Theorem cdlemefr44

Description: Value of f(r) when r is an atom not under pq, using more compact hypotheses. TODO: eliminate and use cdlemefr45 instead? TODO: FIX COMMENT. (Contributed by NM, 31-Mar-2013)

	Ref	Expression
Hypotheses	cdlemef44.b	$$⊢B={\mathrm{Base}}_K$$
	cdlemef44.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdlemef44.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdlemef44.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdlemef44.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdlemef44.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdlemef44.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdlemef44.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdlemef44.o	$$⊢O=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),I,⦋s/t⦌D\right)\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right)$$
	cdlemef44.f	$$⊢F=\left(x\in B⟼if\left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right),O,x\right)\right)$$
Assertion	cdlemefr44	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to F\left(R\right)=⦋R/t⦌D$$

Proof

Step	Hyp	Ref	Expression
1		cdlemef44.b	$$⊢B={\mathrm{Base}}_K$$
2		cdlemef44.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdlemef44.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdlemef44.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdlemef44.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdlemef44.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdlemef44.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		cdlemef44.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdlemef44.o	$$⊢O=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),I,⦋s/t⦌D\right)\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right)$$
10		cdlemef44.f	$$⊢F=\left(x\in B⟼if\left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right),O,x\right)\right)$$
11		eqid	$$⊢\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
12		biid	$$⊢s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\leftrightarrow s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
13		vex	$$⊢s\in \mathrm{V}$$
14	8 11	cdleme31sc	$$⊢s\in \mathrm{V}\to ⦋s/t⦌D=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
15	13 14	ax-mp	$$⊢⦋s/t⦌D=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
16	12 15	ifbieq2i	$$⊢if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),I,⦋s/t⦌D\right)=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),I,\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)\right)$$
17		eqid	$$⊢\left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)$$
18	1 2 3 4 5 6 7 11 16 9 10 17	cdlemefr31fv1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to F\left(R\right)=\left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)$$
19		simp2rl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to R\in A$$
20	8 17	cdleme31sc	$$⊢R\in A\to ⦋R/t⦌D=\left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)$$
21	19 20	syl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to ⦋R/t⦌D=\left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)$$
22	18 21	eqtr4d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to F\left(R\right)=⦋R/t⦌D$$