Theorem cdleme32snaw

Description: Show that [_ R / s ]_ N is an atom not under W . (Contributed by NM, 6-Mar-2013)

	Ref	Expression
Hypotheses	cdleme32.b	$$⊢B={\mathrm{Base}}_K$$
	cdleme32.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme32.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme32.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme32.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme32.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme32.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme32.c	$$⊢C=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme32.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme32.e	$$⊢E=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme32.i	$$⊢I=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=E\right)\right)$$
	cdleme32.n	$$⊢N=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),I,C\right)$$
Assertion	cdleme32snaw	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\right)\to \left(⦋R/s⦌N\in A\wedge \neg ⦋R/s⦌N\underset{\dot{}}{\le }W\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdleme32.b	$$⊢B={\mathrm{Base}}_K$$
2		cdleme32.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdleme32.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdleme32.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdleme32.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdleme32.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdleme32.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		cdleme32.c	$$⊢C=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme32.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10		cdleme32.e	$$⊢E=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
11		cdleme32.i	$$⊢I=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=E\right)\right)$$
12		cdleme32.n	$$⊢N=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),I,C\right)$$
13		eqid	$$⊢\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
14		eqid	$$⊢\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)$$
15	1 2 3 4 5 6 7 9 10 11 12 13 14	cdlemefs32sn1aw	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(⦋R/s⦌N\in A\wedge \neg ⦋R/s⦌N\underset{\dot{}}{\le }W\right)$$
16	15	3expa	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(⦋R/s⦌N\in A\wedge \neg ⦋R/s⦌N\underset{\dot{}}{\le }W\right)$$
17	1 2 3 4 5 6 7 8 12	cdlemefr32sn2aw	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(⦋R/s⦌N\in A\wedge \neg ⦋R/s⦌N\underset{\dot{}}{\le }W\right)$$
18	17	3expa	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(⦋R/s⦌N\in A\wedge \neg ⦋R/s⦌N\underset{\dot{}}{\le }W\right)$$
19	16 18	pm2.61dan	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\right)\to \left(⦋R/s⦌N\in A\wedge \neg ⦋R/s⦌N\underset{\dot{}}{\le }W\right)$$