Theorem dalemsly

Description: Lemma for dath . Frequently-used utility lemma. (Contributed by NM, 15-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	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalemsly.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
Assertion	dalemsly	$$⊢\left(\phi \wedge Y=Z\right)\to S\underset{\dot{}}{\le }Y$$

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		dalemsly.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
6	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
7	1 4	dalemseb	$$⊢\phi \to S\in {\mathrm{Base}}_K$$
8	1 3 4	dalemtjueb	$$⊢\phi \to T\underset{\dot{}}{\vee }U\in {\mathrm{Base}}_K$$
9		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
10	9 2 3	latlej1	$$⊢\left(K\in \mathrm{Lat}\wedge S\in {\mathrm{Base}}_K\wedge T\underset{\dot{}}{\vee }U\in {\mathrm{Base}}_K\right)\to S\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }\left(T\underset{\dot{}}{\vee }U\right)\right)$$
11	6 7 8 10	syl3anc	$$⊢\phi \to S\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }\left(T\underset{\dot{}}{\vee }U\right)\right)$$
12	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
13	1	dalemsea	$$⊢\phi \to S\in A$$
14	1	dalemtea	$$⊢\phi \to T\in A$$
15	1	dalemuea	$$⊢\phi \to U\in A$$
16	3 4	hlatjass	$$⊢\left(K\in \mathrm{HL}\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\to \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U=S\underset{\dot{}}{\vee }\left(T\underset{\dot{}}{\vee }U\right)$$
17	12 13 14 15 16	syl13anc	$$⊢\phi \to \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U=S\underset{\dot{}}{\vee }\left(T\underset{\dot{}}{\vee }U\right)$$
18	11 17	breqtrrd	$$⊢\phi \to S\underset{\dot{}}{\le }\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U\right)$$
19	18 5	breqtrrdi	$$⊢\phi \to S\underset{\dot{}}{\le }Z$$
20	19	adantr	$$⊢\left(\phi \wedge Y=Z\right)\to S\underset{\dot{}}{\le }Z$$
21		simpr	$$⊢\left(\phi \wedge Y=Z\right)\to Y=Z$$
22	20 21	breqtrrd	$$⊢\left(\phi \wedge Y=Z\right)\to S\underset{\dot{}}{\le }Y$$