr}\left(G\right)$$
7	6 2	eqtr4i	$$⊢\mathrm{Cntz}\left(G\right)\left({\mathrm{Base}}_G\right)=Z$$
8	1 5	oppgbas	$$⊢{\mathrm{Base}}_G={\mathrm{Base}}_O$$
9		eqid	$$⊢\mathrm{Cntz}\left(O\right)=\mathrm{Cntz}\left(O\right)$$
10	8 9	cntrval	$$⊢\mathrm{Cntz}\left(O\right)\left({\mathrm{Base}}_G\right)=\mathrm{Cntr}\left(O\right)$$
11	4 7 10	3eqtr3i	$$⊢Z=\mathrm{Cntr}\left(O\right)$$