Theorem cdleme32d

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 20-Feb-2013)

	Ref	Expression
Hypotheses	cdleme32.b	$$⊢B={\mathrm{Base}}_K$$
	cdleme32.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme32.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme32.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme32.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme32.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme32.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme32.c	$$⊢C=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme32.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme32.e	$$⊢E=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme32.i	$$⊢I=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=E\right)\right)$$
	cdleme32.n	$$⊢N=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),I,C\right)$$
	cdleme32.o	$$⊢O=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=N\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right)$$
	cdleme32.f	$$⊢F=\left(x\in B⟼if\left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right),O,x\right)\right)$$
Assertion	cdleme32d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge X\underset{\dot{}}{\le }Y\right)\to F\left(X\right)\underset{\dot{}}{\le }F\left(Y\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdleme32.b	$$⊢B={\mathrm{Base}}_K$$
2		cdleme32.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdleme32.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdleme32.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdleme32.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdleme32.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdleme32.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		cdleme32.c	$$⊢C=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme32.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10		cdleme32.e	$$⊢E=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
11		cdleme32.i	$$⊢I=\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=E\right)\right)$$
12		cdleme32.n	$$⊢N=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),I,C\right)$$
13		cdleme32.o	$$⊢O=\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=N\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right)$$
14		cdleme32.f	$$⊢F=\left(x\in B⟼if\left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right),O,x\right)\right)$$
15		simp11	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
16		simp21	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge X\underset{\dot{}}{\le }Y\right)\to X\in B$$
17		simp23r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \neg X\underset{\dot{}}{\le }W$$
18	1 2 3 4 5 6	lhpmcvr2	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(X\in B\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to \exists s\in A\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)$$
19	15 16 17 18	syl12anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \exists s\in A\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)$$
20		nfv	$$⊢Ⅎs\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge X\underset{\dot{}}{\le }Y\right)$$
21		nfcv	$$⊢\underset{\_}{Ⅎ}sB$$
22		nfv	$$⊢Ⅎs\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right)$$
23		nfra1	$$⊢Ⅎs\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=N\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)$$
24	23 21	nfriota	$$⊢\underset{\_}{Ⅎ}s\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=N\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right)$$
25	13 24	nfcxfr	$$⊢\underset{\_}{Ⅎ}sO$$
26		nfcv	$$⊢\underset{\_}{Ⅎ}sx$$
27	22 25 26	nfif	$$⊢\underset{\_}{Ⅎ}sif\left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right),O,x\right)$$
28	21 27	nfmpt	$$⊢\underset{\_}{Ⅎ}s\left(x\in B⟼if\left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right),O,x\right)\right)$$
29	14 28	nfcxfr	$$⊢\underset{\_}{Ⅎ}sF$$
30		nfcv	$$⊢\underset{\_}{Ⅎ}sX$$
31	29 30	nffv	$$⊢\underset{\_}{Ⅎ}sF\left(X\right)$$
32		nfcv	$$⊢\underset{\_}{Ⅎ}s\underset{\dot{}}{\le }$$
33		nfcv	$$⊢\underset{\_}{Ⅎ}sY$$
34	29 33	nffv	$$⊢\underset{\_}{Ⅎ}sF\left(Y\right)$$
35	31 32 34	nfbr	$$⊢ℲsF\left(X\right)\underset{\dot{}}{\le }F\left(Y\right)$$
36		simpl1	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge X\underset{\dot{}}{\le }Y\right)\wedge \left(s\in A\wedge \left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\right)\to \left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)$$
37		simpl2	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge X\underset{\dot{}}{\le }Y\right)\wedge \left(s\in A\wedge \left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\right)\to \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)$$
38		simprl	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge X\underset{\dot{}}{\le }Y\right)\wedge \left(s\in A\wedge \left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\right)\to s\in A$$
39		simprrl	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge X\underset{\dot{}}{\le }Y\right)\wedge \left(s\in A\wedge \left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\right)\to \neg s\underset{\dot{}}{\le }W$$
40	38 39	jca	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge X\underset{\dot{}}{\le }Y\right)\wedge \left(s\in A\wedge \left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\right)\to \left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)$$
41		simprrr	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge X\underset{\dot{}}{\le }Y\right)\wedge \left(s\in A\wedge \left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\right)\to s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X$$
42		simpl3	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge X\underset{\dot{}}{\le }Y\right)\wedge \left(s\in A\wedge \left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\right)\to X\underset{\dot{}}{\le }Y$$
43	1 2 3 4 5 6 7 8 9 10 11 12 13 14	cdleme32c	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge \left(\left(s\in A\wedge \neg s\underset{\dot{}}{\le }W\right)\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\wedge X\underset{\dot{}}{\le }Y\right)\right)\to F\left(X\right)\underset{\dot{}}{\le }F\left(Y\right)$$
44	36 37 40 41 42 43	syl113anc	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge X\underset{\dot{}}{\le }Y\right)\wedge \left(s\in A\wedge \left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\right)\right)\to F\left(X\right)\underset{\dot{}}{\le }F\left(Y\right)$$
45	44	exp32	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \left(s\in A\to \left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to F\left(X\right)\underset{\dot{}}{\le }F\left(Y\right)\right)\right)$$
46	20 35 45	rexlimd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \left(\exists s\in A\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to F\left(X\right)\underset{\dot{}}{\le }F\left(Y\right)\right)$$
47	19 46	mpd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(X\in B\wedge Y\in B\wedge \left(P\ne Q\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\wedge X\underset{\dot{}}{\le }Y\right)\to F\left(X\right)\underset{\dot{}}{\le }F\left(Y\right)$$