Theorem paddssw2

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	paddssw2	$$⊢\left(K\in B\wedge \left(X\subseteq A\wedge Y\subseteq A\wedge Z\subseteq A\right)\right)\to \left(X\underset{\dot{}}{+}Y\subseteq Z\to \left(X\subseteq Z\wedge Y\subseteq Z\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		paddssw.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
2		paddssw.p	$$⊢\underset{\dot{}}{+}={+}_{𝑃}\left(K\right)$$
3	1 2	sspadd1	$$⊢\left(K\in B\wedge X\subseteq A\wedge Y\subseteq A\right)\to X\subseteq X\underset{\dot{}}{+}Y$$
4	3	3adant3r3	$$⊢\left(K\in B\wedge \left(X\subseteq A\wedge Y\subseteq A\wedge Z\subseteq A\right)\right)\to X\subseteq X\underset{\dot{}}{+}Y$$
5		sstr	$$⊢\left(X\subseteq X\underset{\dot{}}{+}Y\wedge X\underset{\dot{}}{+}Y\subseteq Z\right)\to X\subseteq Z$$
6	4 5	sylan	$$⊢\left(\left(K\in B\wedge \left(X\subseteq A\wedge Y\subseteq A\wedge Z\subseteq A\right)\right)\wedge X\underset{\dot{}}{+}Y\subseteq Z\right)\to X\subseteq Z$$
7	6	ex	$$⊢\left(K\in B\wedge \left(X\subseteq A\wedge Y\subseteq A\wedge Z\subseteq A\right)\right)\to \left(X\underset{\dot{}}{+}Y\subseteq Z\to X\subseteq Z\right)$$
8		simpl	$$⊢\left(K\in B\wedge \left(X\subseteq A\wedge Y\subseteq A\wedge Z\subseteq A\right)\right)\to K\in B$$
9		simpr2	$$⊢\left(K\in B\wedge \left(X\subseteq A\wedge Y\subseteq A\wedge Z\subseteq A\right)\right)\to Y\subseteq A$$
10		simpr1	$$⊢\left(K\in B\wedge \left(X\subseteq A\wedge Y\subseteq A\wedge Z\subseteq A\right)\right)\to X\subseteq A$$
11	1 2	sspadd2	$$⊢\left(K\in B\wedge Y\subseteq A\wedge X\subseteq A\right)\to Y\subseteq X\underset{\dot{}}{+}Y$$
12	8 9 10 11	syl3anc	$$⊢\left(K\in B\wedge \left(X\subseteq A\wedge Y\subseteq A\wedge Z\subseteq A\right)\right)\to Y\subseteq X\underset{\dot{}}{+}Y$$
13		sstr	$$⊢\left(Y\subseteq X\underset{\dot{}}{+}Y\wedge X\underset{\dot{}}{+}Y\subseteq Z\right)\to Y\subseteq Z$$
14	12 13	sylan	$$⊢\left(\left(K\in B\wedge \left(X\subseteq A\wedge Y\subseteq A\wedge Z\subseteq A\right)\right)\wedge X\underset{\dot{}}{+}Y\subseteq Z\right)\to Y\subseteq Z$$
15	14	ex	$$⊢\left(K\in B\wedge \left(X\subseteq A\wedge Y\subseteq A\wedge Z\subseteq A\right)\right)\to \left(X\underset{\dot{}}{+}Y\subseteq Z\to Y\subseteq Z\right)$$
16	7 15	jcad	$$⊢\left(K\in B\wedge \left(X\subseteq A\wedge Y\subseteq A\wedge Z\subseteq A\right)\right)\to \left(X\underset{\dot{}}{+}Y\subseteq Z\to \left(X\subseteq Z\wedge Y\subseteq Z\right)\right)$$