Theorem elpadd2at

Description: Membership in a projective subspace sum of two points. (Contributed by NM, 29-Jan-2012)

	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	elpadd2at	$$⊢\left(K\in \mathrm{Lat}\wedge Q\in A\wedge R\in A\right)\to \left(S\in \left(\left\{Q\right\}\underset{\dot{}}{+}\left\{R\right\}\right)\leftrightarrow \left(S\in A\wedge S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)$$

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		simp1	$$⊢\left(K\in \mathrm{Lat}\wedge Q\in A\wedge R\in A\right)\to K\in \mathrm{Lat}$$
6		simp2	$$⊢\left(K\in \mathrm{Lat}\wedge Q\in A\wedge R\in A\right)\to Q\in A$$
7	6	snssd	$$⊢\left(K\in \mathrm{Lat}\wedge Q\in A\wedge R\in A\right)\to \left\{Q\right\}\subseteq A$$
8		simp3	$$⊢\left(K\in \mathrm{Lat}\wedge Q\in A\wedge R\in A\right)\to R\in A$$
9		snnzg	$$⊢Q\in A\to \left\{Q\right\}\ne \varnothing$$
10	9	3ad2ant2	$$⊢\left(K\in \mathrm{Lat}\wedge Q\in A\wedge R\in A\right)\to \left\{Q\right\}\ne \varnothing$$
11	1 2 3 4	elpaddat	$$⊢\left(\left(K\in \mathrm{Lat}\wedge \left\{Q\right\}\subseteq A\wedge R\in A\right)\wedge \left\{Q\right\}\ne \varnothing \right)\to \left(S\in \left(\left\{Q\right\}\underset{\dot{}}{+}\left\{R\right\}\right)\leftrightarrow \left(S\in A\wedge \exists r\in \left\{Q\right\}S\underset{\dot{}}{\le }\left(r\underset{\dot{}}{\vee }R\right)\right)\right)$$
12	5 7 8 10 11	syl31anc	$$⊢\left(K\in \mathrm{Lat}\wedge Q\in A\wedge R\in A\right)\to \left(S\in \left(\left\{Q\right\}\underset{\dot{}}{+}\left\{R\right\}\right)\leftrightarrow \left(S\in A\wedge \exists r\in \left\{Q\right\}S\underset{\dot{}}{\le }\left(r\underset{\dot{}}{\vee }R\right)\right)\right)$$
13		oveq1	$$⊢r=Q\to r\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R$$
14	13	breq2d	$$⊢r=Q\to \left(S\underset{\dot{}}{\le }\left(r\underset{\dot{}}{\vee }R\right)\leftrightarrow S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)$$
15	14	rexsng	$$⊢Q\in A\to \left(\exists r\in \left\{Q\right\}S\underset{\dot{}}{\le }\left(r\underset{\dot{}}{\vee }R\right)\leftrightarrow S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)$$
16	15	3ad2ant2	$$⊢\left(K\in \mathrm{Lat}\wedge Q\in A\wedge R\in A\right)\to \left(\exists r\in \left\{Q\right\}S\underset{\dot{}}{\le }\left(r\underset{\dot{}}{\vee }R\right)\leftrightarrow S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)$$
17	16	anbi2d	$$⊢\left(K\in \mathrm{Lat}\wedge Q\in A\wedge R\in A\right)\to \left(\left(S\in A\wedge \exists r\in \left\{Q\right\}S\underset{\dot{}}{\le }\left(r\underset{\dot{}}{\vee }R\right)\right)\leftrightarrow \left(S\in A\wedge S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)$$
18	12 17	bitrd	$$⊢\left(K\in \mathrm{Lat}\wedge Q\in A\wedge R\in A\right)\to \left(S\in \left(\left\{Q\right\}\underset{\dot{}}{+}\left\{R\right\}\right)\leftrightarrow \left(S\in A\wedge S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\right)\right)$$