Theorem cdlemg8d

Description: TODO: FIX COMMENT. (Contributed by NM, 29-Apr-2013)

	Ref	Expression
Hypotheses	cdlemg8.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdlemg8.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdlemg8.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdlemg8.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdlemg8.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdlemg8.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
Assertion	cdlemg8d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)=P\underset{\dot{}}{\vee }Q\wedge F\left(G\left(P\right)\right)\ne P\right)\right)\to \left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }W=\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)\underset{\dot{}}{\wedge }W$$

Proof

Step	Hyp	Ref	Expression
1		cdlemg8.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdlemg8.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdlemg8.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdlemg8.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdlemg8.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdlemg8.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
7	1 2 3 4 5 6	cdlemg8b	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)=P\underset{\dot{}}{\vee }Q\wedge F\left(G\left(P\right)\right)\ne P\right)\right)\to P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)=P\underset{\dot{}}{\vee }Q$$
8	1 2 3 4 5 6	cdlemg8c	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)=P\underset{\dot{}}{\vee }Q\wedge F\left(G\left(P\right)\right)\ne P\right)\right)\to Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)=P\underset{\dot{}}{\vee }Q$$
9	7 8	eqtr4d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)=P\underset{\dot{}}{\vee }Q\wedge F\left(G\left(P\right)\right)\ne P\right)\right)\to P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)=Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)$$
10	9	oveq1d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)=P\underset{\dot{}}{\vee }Q\wedge F\left(G\left(P\right)\right)\ne P\right)\right)\to \left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }W=\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)\underset{\dot{}}{\wedge }W$$