Theorem cdlemg2cex

Description: Any translation is one of our F s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf ? (Contributed by NM, 22-Apr-2013)

	Ref	Expression
Hypotheses	cdlemg2.b	$$⊢B={\mathrm{Base}}_K$$
	cdlemg2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdlemg2.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdlemg2.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdlemg2.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdlemg2.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdlemg2.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
	cdlemg2ex.u	$$⊢U=\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }W$$
	cdlemg2ex.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(q\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdlemg2ex.e	$$⊢E=\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdlemg2ex.g	$$⊢G=\left(x\in B⟼if\left(\left(p\ne q\wedge \neg x\underset{\dot{}}{\le }W\right),\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=if\left(s\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right),\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right)\right)\to y=E\right)\right),⦋s/t⦌D\right)\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right),x\right)\right)$$
Assertion	cdlemg2cex	$$⊢\left(K\in \mathrm{HL}\wedge W\in H\right)\to \left(F\in T\leftrightarrow \exists p\in A\exists q\in A\left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\wedge F=G\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdlemg2.b	$$⊢B={\mathrm{Base}}_K$$
2		cdlemg2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdlemg2.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdlemg2.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdlemg2.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdlemg2.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdlemg2.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
8		cdlemg2ex.u	$$⊢U=\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }W$$
9		cdlemg2ex.d	$$⊢D=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(q\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10		cdlemg2ex.e	$$⊢E=\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }\left(D\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
11		cdlemg2ex.g	$$⊢G=\left(x\in B⟼if\left(\left(p\ne q\wedge \neg x\underset{\dot{}}{\le }W\right),\left(\iota z\in B|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=if\left(s\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right),\left(\iota y\in B|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right)\right)\to y=E\right)\right),⦋s/t⦌D\right)\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right),x\right)\right)$$
12	2 5 6 7	cdlemg1cex	$$⊢\left(K\in \mathrm{HL}\wedge W\in H\right)\to \left(F\in T\leftrightarrow \exists p\in A\exists q\in A\left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\wedge F=\left(\iota f\in T|f\left(p\right)=q\right)\right)\right)$$
13		simplll	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(p\in A\wedge q\in A\right)\right)\wedge \left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\right)\right)\to K\in \mathrm{HL}$$
14		simpllr	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(p\in A\wedge q\in A\right)\right)\wedge \left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\right)\right)\to W\in H$$
15		simplrl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(p\in A\wedge q\in A\right)\right)\wedge \left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\right)\right)\to p\in A$$
16		simprl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(p\in A\wedge q\in A\right)\right)\wedge \left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\right)\right)\to \neg p\underset{\dot{}}{\le }W$$
17		simplrr	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(p\in A\wedge q\in A\right)\right)\wedge \left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\right)\right)\to q\in A$$
18		simprr	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(p\in A\wedge q\in A\right)\right)\wedge \left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\right)\right)\to \neg q\underset{\dot{}}{\le }W$$
19		eqid	$$⊢\left(\iota f\in T|f\left(p\right)=q\right)=\left(\iota f\in T|f\left(p\right)=q\right)$$
20	1 2 3 4 5 6 8 9 10 11 7 19	cdlemg1b2	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(p\in A\wedge \neg p\underset{\dot{}}{\le }W\right)\wedge \left(q\in A\wedge \neg q\underset{\dot{}}{\le }W\right)\right)\to \left(\iota f\in T|f\left(p\right)=q\right)=G$$
21	13 14 15 16 17 18 20	syl222anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(p\in A\wedge q\in A\right)\right)\wedge \left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\right)\right)\to \left(\iota f\in T|f\left(p\right)=q\right)=G$$
22	21	eqeq2d	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(p\in A\wedge q\in A\right)\right)\wedge \left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\right)\right)\to \left(F=\left(\iota f\in T|f\left(p\right)=q\right)\leftrightarrow F=G\right)$$
23	22	pm5.32da	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(p\in A\wedge q\in A\right)\right)\to \left(\left(\left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\right)\wedge F=\left(\iota f\in T|f\left(p\right)=q\right)\right)\leftrightarrow \left(\left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\right)\wedge F=G\right)\right)$$
24		df-3an	$$⊢\left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\wedge F=\left(\iota f\in T|f\left(p\right)=q\right)\right)\leftrightarrow \left(\left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\right)\wedge F=\left(\iota f\in T|f\left(p\right)=q\right)\right)$$
25		df-3an	$$⊢\left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\wedge F=G\right)\leftrightarrow \left(\left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\right)\wedge F=G\right)$$
26	23 24 25	3bitr4g	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(p\in A\wedge q\in A\right)\right)\to \left(\left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\wedge F=\left(\iota f\in T|f\left(p\right)=q\right)\right)\leftrightarrow \left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\wedge F=G\right)\right)$$
27	26	2rexbidva	$$⊢\left(K\in \mathrm{HL}\wedge W\in H\right)\to \left(\exists p\in A\exists q\in A\left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\wedge F=\left(\iota f\in T|f\left(p\right)=q\right)\right)\leftrightarrow \exists p\in A\exists q\in A\left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\wedge F=G\right)\right)$$
28	12 27	bitrd	$$⊢\left(K\in \mathrm{HL}\wedge W\in H\right)\to \left(F\in T\leftrightarrow \exists p\in A\exists q\in A\left(\neg p\underset{\dot{}}{\le }W\wedge \neg q\underset{\dot{}}{\le }W\wedge F=G\right)\right)$$