Theorem cdleme35h

Description: Part of proof of Lemma E in Crawley p. 113. Show that f(x) is one-to-one outside of P .\/ Q line. TODO: FIX COMMENT. (Contributed by NM, 11-Mar-2013)

	Ref	Expression
Hypotheses	cdleme35.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme35.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme35.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme35.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme35.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme35.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme35.f	$$⊢F=\left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme35.g	$$⊢G=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
Assertion	cdleme35h	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\right)\to R=S$$

Proof

Step	Hyp	Ref	Expression
1		cdleme35.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme35.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme35.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme35.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme35.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme35.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme35.f	$$⊢F=\left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }W\right)\right)$$
8		cdleme35.g	$$⊢G=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		oveq1	$$⊢F=G\to F\underset{\dot{}}{\vee }U=G\underset{\dot{}}{\vee }U$$
10		oveq2	$$⊢F=G\to Q\underset{\dot{}}{\vee }F=Q\underset{\dot{}}{\vee }G$$
11	10	oveq1d	$$⊢F=G\to \left(Q\underset{\dot{}}{\vee }F\right)\underset{\dot{}}{\wedge }W=\left(Q\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }W$$
12	11	oveq2d	$$⊢F=G\to P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }F\right)\underset{\dot{}}{\wedge }W\right)=P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }W\right)$$
13	9 12	oveq12d	$$⊢F=G\to \left(F\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }F\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(G\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }W\right)\right)$$
14	13	3ad2ant3	$$⊢\left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\to \left(F\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }F\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(G\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }W\right)\right)$$
15	14	3ad2ant3	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\right)\to \left(F\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }F\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(G\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }W\right)\right)$$
16		simp1	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\right)\to \left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)$$
17		simp21	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\right)\to P\ne Q$$
18		simp22	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\right)\to \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)$$
19		simp31	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\right)\to \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
20	1 2 3 4 5 6 7	cdleme35g	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\right)\wedge \neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(F\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }F\right)\underset{\dot{}}{\wedge }W\right)\right)=R$$
21	16 17 18 19 20	syl121anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\right)\to \left(F\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }F\right)\underset{\dot{}}{\wedge }W\right)\right)=R$$
22		simp23	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\right)\to \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)$$
23		simp32	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\right)\to \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
24	1 2 3 4 5 6 8	cdleme35g	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(G\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }W\right)\right)=S$$
25	16 17 22 23 24	syl121anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\right)\to \left(G\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }W\right)\right)=S$$
26	15 21 25	3eqtr3d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(P\ne Q\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\wedge \left(\neg R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge F=G\right)\right)\to R=S$$