Theorem elpadd2at2

Description: Membership in a projective subspace sum of two points. (Contributed by NM, 8-Mar-2012)

	Ref	Expression
Hypotheses	paddfval.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	paddfval.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	paddfval.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	paddfval.p	$$⊢\underset{\dot{}}{+}={+}_{𝑃}\left(K\right)$$
Assertion	elpadd2at2	$$⊢\left(K\in \mathrm{Lat}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\right)\to \left(S\in \left(\left\{Q\right\}\underset{\dot{}}{+}\left\{R\right\}\right)\leftrightarrow S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		paddfval.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		paddfval.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		paddfval.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4		paddfval.p	$$⊢\underset{\dot{}}{+}={+}_{𝑃}\left(K\right)$$
5	1 2 3 4	elpadd2at	$$⊢\left(K\in \mathrm{Lat}\wedge Q\in A\wedge R\in A\right)\to \left(S\in \left(\left\{Q\right\}\underset{\dot{}}{+}\left\{R\right\}\right)\leftrightarrow \left(S\in A\wedge S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)$$
6	5	3adant3r3	$$⊢\left(K\in \mathrm{Lat}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\right)\to \left(S\in \left(\left\{Q\right\}\underset{\dot{}}{+}\left\{R\right\}\right)\leftrightarrow \left(S\in A\wedge S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)$$
7		simpr3	$$⊢\left(K\in \mathrm{Lat}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\right)\to S\in A$$
8	7	biantrurd	$$⊢\left(K\in \mathrm{Lat}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\right)\to \left(S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\leftrightarrow \left(S\in A\wedge S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)$$
9	6 8	bitr4d	$$⊢\left(K\in \mathrm{Lat}\wedge \left(Q\in A\wedge R\in A\wedge S\in A\right)\right)\to \left(S\in \left(\left\{Q\right\}\underset{\dot{}}{+}\left\{R\right\}\right)\leftrightarrow S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)$$