Theorem cdlemeg49le

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 9-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	cdlemeg49le	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to G\left(X\right)\underset{\dot{}}{\le }G\left(Y\right)$$

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		simp11	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
12		simp13	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)$$
13		simp12	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
14		simp2	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \left(X\in B\wedge Y\in B\right)$$
15		simp3	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to X\underset{\dot{}}{\le }Y$$
16		vex	$$⊢u\in \mathrm{V}$$
17		eqid	$$⊢\left(u\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }u\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(u\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }u\right)\underset{\dot{}}{\wedge }W\right)\right)$$
18	8 17	cdleme31sc	$$⊢u\in \mathrm{V}\to ⦋u/v⦌N=\left(u\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }u\right)\underset{\dot{}}{\wedge }W\right)\right)$$
19	16 18	ax-mp	$$⊢⦋u/v⦌N=\left(u\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }u\right)\underset{\dot{}}{\wedge }W\right)\right)$$
20		eqid	$$⊢\left(\iota b\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\to b=O\right)\right)=\left(\iota b\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\to b=O\right)\right)$$
21		eqid	$$⊢if\left(u\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right),\left(\iota b\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\to b=O\right)\right),⦋u/v⦌N\right)=if\left(u\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right),\left(\iota b\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\to b=O\right)\right),⦋u/v⦌N\right)$$
22		eqid	$$⊢\left(\iota c\in B|\forall u\in A\left(\left(\neg u\underset{\dot{}}{\le }W\wedge u\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)=a\right)\to c=if\left(u\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right),\left(\iota b\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\to b=O\right)\right),⦋u/v⦌N\right)\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)\right)\right)=\left(\iota c\in B|\forall u\in A\left(\left(\neg u\underset{\dot{}}{\le }W\wedge u\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)=a\right)\to c=if\left(u\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right),\left(\iota b\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\to b=O\right)\right),⦋u/v⦌N\right)\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)\right)\right)$$
23	1 2 3 4 5 6 7 19 8 9 20 21 22 10	cdleme32le	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to G\left(X\right)\underset{\dot{}}{\le }G\left(Y\right)$$
24	11 12 13 14 15 23	syl311anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to G\left(X\right)\underset{\dot{}}{\le }G\left(Y\right)$$