Theorem cdleme29c

Description: Transform cdleme28b . (Compare cdleme25c .) TODO: FIX COMMENT. (Contributed by NM, 8-Feb-2013)

	Ref	Expression
Hypotheses	cdleme26.b	$$⊢B={\mathrm{Base}}_K$$
	cdleme26.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme26.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme26.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme26.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme26.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme27.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme27.f	$$⊢F=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme27.z	$$⊢Z=\left(z\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme27.n	$$⊢N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme27.d	$$⊢D=\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\right)$$
	cdleme27.c	$$⊢C=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),D,F\right)$$
Assertion	cdleme29c	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge P\ne Q\wedge \left(X\in B\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to \exists !v\in B\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to v=C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdleme26.b	$$⊢B={\mathrm{Base}}_K$$
2		cdleme26.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdleme26.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdleme26.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdleme26.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdleme26.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdleme27.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		cdleme27.f	$$⊢F=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme27.z	$$⊢Z=\left(z\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10		cdleme27.n	$$⊢N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
11		cdleme27.d	$$⊢D=\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\right)$$
12		cdleme27.c	$$⊢C=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),D,F\right)$$
13	1 2 3 4 5 6 7 8 9 10 11 12	cdleme29b	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge P\ne Q\wedge \left(X\in B\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to \exists v\in B\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to v=C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)$$
14		simp11	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge P\ne Q\wedge \left(X\in B\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
15		simp3	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge P\ne Q\wedge \left(X\in B\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to \left(X\in B\wedge \neg X\underset{\dot{}}{\le }W\right)$$
16	1 2 3 4 5 6	lhpmcvr2	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(X\in B\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to \exists s\in A\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)$$
17	14 15 16	syl2anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge P\ne Q\wedge \left(X\in B\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to \exists s\in A\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)$$
18		reusv1	$$⊢\exists s\in A\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to \left(\exists !v\in B\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to v=C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\leftrightarrow \exists v\in B\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to v=C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)$$
19	17 18	syl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge P\ne Q\wedge \left(X\in B\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to \left(\exists !v\in B\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to v=C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\leftrightarrow \exists v\in B\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to v=C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)\right)$$
20	13 19	mpbird	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge P\ne Q\wedge \left(X\in B\wedge \neg X\underset{\dot{}}{\le }W\right)\right)\to \exists !v\in B\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)=X\right)\to v=C\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }W\right)\right)$$