Theorem dalem17

Description: Lemma for dath . When planes Y and Z are equal, the center of perspectivity C is in Y . (Contributed by NM, 1-Aug-2012)

	Ref	Expression
Hypotheses	dalema.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(Y\in O\wedge Z\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\right)$$
	dalemc.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	dalemc.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	dalemc.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem17.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem17.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem17.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
Assertion	dalem17	$$⊢\left(\phi \wedge Y=Z\right)\to C\underset{\dot{}}{\le }Y$$

Proof

Step	Hyp	Ref	Expression
1		dalema.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		dalemc.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		dalemc.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		dalemc.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		dalem17.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
6		dalem17.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
7		dalem17.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
8	1	dalemclrju	$$⊢\phi \to C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)$$
9	8	adantr	$$⊢\left(\phi \wedge Y=Z\right)\to C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)$$
10	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
11	1 3 4	dalempjqeb	$$⊢\phi \to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
12	1 4	dalemreb	$$⊢\phi \to R\in {\mathrm{Base}}_K$$
13		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
14	13 2 3	latlej2	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\wedge R\in {\mathrm{Base}}_K\right)\to R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
15	10 11 12 14	syl3anc	$$⊢\phi \to R\underset{\dot{}}{\le }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R\right)$$
16	15 6	breqtrrdi	$$⊢\phi \to R\underset{\dot{}}{\le }Y$$
17	16	adantr	$$⊢\left(\phi \wedge Y=Z\right)\to R\underset{\dot{}}{\le }Y$$
18	1 3 4	dalemsjteb	$$⊢\phi \to S\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
19	1 4	dalemueb	$$⊢\phi \to U\in {\mathrm{Base}}_K$$
20	13 2 3	latlej2	$$⊢\left(K\in \mathrm{Lat}\wedge S\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\wedge U\in {\mathrm{Base}}_K\right)\to U\underset{\dot{}}{\le }\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\right)$$
21	10 18 19 20	syl3anc	$$⊢\phi \to U\underset{\dot{}}{\le }\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\right)$$
22	21 7	breqtrrdi	$$⊢\phi \to U\underset{\dot{}}{\le }Z$$
23	22	adantr	$$⊢\left(\phi \wedge Y=Z\right)\to U\underset{\dot{}}{\le }Z$$
24		simpr	$$⊢\left(\phi \wedge Y=Z\right)\to Y=Z$$
25	23 24	breqtrrd	$$⊢\left(\phi \wedge Y=Z\right)\to U\underset{\dot{}}{\le }Y$$
26	1 5	dalemyeb	$$⊢\phi \to Y\in {\mathrm{Base}}_K$$
27	13 2 3	latjle12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(R\in {\mathrm{Base}}_K\wedge U\in {\mathrm{Base}}_K\wedge Y\in {\mathrm{Base}}_K\right)\right)\to \left(\left(R\underset{\dot{}}{\le }Y\wedge U\underset{\dot{}}{\le }Y\right)\leftrightarrow \left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\le }Y\right)$$
28	10 12 19 26 27	syl13anc	$$⊢\phi \to \left(\left(R\underset{\dot{}}{\le }Y\wedge U\underset{\dot{}}{\le }Y\right)\leftrightarrow \left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\le }Y\right)$$
29	28	adantr	$$⊢\left(\phi \wedge Y=Z\right)\to \left(\left(R\underset{\dot{}}{\le }Y\wedge U\underset{\dot{}}{\le }Y\right)\leftrightarrow \left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\le }Y\right)$$
30	17 25 29	mpbi2and	$$⊢\left(\phi \wedge Y=Z\right)\to \left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\le }Y$$
31	1 4	dalemceb	$$⊢\phi \to C\in {\mathrm{Base}}_K$$
32	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
33	1	dalemrea	$$⊢\phi \to R\in A$$
34	1	dalemuea	$$⊢\phi \to U\in A$$
35	13 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge R\in A\wedge U\in A\right)\to R\underset{\dot{}}{\vee }U\in {\mathrm{Base}}_K$$
36	32 33 34 35	syl3anc	$$⊢\phi \to R\underset{\dot{}}{\vee }U\in {\mathrm{Base}}_K$$
37	13 2	lattr	$$⊢\left(K\in \mathrm{Lat}\wedge \left(C\in {\mathrm{Base}}_K\wedge R\underset{\dot{}}{\vee }U\in {\mathrm{Base}}_K\wedge Y\in {\mathrm{Base}}_K\right)\right)\to \left(\left(C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\wedge \left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\le }Y\right)\to C\underset{\dot{}}{\le }Y\right)$$
38	10 31 36 26 37	syl13anc	$$⊢\phi \to \left(\left(C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\wedge \left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\le }Y\right)\to C\underset{\dot{}}{\le }Y\right)$$
39	38	adantr	$$⊢\left(\phi \wedge Y=Z\right)\to \left(\left(C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\wedge \left(R\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\le }Y\right)\to C\underset{\dot{}}{\le }Y\right)$$
40	9 30 39	mp2and	$$⊢\left(\phi \wedge Y=Z\right)\to C\underset{\dot{}}{\le }Y$$