Theorem elpaddatiN

Description: Consequence of membership in a projective subspace sum with a point. (Contributed by NM, 2-Feb-2012) (New usage is discouraged.)

	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	elpaddatiN	$$⊢\left(\left(K\in \mathrm{Lat}\wedge X\subseteq A\wedge Q\in A\right)\wedge \left(X\ne \varnothing \wedge R\in \left(X\underset{\dot{}}{+}\left\{Q\right\}\right)\right)\right)\to \exists p\in XR\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }Q\right)$$

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	elpaddat	$$⊢\left(\left(K\in \mathrm{Lat}\wedge X\subseteq A\wedge Q\in A\right)\wedge X\ne \varnothing \right)\to \left(R\in \left(X\underset{\dot{}}{+}\left\{Q\right\}\right)\leftrightarrow \left(R\in A\wedge \exists p\in XR\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }Q\right)\right)\right)$$
6		simpr	$$⊢\left(R\in A\wedge \exists p\in XR\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }Q\right)\right)\to \exists p\in XR\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }Q\right)$$
7	5 6	syl6bi	$$⊢\left(\left(K\in \mathrm{Lat}\wedge X\subseteq A\wedge Q\in A\right)\wedge X\ne \varnothing \right)\to \left(R\in \left(X\underset{\dot{}}{+}\left\{Q\right\}\right)\to \exists p\in XR\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }Q\right)\right)$$
8	7	impr	$$⊢\left(\left(K\in \mathrm{Lat}\wedge X\subseteq A\wedge Q\in A\right)\wedge \left(X\ne \varnothing \wedge R\in \left(X\underset{\dot{}}{+}\left\{Q\right\}\right)\right)\right)\to \exists p\in XR\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }Q\right)$$