Theorem cdleme18c

Description: Part of proof of Lemma E in Crawley p. 114, 2nd sentence of 4th paragraph. F , G represent f(s), f_s(q) respectively. We show -. f_s(q) = p whenever p \/ q has three atoms under it (implied by the negated existential condition). (Contributed by NM, 10-Nov-2012)

	Ref	Expression
Hypotheses	cdleme18.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme18.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme18.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme18.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme18.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme18.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme18.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme18.g	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
Assertion	cdleme18c	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to G=P$$

Proof

Step	Hyp	Ref	Expression
1		cdleme18.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme18.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme18.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme18.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme18.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme18.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme18.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
8		cdleme18.g	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		simp31	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to P\ne Q$$
10		simp32	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
11	9 10	jca	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
12	1 2 3 4 5 6 7 8	cdleme18b	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to G\ne Q$$
13	11 12	syld3an3	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to G\ne Q$$
14	13	neneqd	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \neg G=Q$$
15		simp1l	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to K\in \mathrm{HL}$$
16		simp1r	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to W\in H$$
17		simp21l	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to P\in A$$
18		simp22l	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to Q\in A$$
19		simp23l	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to S\in A$$
20	1 2 3 4 5 6 7 8	cdleme4a	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge Q\in A\wedge Q\in A\right)\wedge S\in A\right)\to G\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
21	15 16 17 18 18 19 20	syl231anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to G\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
22		simp33	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)$$
23		simp1	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
24		simp21	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)$$
25		simp22	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)$$
26		simp23	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)$$
27	1 2 4	hlatlej2	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
28	15 17 18 27	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
29	1 2 3 4 5 6 7 8	cdleme7ga	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge Q\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to G\in A$$
30	23 24 25 25 26 9 28 10 29	syl323anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to G\in A$$
31	1 2 3 4 5 6 7 8	cdleme18a	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg G\underset{\dot{}}{\le }W$$
32	11 31	syld3an3	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \neg G\underset{\dot{}}{\le }W$$
33	1 2 4	cdleme0nex	$$⊢\left(\left(K\in \mathrm{HL}\wedge G\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\wedge \left(P\in A\wedge Q\in A\wedge P\ne Q\right)\wedge \left(G\in A\wedge \neg G\underset{\dot{}}{\le }W\right)\right)\to \left(G=P\vee G=Q\right)$$
34	15 21 22 17 18 9 30 32 33	syl332anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \left(G=P\vee G=Q\right)$$
35	34	ord	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to \left(\neg G=P\to G=Q\right)$$
36	14 35	mt3d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg \exists r\in A\left(\neg r\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }r=Q\underset{\dot{}}{\vee }r\right)\right)\right)\to G=P$$