Theorem cdlemg18

Description: Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-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	cdlemg18	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(F\in T\wedge G\in T\right)\wedge P\ne Q\wedge G\left(P\right)\ne P\right)\wedge \left(R\left(G\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)\right)\underset{\dot{}}{\le }W$$

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		simp11	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(F\in T\wedge G\in T\right)\wedge P\ne Q\wedge G\left(P\right)\ne P\right)\wedge \left(R\left(G\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
9		simp21r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(F\in T\wedge G\in T\right)\wedge P\ne Q\wedge G\left(P\right)\ne P\right)\wedge \left(R\left(G\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to G\in T$$
10		simp12	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(F\in T\wedge G\in T\right)\wedge P\ne Q\wedge G\left(P\right)\ne P\right)\wedge \left(R\left(G\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
11	1 2 3 4 5 6 7	cdlemg18d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(F\in T\wedge G\in T\right)\wedge P\ne Q\wedge G\left(P\right)\ne P\right)\wedge \left(R\left(G\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)\in A$$
12		simp23	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(F\in T\wedge G\in T\right)\wedge P\ne Q\wedge G\left(P\right)\ne P\right)\wedge \left(R\left(G\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to G\left(P\right)\ne P$$
13		simp1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(F\in T\wedge G\in T\right)\wedge P\ne Q\wedge G\left(P\right)\ne P\right)\wedge \left(R\left(G\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)$$
14		simp21l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(F\in T\wedge G\in T\right)\wedge P\ne Q\wedge G\left(P\right)\ne P\right)\wedge \left(R\left(G\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to F\in T$$
15		simp22	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(F\in T\wedge G\in T\right)\wedge P\ne Q\wedge G\left(P\right)\ne P\right)\wedge \left(R\left(G\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to P\ne Q$$
16		simp31	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(F\in T\wedge G\in T\right)\wedge P\ne Q\wedge G\left(P\right)\ne P\right)\wedge \left(R\left(G\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to R\left(G\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
17		simp33	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(F\in T\wedge G\in T\right)\wedge P\ne Q\wedge G\left(P\right)\ne P\right)\wedge \left(R\left(G\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)$$
18	1 2 3 4 5 6 7	cdlemg17	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(F\in T\wedge G\in T\wedge P\ne Q\right)\wedge \left(G\left(P\right)\ne P\wedge R\left(G\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to G\left(\left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)\right)=\left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)$$
19	13 14 9 15 12 16 17 18	syl133anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(F\in T\wedge G\in T\right)\wedge P\ne Q\wedge G\left(P\right)\ne P\right)\wedge \left(R\left(G\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to G\left(\left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)\right)=\left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)$$
20	1 4 5 6	ltrnatlw	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(G\in T\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)\in A\right)\wedge \left(G\left(P\right)\ne P\wedge G\left(\left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)\right)=\left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)\right)\underset{\dot{}}{\le }W$$
21	8 9 10 11 12 19 20	syl132anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(F\in T\wedge G\in T\right)\wedge P\ne Q\wedge G\left(P\right)\ne P\right)\wedge \left(R\left(G\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F\left(G\left(P\right)\right)\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\ne P\underset{\dot{}}{\vee }Q\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \left(\left(P\underset{\dot{}}{\vee }F\left(G\left(P\right)\right)\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }F\left(G\left(Q\right)\right)\right)\right)\underset{\dot{}}{\le }W$$