Theorem cdleme17d4

Description: TODO: FIX COMMENT. (Contributed by NM, 11-Apr-2013)

	Ref	Expression
Hypotheses	cdlemef46.b	$$⊢B={\mathrm{Base}}_K$$
	cdlemef46.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdlemef46.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdlemef46.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdlemef46.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdlemef46.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdlemef46.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdlemef46.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdlemefs46.e	$$⊢E=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdlemef46.f	$$⊢F=\left(x\in B⟼if\left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right),\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=E\right)\right),⦋s/t⦌D\right)\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right),x\right)\right)$$
Assertion	cdleme17d4	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge P=Q\right)\to F\left(P\right)=Q$$

Proof

Step	Hyp	Ref	Expression
1		cdlemef46.b	$$⊢B={\mathrm{Base}}_K$$
2		cdlemef46.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdlemef46.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdlemef46.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdlemef46.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdlemef46.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdlemef46.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		cdlemef46.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdlemefs46.e	$$⊢E=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10		cdlemef46.f	$$⊢F=\left(x\in B⟼if\left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right),\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=E\right)\right),⦋s/t⦌D\right)\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right),x\right)\right)$$
11		simp2l	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\to P\in A$$
12	1 5	atbase	$$⊢P\in A\to P\in B$$
13	11 12	syl	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\to P\in B$$
14	10	cdleme31id	$$⊢\left(P\in B\wedge P=Q\right)\to F\left(P\right)=P$$
15	13 14	sylan	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge P=Q\right)\to F\left(P\right)=P$$
16		simpr	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge P=Q\right)\to P=Q$$
17	15 16	eqtrd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge P=Q\right)\to F\left(P\right)=Q$$