Theorem diaval

Description: The partial isomorphism A for a lattice K . Definition of isomorphism map in Crawley p. 120 line 24. (Contributed by NM, 15-Oct-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	diaval	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to I\left(X\right)=\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }X\right\}$$

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	diafval	$$⊢\left(K\in V\wedge W\in H\right)\to I=\left(x\in \left\{y\in B|y\underset{\dot{}}{\le }W\right\}⟼\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }x\right\}\right)$$
8	7	adantr	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to I=\left(x\in \left\{y\in B|y\underset{\dot{}}{\le }W\right\}⟼\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }x\right\}\right)$$
9	8	fveq1d	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to I\left(X\right)=\left(x\in \left\{y\in B|y\underset{\dot{}}{\le }W\right\}⟼\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }x\right\}\right)\left(X\right)$$
10		simpr	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)$$
11		breq1	$$⊢y=X\to \left(y\underset{\dot{}}{\le }W\leftrightarrow X\underset{\dot{}}{\le }W\right)$$
12	11	elrab	$$⊢X\in \left\{y\in B|y\underset{\dot{}}{\le }W\right\}\leftrightarrow \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)$$
13	10 12	sylibr	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to X\in \left\{y\in B|y\underset{\dot{}}{\le }W\right\}$$
14		breq2	$$⊢x=X\to \left(R\left(f\right)\underset{\dot{}}{\le }x\leftrightarrow R\left(f\right)\underset{\dot{}}{\le }X\right)$$
15	14	rabbidv	$$⊢x=X\to \left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }x\right\}=\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }X\right\}$$
16		eqid	$$⊢\left(x\in \left\{y\in B|y\underset{\dot{}}{\le }W\right\}⟼\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }x\right\}\right)=\left(x\in \left\{y\in B|y\underset{\dot{}}{\le }W\right\}⟼\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }x\right\}\right)$$
17	4	fvexi	$$⊢T\in \mathrm{V}$$
18	17	rabex	$$⊢\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }X\right\}\in \mathrm{V}$$
19	15 16 18	fvmpt	$$⊢X\in \left\{y\in B|y\underset{\dot{}}{\le }W\right\}\to \left(x\in \left\{y\in B|y\underset{\dot{}}{\le }W\right\}⟼\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }x\right\}\right)\left(X\right)=\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }X\right\}$$
20	13 19	syl	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to \left(x\in \left\{y\in B|y\underset{\dot{}}{\le }W\right\}⟼\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }x\right\}\right)\left(X\right)=\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }X\right\}$$
21	9 20	eqtrd	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to I\left(X\right)=\left\{f\in T|R\left(f\right)\underset{\dot{}}{\le }X\right\}$$