Theorem cdleme38m

Description: Part of proof of Lemma E in Crawley p. 113. Show that f(x) is one-to-one on P .\/ Q line. TODO: FIX COMMENT. (Contributed by NM, 13-Mar-2013)

	Ref	Expression
Hypotheses	cdleme38.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme38.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme38.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme38.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme38.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme38.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme38.e	$$⊢E=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme38.d	$$⊢D=\left(u\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }u\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme38.v	$$⊢V=\left(t\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\wedge }W$$
	cdleme38.x	$$⊢X=\left(u\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\wedge }W$$
	cdleme38.f	$$⊢F=\left(R\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme38.g	$$⊢G=\left(S\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
Assertion	cdleme38m	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to R=S$$

Proof

Step	Hyp	Ref	Expression
1		cdleme38.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme38.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme38.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme38.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme38.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme38.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme38.e	$$⊢E=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
8		cdleme38.d	$$⊢D=\left(u\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }u\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme38.v	$$⊢V=\left(t\underset{\dot{}}{\vee }E\right)\underset{\dot{}}{\wedge }W$$
10		cdleme38.x	$$⊢X=\left(u\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\wedge }W$$
11		cdleme38.f	$$⊢F=\left(R\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)$$
12		cdleme38.g	$$⊢G=\left(S\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
13		simp1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)$$
14		simp2	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)$$
15		simp311	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
16		simp312	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
17		simp313	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to F=G$$
18	15 16	jca	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
19		simp32	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
20		simp33	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
21		eqid	$$⊢\left(S\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(S\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
22	1 2 3 4 5 6 7 8 9 10 21 12	cdleme37m	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(S\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)=G$$
23	13 14 18 19 20 22	syl113anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(S\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)=G$$
24	17 23	eqtr4d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to F=\left(S\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
25	15 16 24	3jca	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to \left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=\left(S\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)\right)$$
26		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
27	26 1 2 3 4 5 6 7 9 11 21	cdleme36m	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=\left(S\underset{\dot{}}{\vee }V\right)\underset{\dot{}}{\wedge }\left(E\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to R=S$$
28	13 14 25 19 27	syl112anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\wedge \left(\left(t\in A\wedge \neg t\underset{\dot{}}{\le }W\right)\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\left(u\in A\wedge \neg u\underset{\dot{}}{\le }W\right)\wedge \neg u\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\right)\to R=S$$