Theorem dalem23

Description: Lemma for dath . Show that auxiliary atom G is an atom. (Contributed by NM, 2-Aug-2012)

	Ref	Expression
Hypotheses	dalem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(Y\in O\wedge Z\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\right)$$
	dalem.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	dalem.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	dalem.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem.ps	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
	dalem23.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem23.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem23.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem23.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem23.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
Assertion	dalem23	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\in A$$

Proof

Step	Hyp	Ref	Expression
1		dalem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(Y\in O\wedge Z\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\right)$$
2		dalem.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		dalem.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		dalem.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		dalem.ps	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
6		dalem23.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
7		dalem23.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
8		dalem23.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
9		dalem23.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
10		dalem23.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
11	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
12	11	adantr	$$⊢\left(\phi \wedge \psi \right)\to K\in \mathrm{HL}$$
13	5	dalemccea	$$⊢\psi \to c\in A$$
14	13	adantl	$$⊢\left(\phi \wedge \psi \right)\to c\in A$$
15	1	dalempea	$$⊢\phi \to P\in A$$
16	15	adantr	$$⊢\left(\phi \wedge \psi \right)\to P\in A$$
17	5	dalemddea	$$⊢\psi \to d\in A$$
18	17	adantl	$$⊢\left(\phi \wedge \psi \right)\to d\in A$$
19	1	dalemsea	$$⊢\phi \to S\in A$$
20	19	adantr	$$⊢\left(\phi \wedge \psi \right)\to S\in A$$
21	3 4	hlatj4	$$⊢\left(K\in \mathrm{HL}\wedge \left(c\in A\wedge P\in A\right)\wedge \left(d\in A\wedge S\in A\right)\right)\to \left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }\left(d\underset{\dot{}}{\vee }S\right)=\left(c\underset{\dot{}}{\vee }d\right)\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }S\right)$$
22	12 14 16 18 20 21	syl122anc	$$⊢\left(\phi \wedge \psi \right)\to \left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }\left(d\underset{\dot{}}{\vee }S\right)=\left(c\underset{\dot{}}{\vee }d\right)\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }S\right)$$
23	22	3adant2	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }\left(d\underset{\dot{}}{\vee }S\right)=\left(c\underset{\dot{}}{\vee }d\right)\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }S\right)$$
24	1 2 3 4 5 7 8 9	dalem22	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(c\underset{\dot{}}{\vee }d\right)\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }S\right)\in O$$
25	23 24	eqeltrd	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }\left(d\underset{\dot{}}{\vee }S\right)\in O$$
26	11	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{HL}$$
27	1 2 3 4 7 8	dalemply	$$⊢\phi \to P\underset{\dot{}}{\le }Y$$
28	5	dalem-ccly	$$⊢\psi \to \neg c\underset{\dot{}}{\le }Y$$
29		nbrne2	$$⊢\left(P\underset{\dot{}}{\le }Y\wedge \neg c\underset{\dot{}}{\le }Y\right)\to P\ne c$$
30	27 28 29	syl2an	$$⊢\left(\phi \wedge \psi \right)\to P\ne c$$
31	30	necomd	$$⊢\left(\phi \wedge \psi \right)\to c\ne P$$
32		eqid	$$⊢\mathrm{LLines}\left(K\right)=\mathrm{LLines}\left(K\right)$$
33	3 4 32	llni2	$$⊢\left(\left(K\in \mathrm{HL}\wedge c\in A\wedge P\in A\right)\wedge c\ne P\right)\to c\underset{\dot{}}{\vee }P\in \mathrm{LLines}\left(K\right)$$
34	12 14 16 31 33	syl31anc	$$⊢\left(\phi \wedge \psi \right)\to c\underset{\dot{}}{\vee }P\in \mathrm{LLines}\left(K\right)$$
35	34	3adant2	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\underset{\dot{}}{\vee }P\in \mathrm{LLines}\left(K\right)$$
36	17	3ad2ant3	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to d\in A$$
37	19	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to S\in A$$
38	1 2 3 4 9	dalemsly	$$⊢\left(\phi \wedge Y=Z\right)\to S\underset{\dot{}}{\le }Y$$
39	38	3adant3	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to S\underset{\dot{}}{\le }Y$$
40	5	dalem-ddly	$$⊢\psi \to \neg d\underset{\dot{}}{\le }Y$$
41	40	3ad2ant3	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \neg d\underset{\dot{}}{\le }Y$$
42		nbrne2	$$⊢\left(S\underset{\dot{}}{\le }Y\wedge \neg d\underset{\dot{}}{\le }Y\right)\to S\ne d$$
43	39 41 42	syl2anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to S\ne d$$
44	43	necomd	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to d\ne S$$
45	3 4 32	llni2	$$⊢\left(\left(K\in \mathrm{HL}\wedge d\in A\wedge S\in A\right)\wedge d\ne S\right)\to d\underset{\dot{}}{\vee }S\in \mathrm{LLines}\left(K\right)$$
46	26 36 37 44 45	syl31anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to d\underset{\dot{}}{\vee }S\in \mathrm{LLines}\left(K\right)$$
47	3 6 4 32 7	2llnmj	$$⊢\left(K\in \mathrm{HL}\wedge c\underset{\dot{}}{\vee }P\in \mathrm{LLines}\left(K\right)\wedge d\underset{\dot{}}{\vee }S\in \mathrm{LLines}\left(K\right)\right)\to \left(\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)\in A\leftrightarrow \left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }\left(d\underset{\dot{}}{\vee }S\right)\in O\right)$$
48	26 35 46 47	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)\in A\leftrightarrow \left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }\left(d\underset{\dot{}}{\vee }S\right)\in O\right)$$
49	25 48	mpbird	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)\in A$$
50	10 49	eqeltrid	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\in A$$