Theorem cdleme27b

Description: Lemma for cdleme27N . (Contributed by NM, 3-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	$$⊢C=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),D,F\right)$$
	cdleme27.g	$$⊢G=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme27.o	$$⊢O=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme27.e	$$⊢E=\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=O\right)\right)$$
	cdleme27.y	$$⊢Y=if\left(t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),E,G\right)$$
Assertion	cdleme27b	$$⊢s=t\to C=Y$$

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		cdleme27.g	$$⊢G=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
14		cdleme27.o	$$⊢O=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
15		cdleme27.e	$$⊢E=\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=O\right)\right)$$
16		cdleme27.y	$$⊢Y=if\left(t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),E,G\right)$$
17		breq1	$$⊢s=t\to \left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\leftrightarrow t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
18		oveq1	$$⊢s=t\to s\underset{\dot{}}{\vee }z=t\underset{\dot{}}{\vee }z$$
19	18	oveq1d	$$⊢s=t\to \left(s\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W=\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W$$
20	19	oveq2d	$$⊢s=t\to Z\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)=Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)$$
21	20	oveq2d	$$⊢s=t\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(Z\underset{\dot{}}{\vee }\left(\left(t\underset{\dot{}}{\vee }z\right)\underset{\dot{}}{\wedge }W\right)\right)$$
22	21 10 14	3eqtr4g	$$⊢s=t\to N=O$$
23	22	eqeq2d	$$⊢s=t\to \left(u=N\leftrightarrow u=O\right)$$
24	23	imbi2d	$$⊢s=t\to \left(\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\leftrightarrow \left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=O\right)\right)$$
25	24	ralbidv	$$⊢s=t\to \left(\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\leftrightarrow \forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=O\right)\right)$$
26	25	riotabidv	$$⊢s=t\to \left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\right)=\left(\iota u\in B|\forall z\in A\left(\left(\neg z\underset{\dot{}}{\le }W\wedge \neg z\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=O\right)\right)$$
27	26 11 15	3eqtr4g	$$⊢s=t\to D=E$$
28		oveq1	$$⊢s=t\to s\underset{\dot{}}{\vee }U=t\underset{\dot{}}{\vee }U$$
29		oveq2	$$⊢s=t\to P\underset{\dot{}}{\vee }s=P\underset{\dot{}}{\vee }t$$
30	29	oveq1d	$$⊢s=t\to \left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W=\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W$$
31	30	oveq2d	$$⊢s=t\to Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)=Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)$$
32	28 31	oveq12d	$$⊢s=t\to \left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
33	32 8 13	3eqtr4g	$$⊢s=t\to F=G$$
34	17 27 33	ifbieq12d	$$⊢s=t\to if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),D,F\right)=if\left(t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),E,G\right)$$
35	34 12 16	3eqtr4g	$$⊢s=t\to C=Y$$