Theorem lsmunss

Description: Union of subgroups is a subset of subgroup sum. (Contributed by NM, 6-Feb-2014) (Proof shortened by Mario Carneiro, 21-Jun-2014)

	Ref	Expression
Hypothesis	lsmub1.p	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(G\right)$$
Assertion	lsmunss	$$⊢\left(T\in \mathrm{SubGrp}\left(G\right)\wedge U\in \mathrm{SubGrp}\left(G\right)\right)\to T\cup U\subseteq T\underset{\dot{}}{\oplus }U$$

Proof

Step	Hyp	Ref	Expression
1		lsmub1.p	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(G\right)$$
2	1	lsmub1	$$⊢\left(T\in \mathrm{SubGrp}\left(G\right)\wedge U\in \mathrm{SubGrp}\left(G\right)\right)\to T\subseteq T\underset{\dot{}}{\oplus }U$$
3	1	lsmub2	$$⊢\left(T\in \mathrm{SubGrp}\left(G\right)\wedge U\in \mathrm{SubGrp}\left(G\right)\right)\to U\subseteq T\underset{\dot{}}{\oplus }U$$
4	2 3	unssd	$$⊢\left(T\in \mathrm{SubGrp}\left(G\right)\wedge U\in \mathrm{SubGrp}\left(G\right)\right)\to T\cup U\subseteq T\underset{\dot{}}{\oplus }U$$