Theorem elpadd

Description: Member of a projective subspace sum. (Contributed by NM, 29-Dec-2011)

	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	elpadd	$$⊢\left(K\in B\wedge X\subseteq A\wedge Y\subseteq A\right)\to \left(S\in \left(X\underset{\dot{}}{+}Y\right)\leftrightarrow \left(\left(S\in X\vee S\in Y\right)\vee \left(S\in A\wedge \exists q\in X\exists r\in YS\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right)\right)\right)$$

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	paddval	$$⊢\left(K\in B\wedge X\subseteq A\wedge Y\subseteq A\right)\to X\underset{\dot{}}{+}Y=\left(X\cup Y\right)\cup \left\{p\in A|\exists q\in X\exists r\in Yp\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right\}$$
6	5	eleq2d	$$⊢\left(K\in B\wedge X\subseteq A\wedge Y\subseteq A\right)\to \left(S\in \left(X\underset{\dot{}}{+}Y\right)\leftrightarrow S\in \left(\left(X\cup Y\right)\cup \left\{p\in A|\exists q\in X\exists r\in Yp\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right\}\right)\right)$$
7		elun	$$⊢S\in \left(\left(X\cup Y\right)\cup \left\{p\in A|\exists q\in X\exists r\in Yp\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right\}\right)\leftrightarrow \left(S\in \left(X\cup Y\right)\vee S\in \left\{p\in A|\exists q\in X\exists r\in Yp\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right\}\right)$$
8		elun	$$⊢S\in \left(X\cup Y\right)\leftrightarrow \left(S\in X\vee S\in Y\right)$$
9		breq1	$$⊢p=S\to \left(p\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\leftrightarrow S\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right)$$
10	9	2rexbidv	$$⊢p=S\to \left(\exists q\in X\exists r\in Yp\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\leftrightarrow \exists q\in X\exists r\in YS\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right)$$
11	10	elrab	$$⊢S\in \left\{p\in A|\exists q\in X\exists r\in Yp\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right\}\leftrightarrow \left(S\in A\wedge \exists q\in X\exists r\in YS\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right)$$
12	8 11	orbi12i	$$⊢\left(S\in \left(X\cup Y\right)\vee S\in \left\{p\in A|\exists q\in X\exists r\in Yp\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right\}\right)\leftrightarrow \left(\left(S\in X\vee S\in Y\right)\vee \left(S\in A\wedge \exists q\in X\exists r\in YS\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right)\right)$$
13	7 12	bitri	$$⊢S\in \left(\left(X\cup Y\right)\cup \left\{p\in A|\exists q\in X\exists r\in Yp\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right\}\right)\leftrightarrow \left(\left(S\in X\vee S\in Y\right)\vee \left(S\in A\wedge \exists q\in X\exists r\in YS\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right)\right)$$
14	6 13	syl6bb	$$⊢\left(K\in B\wedge X\subseteq A\wedge Y\subseteq A\right)\to \left(S\in \left(X\underset{\dot{}}{+}Y\right)\leftrightarrow \left(\left(S\in X\vee S\in Y\right)\vee \left(S\in A\wedge \exists q\in X\exists r\in YS\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right)\right)\right)$$