Theorem 4atlem10b

Description: Lemma for 4at . Substitute V for R (cont.). (Contributed by NM, 10-Jul-2012)

	Ref	Expression
Hypotheses	4at.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	4at.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	4at.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	4atlem10b	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\wedge V\in A\right)\wedge \left(W\in A\wedge \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\wedge S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }S\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)$$

Proof

Step	Hyp	Ref	Expression
1		4at.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		4at.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		4at.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
4		simprr	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\wedge V\in A\right)\wedge \left(W\in A\wedge \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\wedge S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\right)\right)\to S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)$$
5		simprl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\wedge V\in A\right)\wedge \left(W\in A\wedge \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\wedge S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\right)\right)\to R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)$$
6		simpl1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\wedge V\in A\right)\wedge \left(W\in A\wedge \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\wedge S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\right)\right)\to \left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)$$
7		simpl21	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\wedge V\in A\right)\wedge \left(W\in A\wedge \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\wedge S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\right)\right)\to R\in A$$
8		simpl23	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\wedge V\in A\right)\wedge \left(W\in A\wedge \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\wedge S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\right)\right)\to V\in A$$
9		simpl31	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\wedge V\in A\right)\wedge \left(W\in A\wedge \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\wedge S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\right)\right)\to W\in A$$
10		simpl32	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\wedge V\in A\right)\wedge \left(W\in A\wedge \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\wedge S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\right)\right)\to \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)$$
11	1 2 3	4atlem10a	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge V\in A\wedge W\in A\right)\wedge \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)\right)\to \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\leftrightarrow \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }W\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)$$
12	6 7 8 9 10 11	syl131anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\wedge V\in A\right)\wedge \left(W\in A\wedge \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\wedge S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\right)\right)\to \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\leftrightarrow \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }W\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)$$
13	5 12	mpbid	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\wedge V\in A\right)\wedge \left(W\in A\wedge \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\wedge S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }W\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)$$
14	4 13	breqtrrd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\wedge V\in A\right)\wedge \left(W\in A\wedge \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\wedge S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\right)\right)\to S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }W\right)\right)$$
15		simpl22	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\wedge V\in A\right)\wedge \left(W\in A\wedge \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\wedge S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\right)\right)\to S\in A$$
16		simpl33	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\wedge V\in A\right)\wedge \left(W\in A\wedge \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\wedge S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\right)\right)\to \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
17	1 2 3	4atlem9	$$⊢\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\wedge W\in A\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\to \left(S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }W\right)\right)\leftrightarrow \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }S\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }W\right)\right)$$
18	6 7 15 9 16 17	syl131anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\wedge V\in A\right)\wedge \left(W\in A\wedge \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\wedge S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\right)\right)\to \left(S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }W\right)\right)\leftrightarrow \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }S\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }W\right)\right)$$
19	14 18	mpbid	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\wedge V\in A\right)\wedge \left(W\in A\wedge \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\wedge S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }S\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }W\right)$$
20	19 13	eqtrd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\wedge \left(R\in A\wedge S\in A\wedge V\in A\right)\wedge \left(W\in A\wedge \neg R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }W\right)\wedge \neg S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)\right)\right)\wedge \left(R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\wedge S\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)\right)\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(R\underset{\dot{}}{\vee }S\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }\left(V\underset{\dot{}}{\vee }W\right)$$