Theorem cdlemg37

Description: Use cdlemg8 to eliminate the =/= ( P .\/ Q ) condition of cdlemg24 . (Contributed by NM, 31-May-2013)

	Ref	Expression
Hypotheses	cdlemg12.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdlemg12.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdlemg12.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdlemg12.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdlemg12.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdlemg12.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
	cdlemg12b.r	$$⊢R=\mathrm{trL}\left(K\right)\left(W\right)$$
Assertion	cdlemg37	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge P\ne Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }W=\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)\underset{\dot{}}{\wedge }W$$

Proof

Step	Hyp	Ref	Expression
1		cdlemg12.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdlemg12.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdlemg12.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdlemg12.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdlemg12.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdlemg12.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
7		cdlemg12b.r	$$⊢R=\mathrm{trL}\left(K\right)\left(W\right)$$
8		simpl1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge P\ne Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)=P\underset{\dot{}}{\vee }Q\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
9		simpl2	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge P\ne Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)=P\underset{\dot{}}{\vee }Q\right)\to \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)$$
10		simpl31	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge P\ne Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)=P\underset{\dot{}}{\vee }Q\right)\to G\in T$$
11		simpr	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge P\ne Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)=P\underset{\dot{}}{\vee }Q\right)\to F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)=P\underset{\dot{}}{\vee }Q$$
12	1 2 3 4 5 6	cdlemg8	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)=P\underset{\dot{}}{\vee }Q\right)\right)\to \left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }W=\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)\underset{\dot{}}{\wedge }W$$
13	8 9 10 11 12	syl112anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge P\ne Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)=P\underset{\dot{}}{\vee }Q\right)\to \left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }W=\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)\underset{\dot{}}{\wedge }W$$
14		simpl1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge P\ne Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
15		simpl21	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge P\ne Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\right)\to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
16		simpl22	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge P\ne Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\right)\to \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)$$
17		simpl23	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge P\ne Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\right)\to F\in T$$
18		simpl31	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge P\ne Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\right)\to G\in T$$
19		simpl32	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge P\ne Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\right)\to P\ne Q$$
20		simpr	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge P\ne Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\right)\to F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q$$
21		simpl33	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge P\ne Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\right)\to \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)$$
22	1 2 3 4 5 6 7	cdlemg24	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(F\in T\wedge G\in T\wedge P\ne Q\right)\wedge \left(F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }W=\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)\underset{\dot{}}{\wedge }W$$
23	14 15 16 17 18 19 20 21 22	syl332anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge P\ne Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\right)\to \left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }W=\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)\underset{\dot{}}{\wedge }W$$
24	13 23	pm2.61dane	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge P\ne Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }W=\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)\underset{\dot{}}{\wedge }W$$