Theorem 4atexlemex2

Description: Lemma for 4atexlem7 . Show that when C =/= S , C satisfies the existence condition of the consequent. (Contributed by NM, 25-Nov-2012)

	Ref	Expression
Hypotheses	4thatlem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\wedge \left(T\in A\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)$$
	4thatlem0.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	4thatlem0.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	4thatlem0.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	4thatlem0.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	4thatlem0.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	4thatlem0.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	4thatlem0.v	$$⊢V=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
	4thatlem0.c	$$⊢C=\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)$$
Assertion	4atexlemex2	$$⊢\left(\phi \wedge C\ne S\right)\to \exists z\in A\left(\neg z\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)$$

Proof

Step	Hyp	Ref	Expression
1		4thatlem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(S\in A\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }R=Q\underset{\dot{}}{\vee }R\right)\wedge \left(T\in A\wedge U\underset{\dot{}}{\vee }T=V\underset{\dot{}}{\vee }T\right)\right)\wedge \left(P\ne Q\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)$$
2		4thatlem0.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		4thatlem0.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		4thatlem0.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		4thatlem0.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		4thatlem0.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		4thatlem0.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		4thatlem0.v	$$⊢V=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
9		4thatlem0.c	$$⊢C=\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)$$
10	1 2 3 4 5 6 7 8 9	4atexlemc	$$⊢\phi \to C\in A$$
11	10	adantr	$$⊢\left(\phi \wedge C\ne S\right)\to C\in A$$
12	1 2 3 4 5 6 7 8 9	4atexlemnclw	$$⊢\phi \to \neg C\underset{\dot{}}{\le }W$$
13	12	adantr	$$⊢\left(\phi \wedge C\ne S\right)\to \neg C\underset{\dot{}}{\le }W$$
14	1 2 3 4 5 6 7 8	4atexlemntlpq	$$⊢\phi \to \neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
15		id	$$⊢C=P\to C=P$$
16	9 15	eqtr3id	$$⊢C=P\to \left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)=P$$
17	16	adantl	$$⊢\left(\phi \wedge C=P\right)\to \left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)=P$$
18	1	4atexlemkl	$$⊢\phi \to K\in \mathrm{Lat}$$
19	1 3 5	4atexlemqtb	$$⊢\phi \to Q\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
20	1 3 5	4atexlempsb	$$⊢\phi \to P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
21		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
22	21 2 4	latmle1	$$⊢\left(K\in \mathrm{Lat}\wedge Q\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\right)\to \left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)$$
23	18 19 20 22	syl3anc	$$⊢\phi \to \left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)$$
24	1	4atexlemk	$$⊢\phi \to K\in \mathrm{HL}$$
25	1	4atexlemq	$$⊢\phi \to Q\in A$$
26	1	4atexlemt	$$⊢\phi \to T\in A$$
27	3 5	hlatjcom	$$⊢\left(K\in \mathrm{HL}\wedge Q\in A\wedge T\in A\right)\to Q\underset{\dot{}}{\vee }T=T\underset{\dot{}}{\vee }Q$$
28	24 25 26 27	syl3anc	$$⊢\phi \to Q\underset{\dot{}}{\vee }T=T\underset{\dot{}}{\vee }Q$$
29	23 28	breqtrd	$$⊢\phi \to \left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }Q\right)$$
30	29	adantr	$$⊢\left(\phi \wedge C=P\right)\to \left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }Q\right)$$
31	17 30	eqbrtrrd	$$⊢\left(\phi \wedge C=P\right)\to P\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }Q\right)$$
32	1	4atexlemkc	$$⊢\phi \to K\in \mathrm{CvLat}$$
33	1	4atexlemp	$$⊢\phi \to P\in A$$
34	1	4atexlempnq	$$⊢\phi \to P\ne Q$$
35	2 3 5	cvlatexch2	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge T\in A\wedge Q\in A\right)\wedge P\ne Q\right)\to \left(P\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }Q\right)\to T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
36	32 33 26 25 34 35	syl131anc	$$⊢\phi \to \left(P\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }Q\right)\to T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
37	36	adantr	$$⊢\left(\phi \wedge C=P\right)\to \left(P\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }Q\right)\to T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
38	31 37	mpd	$$⊢\left(\phi \wedge C=P\right)\to T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
39	38	ex	$$⊢\phi \to \left(C=P\to T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
40	39	necon3bd	$$⊢\phi \to \left(\neg T\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\to C\ne P\right)$$
41	14 40	mpd	$$⊢\phi \to C\ne P$$
42	41	adantr	$$⊢\left(\phi \wedge C\ne S\right)\to C\ne P$$
43		simpr	$$⊢\left(\phi \wedge C\ne S\right)\to C\ne S$$
44	21 2 4	latmle2	$$⊢\left(K\in \mathrm{Lat}\wedge Q\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\wedge P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\right)\to \left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
45	18 19 20 44	syl3anc	$$⊢\phi \to \left(\left(Q\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }S\right)\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
46	9 45	eqbrtrid	$$⊢\phi \to C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
47	46	adantr	$$⊢\left(\phi \wedge C\ne S\right)\to C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
48	1	4atexlems	$$⊢\phi \to S\in A$$
49	1 2 3 5	4atexlempns	$$⊢\phi \to P\ne S$$
50	5 2 3	cvlsupr2	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge S\in A\wedge C\in A\right)\wedge P\ne S\right)\to \left(P\underset{\dot{}}{\vee }C=S\underset{\dot{}}{\vee }C\leftrightarrow \left(C\ne P\wedge C\ne S\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)$$
51	32 33 48 10 49 50	syl131anc	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }C=S\underset{\dot{}}{\vee }C\leftrightarrow \left(C\ne P\wedge C\ne S\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)$$
52	51	adantr	$$⊢\left(\phi \wedge C\ne S\right)\to \left(P\underset{\dot{}}{\vee }C=S\underset{\dot{}}{\vee }C\leftrightarrow \left(C\ne P\wedge C\ne S\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)$$
53	42 43 47 52	mpbir3and	$$⊢\left(\phi \wedge C\ne S\right)\to P\underset{\dot{}}{\vee }C=S\underset{\dot{}}{\vee }C$$
54		breq1	$$⊢z=C\to \left(z\underset{\dot{}}{\le }W\leftrightarrow C\underset{\dot{}}{\le }W\right)$$
55	54	notbid	$$⊢z=C\to \left(\neg z\underset{\dot{}}{\le }W\leftrightarrow \neg C\underset{\dot{}}{\le }W\right)$$
56		oveq2	$$⊢z=C\to P\underset{\dot{}}{\vee }z=P\underset{\dot{}}{\vee }C$$
57		oveq2	$$⊢z=C\to S\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }C$$
58	56 57	eqeq12d	$$⊢z=C\to \left(P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\leftrightarrow P\underset{\dot{}}{\vee }C=S\underset{\dot{}}{\vee }C\right)$$
59	55 58	anbi12d	$$⊢z=C\to \left(\left(\neg z\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)\leftrightarrow \left(\neg C\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }C=S\underset{\dot{}}{\vee }C\right)\right)$$
60	59	rspcev	$$⊢\left(C\in A\wedge \left(\neg C\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }C=S\underset{\dot{}}{\vee }C\right)\right)\to \exists z\in A\left(\neg z\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)$$
61	11 13 53 60	syl12anc	$$⊢\left(\phi \wedge C\ne S\right)\to \exists z\in A\left(\neg z\underset{\dot{}}{\le }W\wedge P\underset{\dot{}}{\vee }z=S\underset{\dot{}}{\vee }z\right)$$