Theorem cdlemeg47rv

Description: Value of g_s(r) when r is an atom under pq and s is any atom not under pq, using very compact hypotheses. TODO: FIX COMMENT. (Contributed by NM, 3-Apr-2013)

	Ref	Expression
Hypotheses	cdlemef47.b	$$⊢B={\mathrm{Base}}_K$$
	cdlemef47.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdlemef47.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdlemef47.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdlemef47.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdlemef47.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdlemef47.v	$$⊢V=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W$$
	cdlemef47.n	$$⊢N=\left(v\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdlemefs47.o	$$⊢O=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(N\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdlemef47.g	$$⊢G=\left(a\in B⟼if\left(\left(Q\ne P\wedge \neg a\underset{\dot{}}{\le }W\right),\left(\iota c\in B|\forall u\in A\left(\left(\neg u\underset{\dot{}}{\le }W\wedge u\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)=a\right)\to c=if\left(u\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right),\left(\iota b\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\to b=O\right)\right),⦋u/v⦌N\right)\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)\right)\right),a\right)\right)$$
Assertion	cdlemeg47rv	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to G\left(R\right)=⦋R/u⦌⦋S/v⦌O$$

Proof

Step	Hyp	Ref	Expression
1		cdlemef47.b	$$⊢B={\mathrm{Base}}_K$$
2		cdlemef47.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdlemef47.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdlemef47.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdlemef47.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdlemef47.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdlemef47.v	$$⊢V=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W$$
8		cdlemef47.n	$$⊢N=\left(v\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdlemefs47.o	$$⊢O=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(N\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10		cdlemef47.g	$$⊢G=\left(a\in B⟼if\left(\left(Q\ne P\wedge \neg a\underset{\dot{}}{\le }W\right),\left(\iota c\in B|\forall u\in A\left(\left(\neg u\underset{\dot{}}{\le }W\wedge u\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)=a\right)\to c=if\left(u\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right),\left(\iota b\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\to b=O\right)\right),⦋u/v⦌N\right)\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)\right)\right),a\right)\right)$$
11	3 5	cdleme46f2g1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\right)\wedge \left(Q\ne P\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\wedge \neg S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\right)$$
12	1 2 3 4 5 6 7 8 10 9	cdlemefs45	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\right)\wedge \left(Q\ne P\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\wedge \neg S\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\right)\to G\left(R\right)=⦋R/u⦌⦋S/v⦌O$$
13	11 12	syl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to G\left(R\right)=⦋R/u⦌⦋S/v⦌O$$