Theorem lejoin2

Description: A join's second argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011)

	Ref	Expression
Hypotheses	joinval2.b	$$⊢B={\mathrm{Base}}_K$$
	joinval2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	joinval2.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	joinval2.k	$$⊢\phi \to K\in V$$
	joinval2.x	$$⊢\phi \to X\in B$$
	joinval2.y	$$⊢\phi \to Y\in B$$
	joinlem.e	$$⊢\phi \to ⟨X,Y⟩\in \mathrm{dom}\underset{\dot{}}{\vee }$$
Assertion	lejoin2	$$⊢\phi \to Y\underset{\dot{}}{\le }\left(X\underset{\dot{}}{\vee }Y\right)$$

Proof

Step	Hyp	Ref	Expression
1		joinval2.b	$$⊢B={\mathrm{Base}}_K$$
2		joinval2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		joinval2.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		joinval2.k	$$⊢\phi \to K\in V$$
5		joinval2.x	$$⊢\phi \to X\in B$$
6		joinval2.y	$$⊢\phi \to Y\in B$$
7		joinlem.e	$$⊢\phi \to ⟨X,Y⟩\in \mathrm{dom}\underset{\dot{}}{\vee }$$
8	1 2 3 4 5 6 7	joinlem	$$⊢\phi \to \left(\left(X\underset{\dot{}}{\le }\left(X\underset{\dot{}}{\vee }Y\right)\wedge Y\underset{\dot{}}{\le }\left(X\underset{\dot{}}{\vee }Y\right)\right)\wedge \forall z\in B\left(\left(X\underset{\dot{}}{\le }z\wedge Y\underset{\dot{}}{\le }z\right)\to \left(X\underset{\dot{}}{\vee }Y\right)\underset{\dot{}}{\le }z\right)\right)$$
9	8	simplrd	$$⊢\phi \to Y\underset{\dot{}}{\le }\left(X\underset{\dot{}}{\vee }Y\right)$$