Theorem paddvaln0N

Description: Projective subspace sum operation value for nonempty sets. (Contributed by NM, 27-Jan-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	paddvaln0N	$$⊢\left(\left(K\in \mathrm{Lat}\wedge X\subseteq A\wedge Y\subseteq A\right)\wedge \left(X\ne \varnothing \wedge Y\ne \varnothing \right)\right)\to X\underset{\dot{}}{+}Y=\left\{p\in A|\exists q\in X\exists r\in Yp\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right\}$$

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	elpaddn0	$$⊢\left(\left(K\in \mathrm{Lat}\wedge X\subseteq A\wedge Y\subseteq A\right)\wedge \left(X\ne \varnothing \wedge Y\ne \varnothing \right)\right)\to \left(s\in \left(X\underset{\dot{}}{+}Y\right)\leftrightarrow \left(s\in A\wedge \exists q\in X\exists r\in Ys\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right)\right)$$
6		breq1	$$⊢p=s\to \left(p\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\leftrightarrow s\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right)$$
7	6	2rexbidv	$$⊢p=s\to \left(\exists q\in X\exists r\in Yp\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\leftrightarrow \exists q\in X\exists r\in Ys\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right)$$
8	7	elrab	$$⊢s\in \left\{p\in A|\exists q\in X\exists r\in Yp\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right\}\leftrightarrow \left(s\in A\wedge \exists q\in X\exists r\in Ys\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right)$$
9	5 8	bitr4di	$$⊢\left(\left(K\in \mathrm{Lat}\wedge X\subseteq A\wedge Y\subseteq A\right)\wedge \left(X\ne \varnothing \wedge Y\ne \varnothing \right)\right)\to \left(s\in \left(X\underset{\dot{}}{+}Y\right)\leftrightarrow s\in \left\{p\in A|\exists q\in X\exists r\in Yp\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right\}\right)$$
10	9	eqrdv	$$⊢\left(\left(K\in \mathrm{Lat}\wedge X\subseteq A\wedge Y\subseteq A\right)\wedge \left(X\ne \varnothing \wedge Y\ne \varnothing \right)\right)\to X\underset{\dot{}}{+}Y=\left\{p\in A|\exists q\in X\exists r\in Yp\underset{\dot{}}{\le }\left(q\underset{\dot{}}{\vee }r\right)\right\}$$