Theorem diatrl

Description: Trace of a member of the partial isomorphism A. (Contributed by NM, 17-Jan-2014)

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

Proof

Step	Hyp	Ref	Expression
1		diatrl.b	$$⊢B={\mathrm{Base}}_K$$
2		diatrl.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		diatrl.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
4		diatrl.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
5		diatrl.r	$$⊢R=\mathrm{trL}\left(K\right)\left(W\right)$$
6		diatrl.i	$$⊢I=\mathrm{DIsoA}\left(K\right)\left(W\right)$$
7	1 2 3 4 5 6	diaelval	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to \left(F\in I\left(X\right)\leftrightarrow \left(F\in T\wedge R\left(F\right)\underset{\dot{}}{\le }X\right)\right)$$
8		simpr	$$⊢\left(F\in T\wedge R\left(F\right)\underset{\dot{}}{\le }X\right)\to R\left(F\right)\underset{\dot{}}{\le }X$$
9	7 8	syl6bi	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\right)\to \left(F\in I\left(X\right)\to R\left(F\right)\underset{\dot{}}{\le }X\right)$$
10	9	3impia	$$⊢\left(\left(K\in V\wedge W\in H\right)\wedge \left(X\in B\wedge X\underset{\dot{}}{\le }W\right)\wedge F\in I\left(X\right)\right)\to R\left(F\right)\underset{\dot{}}{\le }X$$