Theorem cdlemg25zz

Description: cdlemg16zz restated for easier studying. TODO: Discard this after everything is figured out. (Contributed by NM, 26-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	cdlemg25zz	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge \neg R\left(F\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }z\right)\wedge \neg R\left(G\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }z\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }W=\left(z\underset{\dot{}}{\vee }F\left(G\left(z\right)\right)\right)\underset{\dot{}}{\wedge }W$$

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	1 2 3 4 5 6 7	cdlemg16zz	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(z\in A\wedge \neg z\underset{\dot{}}{\le }W\right)\wedge F\in T\right)\wedge \left(G\in T\wedge \neg R\left(F\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }z\right)\wedge \neg R\left(G\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }z\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }W=\left(z\underset{\dot{}}{\vee }F\left(G\left(z\right)\right)\right)\underset{\dot{}}{\wedge }W$$