Theorem cdlemg2jlemOLDN

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. f preserves join: f(r \/ s) = f(r) \/ s, p. 115 10th line from bottom. TODO: Combine with cdlemg2jOLDN ? (Contributed by NM, 22-Apr-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemg2.b	$$⊢B={\mathrm{Base}}_K$$
	cdlemg2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdlemg2.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdlemg2.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdlemg2.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdlemg2.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdlemg2.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
	cdlemg2ex.u	$$⊢U=\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }W$$
	cdlemg2ex.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(q\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdlemg2ex.e	$$⊢E=\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdlemg2ex.g	$$⊢G=\left(x\in B⟼if\left(\left(p\ne q\wedge \neg x\underset{\dot{}}{\le }W\right),\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=if\left(s\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right),\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right)\right)\to y=E\right)\right),⦋s/t⦌D\right)\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right),x\right)\right)$$
Assertion	cdlemg2jlemOLDN	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge F\in T\right)\to F\left(P\underset{\dot{}}{\vee }Q\right)=F\left(P\right)\underset{\dot{}}{\vee }F\left(Q\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdlemg2.b	$$⊢B={\mathrm{Base}}_K$$
2		cdlemg2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdlemg2.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdlemg2.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdlemg2.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdlemg2.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdlemg2.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
8		cdlemg2ex.u	$$⊢U=\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }W$$
9		cdlemg2ex.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		cdlemg2ex.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		cdlemg2ex.g	$$⊢G=\left(x\in B⟼if\left(\left(p\ne q\wedge \neg x\underset{\dot{}}{\le }W\right),\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=if\left(s\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right),\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right)\right)\to y=E\right)\right),⦋s/t⦌D\right)\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right),x\right)\right)$$
12		fveq1	$$⊢F=G\to F\left(P\underset{\dot{}}{\vee }Q\right)=G\left(P\underset{\dot{}}{\vee }Q\right)$$
13		fveq1	$$⊢F=G\to F\left(P\right)=G\left(P\right)$$
14		fveq1	$$⊢F=G\to F\left(Q\right)=G\left(Q\right)$$
15	13 14	oveq12d	$$⊢F=G\to F\left(P\right)\underset{\dot{}}{\vee }F\left(Q\right)=G\left(P\right)\underset{\dot{}}{\vee }G\left(Q\right)$$
16	12 15	eqeq12d	$$⊢F=G\to \left(F\left(P\underset{\dot{}}{\vee }Q\right)=F\left(P\right)\underset{\dot{}}{\vee }F\left(Q\right)\leftrightarrow G\left(P\underset{\dot{}}{\vee }Q\right)=G\left(P\right)\underset{\dot{}}{\vee }G\left(Q\right)\right)$$
17		vex	$$⊢s\in \mathrm{V}$$
18		eqid	$$⊢\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(q\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(q\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
19	9 18	cdleme31sc	$$⊢s\in \mathrm{V}\to ⦋s/t⦌D=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(q\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
20	17 19	ax-mp	$$⊢⦋s/t⦌D=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(q\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
21		eqid	$$⊢\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right)\right)\to y=E\right)\right)=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right)\right)\to y=E\right)\right)$$
22		eqid	$$⊢if\left(s\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right),\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right)\right)\to y=E\right)\right),⦋s/t⦌D\right)=if\left(s\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right),\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right)\right)\to y=E\right)\right),⦋s/t⦌D\right)$$
23		eqid	$$⊢\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=if\left(s\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right),\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right)\right)\to y=E\right)\right),⦋s/t⦌D\right)\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right)=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=if\left(s\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right),\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right)\right)\to y=E\right)\right),⦋s/t⦌D\right)\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right)$$
24	1 2 3 4 5 6 8 20 9 10 21 22 23 11	cdleme42mgN	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(p\in A\wedge \neg p\underset{\dot{}}{\le }W\right)\wedge \left(q\in A\wedge \neg q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\right)\to G\left(P\underset{\dot{}}{\vee }Q\right)=G\left(P\right)\underset{\dot{}}{\vee }G\left(Q\right)$$
25	1 2 3 4 5 6 7 8 9 10 11 16 24	cdlemg2ce	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge F\in T\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\right)\to F\left(P\underset{\dot{}}{\vee }Q\right)=F\left(P\right)\underset{\dot{}}{\vee }F\left(Q\right)$$
26	25	3com23	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge F\in T\right)\to F\left(P\underset{\dot{}}{\vee }Q\right)=F\left(P\right)\underset{\dot{}}{\vee }F\left(Q\right)$$