Theorem paddssw1

Description: Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012)

	Ref	Expression
Hypotheses	paddssw.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	paddssw.p	$$⊢\underset{\dot{}}{+}={+}_{𝑃}\left(K\right)$$
Assertion	paddssw1	$$⊢\left(K\in B\wedge \left(X\subseteq A\wedge Y\subseteq A\wedge Z\subseteq A\right)\right)\to \left(\left(X\subseteq Z\wedge Y\subseteq Z\right)\to X\underset{\dot{}}{+}Y\subseteq Z\underset{\dot{}}{+}Z\right)$$

Proof

Step	Hyp	Ref	Expression
1		paddssw.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
2		paddssw.p	$$⊢\underset{\dot{}}{+}={+}_{𝑃}\left(K\right)$$
3		simpl	$$⊢\left(K\in B\wedge \left(X\subseteq A\wedge Y\subseteq A\wedge Z\subseteq A\right)\right)\to K\in B$$
4		simpr3	$$⊢\left(K\in B\wedge \left(X\subseteq A\wedge Y\subseteq A\wedge Z\subseteq A\right)\right)\to Z\subseteq A$$
5	1 2	paddss12	$$⊢\left(K\in B\wedge Z\subseteq A\wedge Z\subseteq A\right)\to \left(\left(X\subseteq Z\wedge Y\subseteq Z\right)\to X\underset{\dot{}}{+}Y\subseteq Z\underset{\dot{}}{+}Z\right)$$
6	3 4 4 5	syl3anc	$$⊢\left(K\in B\wedge \left(X\subseteq A\wedge Y\subseteq A\wedge Z\subseteq A\right)\right)\to \left(\left(X\subseteq Z\wedge Y\subseteq Z\right)\to X\underset{\dot{}}{+}Y\subseteq Z\underset{\dot{}}{+}Z\right)$$