Theorem cdleme51finvtrN

Description: Part of proof of Lemma E in Crawley p. 113. TODO: fix comment. (Contributed by NM, 14-Apr-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdlemef50.b	$$⊢B={\mathrm{Base}}_K$$
	cdlemef50.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdlemef50.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdlemef50.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdlemef50.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdlemef50.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdlemef50.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdlemef50.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdlemefs50.e	$$⊢E=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdlemef50.f	$$⊢F=\left(x\in B⟼if\left(\left(P\ne Q\wedge \neg x\underset{\dot{}}{\le }W\right),\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=if\left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right),\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to y=E\right)\right),⦋s/t⦌D\right)\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right),x\right)\right)$$
	cdleme50ltrn.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
Assertion	cdleme51finvtrN	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\to F^{-1}\in T$$

Proof

Step	Hyp	Ref	Expression
1		cdlemef50.b	$$⊢B={\mathrm{Base}}_K$$
2		cdlemef50.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdlemef50.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdlemef50.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdlemef50.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdlemef50.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdlemef50.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		cdlemef50.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		cdlemefs50.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		cdlemef50.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		cdleme50ltrn.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
12		eqid	$$⊢\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W$$
13		eqid	$$⊢\left(v\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(v\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
14		eqid	$$⊢\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(\left(\left(v\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(\left(\left(v\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)$$
15		eqid	$$⊢\left(a\in B⟼if\left(\left(Q\ne P\wedge \neg a\underset{\dot{}}{\le }W\right),\left(\iota c\in B|\forall u\in A\left(\left(\neg u\underset{\dot{}}{\le }W\wedge u\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)=a\right)\to c=if\left(u\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right),\left(\iota b\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\to b=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(\left(\left(v\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right),⦋u/v⦌\left(\left(v\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)\right)\right),a\right)\right)=\left(a\in B⟼if\left(\left(Q\ne P\wedge \neg a\underset{\dot{}}{\le }W\right),\left(\iota c\in B|\forall u\in A\left(\left(\neg u\underset{\dot{}}{\le }W\wedge u\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)=a\right)\to c=if\left(u\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right),\left(\iota b\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\to b=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(\left(\left(v\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right),⦋u/v⦌\left(\left(v\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)\right)\right),a\right)\right)$$
16	1 2 3 4 5 6 7 8 9 10 12 13 14 15	cdleme51finvN	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\to F^{-1}=\left(a\in B⟼if\left(\left(Q\ne P\wedge \neg a\underset{\dot{}}{\le }W\right),\left(\iota c\in B|\forall u\in A\left(\left(\neg u\underset{\dot{}}{\le }W\wedge u\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)=a\right)\to c=if\left(u\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right),\left(\iota b\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\to b=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(\left(\left(v\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right),⦋u/v⦌\left(\left(v\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)\right)\right),a\right)\right)$$
17	1 2 3 4 5 6 12 13 14 15 11	cdleme50ltrn	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\right)\to \left(a\in B⟼if\left(\left(Q\ne P\wedge \neg a\underset{\dot{}}{\le }W\right),\left(\iota c\in B|\forall u\in A\left(\left(\neg u\underset{\dot{}}{\le }W\wedge u\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)=a\right)\to c=if\left(u\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right),\left(\iota b\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\to b=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(\left(\left(v\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right),⦋u/v⦌\left(\left(v\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)\right)\right),a\right)\right)\in T$$
18	17	3com23	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\to \left(a\in B⟼if\left(\left(Q\ne P\wedge \neg a\underset{\dot{}}{\le }W\right),\left(\iota c\in B|\forall u\in A\left(\left(\neg u\underset{\dot{}}{\le }W\wedge u\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)=a\right)\to c=if\left(u\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right),\left(\iota b\in B|\forall v\in A\left(\left(\neg v\underset{\dot{}}{\le }W\wedge \neg v\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }P\right)\right)\to b=\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(\left(\left(v\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(u\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right),⦋u/v⦌\left(\left(v\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }\left(\left(Q\underset{\dot{}}{\vee }v\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)\underset{\dot{}}{\vee }\left(a\underset{\dot{}}{\wedge }W\right)\right)\right),a\right)\right)\in T$$
19	16 18	eqeltrd	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\to F^{-1}\in T$$