Theorem diaelval

Description: Member of the partial isomorphism A for a lattice K . (Contributed by NM, 3-Dec-2013)

	Ref	Expression
Hypotheses	diaval.b	$$⊢B={\mathrm{Base}}_K$$
	diaval.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	diaval.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	diaval.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
	diaval.r	$$⊢R=\mathrm{trL}\left(K\right)\left(W\right)$$
	diaval.i	$$⊢I=\mathrm{DIsoA}\left(K\right)\left(W\right)$$
Assertion	diaelval	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to \left(F\in I\left(X\right)\leftrightarrow \left(F\in T\wedge R\left(F\right)\underset{\dot{}}{\le }X\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		diaval.b	$$⊢B={\mathrm{Base}}_K$$
2		diaval.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		diaval.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
4		diaval.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
5		diaval.r	$$⊢R=\mathrm{trL}\left(K\right)\left(W\right)$$
6		diaval.i	$$⊢I=\mathrm{DIsoA}\left(K\right)\left(W\right)$$
7	1 2 3 4 5 6	diaval	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to I\left(X\right)=\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }X\right\}$$
8	7	eleq2d	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to \left(F\in I\left(X\right)\leftrightarrow F\in \left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }X\right\}\right)$$
9		fveq2	$$⊢f=F\to R\left(f\right)=R\left(F\right)$$
10	9	breq1d	$$⊢f=F\to \left(R\left(f\right)\underset{\dot{}}{\le }X\leftrightarrow R\left(F\right)\underset{\dot{}}{\le }X\right)$$
11	10	elrab	$$⊢F\in \left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }X\right\}\leftrightarrow \left(F\in T\wedge R\left(F\right)\underset{\dot{}}{\le }X\right)$$
12	8 11	bitrdi	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to \left(F\in I\left(X\right)\leftrightarrow \left(F\in T\wedge R\left(F\right)\underset{\dot{}}{\le }X\right)\right)$$