Theorem cdleme25cl

Description: Show closure of the unique element in cdleme25c . (Contributed by NM, 2-Feb-2013)

	Ref	Expression
Hypotheses	cdleme24.b	$$⊢B={\mathrm{Base}}_K$$
	cdleme24.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme24.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme24.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme24.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme24.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme24.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme24.f	$$⊢F=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme24.n	$$⊢N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme25cl.i	$$⊢I=\left(\iota u\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge \neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\right)$$
Assertion	cdleme25cl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to I\in B$$

Proof

Step	Hyp	Ref	Expression
1		cdleme24.b	$$⊢B={\mathrm{Base}}_K$$
2		cdleme24.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdleme24.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdleme24.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdleme24.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdleme24.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdleme24.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		cdleme24.f	$$⊢F=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme24.n	$$⊢N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10		cdleme25cl.i	$$⊢I=\left(\iota u\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge \neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\right)$$
11	1 2 3 4 5 6 7 8 9	cdleme25c	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \exists !u\in B\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge \neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)$$
12		riotacl	$$⊢\exists !u\in B\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge \neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\to \left(\iota u\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge \neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\right)\in B$$
13	11 12	syl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(\iota u\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge \neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\right)\in B$$
14	10 13	eqeltrid	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to I\in B$$