Theorem subgbas

Description: The base of the restricted group in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014)

	Ref	Expression
Hypothesis	subggrp.h	$$⊢H=G{\upharpoonright }_{𝑠}S$$
Assertion	subgbas	$$⊢S\in \mathrm{SubGrp}\left(G\right)\to S={\mathrm{Base}}_H$$

Proof

Step	Hyp	Ref	Expression
1		subggrp.h	$$⊢H=G{\upharpoonright }_{𝑠}S$$
2		eqid	$$⊢{\mathrm{Base}}_G={\mathrm{Base}}_G$$
3	2	subgss	$$⊢S\in \mathrm{SubGrp}\left(G\right)\to S\subseteq {\mathrm{Base}}_G$$
4	1 2	ressbas2	$$⊢S\subseteq {\mathrm{Base}}_G\to S={\mathrm{Base}}_H$$
5	3 4	syl	$$⊢S\in \mathrm{SubGrp}\left(G\right)\to S={\mathrm{Base}}_H$$