Theorem 4atexlemex4

Description: Lemma for 4atexlem7 . Show that when C = S , D satisfies the existence condition of the consequent. (Contributed by NM, 26-Nov-2012)

	Ref	Expression
Hypotheses	4thatlem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\wedge \left(T\in A\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)$$
	4thatlem0.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	4thatlem0.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	4thatlem0.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	4thatlem0.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	4thatlem0.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	4thatlem0.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	4thatlem0.v	$$⊢V=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
	4thatlem0.c	$$⊢C=\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)$$
	4thatlem0.d	$$⊢D=\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)$$
Assertion	4atexlemex4	$$⊢\left(\phi \wedge C=S\right)\to \exists z\in A\left(\neg z\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)$$

Proof

Step	Hyp	Ref	Expression
1		4thatlem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\wedge \left(T\in A\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)$$
2		4thatlem0.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		4thatlem0.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		4thatlem0.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		4thatlem0.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		4thatlem0.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		4thatlem0.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		4thatlem0.v	$$⊢V=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
9		4thatlem0.c	$$⊢C=\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)$$
10		4thatlem0.d	$$⊢D=\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)$$
11	1 2 3 5 7	4atexlemswapqr	$$⊢\phi \to \left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }Q=R\underset{\dot{}}{\vee }Q\right)\wedge \left(T\in A\wedge \left(\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne R\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }R\right)\right)\right)$$
12	1 2 3 4 5 6 7 8 9 10	4atexlemcnd	$$⊢\phi \to C\ne D$$
13		pm13.18	$$⊢\left(C=S\wedge C\ne D\right)\to S\ne D$$
14	13	necomd	$$⊢\left(C=S\wedge C\ne D\right)\to D\ne S$$
15	14	expcom	$$⊢C\ne D\to \left(C=S\to D\ne S\right)$$
16	12 15	syl	$$⊢\phi \to \left(C=S\to D\ne S\right)$$
17		biid	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }Q=R\underset{\dot{}}{\vee }Q\right)\wedge \left(T\in A\wedge \left(\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne R\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }R\right)\right)\right)\leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }Q=R\underset{\dot{}}{\vee }Q\right)\wedge \left(T\in A\wedge \left(\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne R\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }R\right)\right)\right)$$
18		eqid	$$⊢\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W=\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W$$
19	17 2 3 4 5 6 18 8 10	4atexlemex2	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }Q=R\underset{\dot{}}{\vee }Q\right)\wedge \left(T\in A\wedge \left(\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne R\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }R\right)\right)\right)\wedge D\ne S\right)\to \exists z\in A\left(\neg z\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)$$
20	11 16 19	syl6an	$$⊢\phi \to \left(C=S\to \exists z\in A\left(\neg z\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\right)$$
21	20	imp	$$⊢\left(\phi \wedge C=S\right)\to \exists z\in A\left(\neg z\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)$$