Theorem dihf11

Description: The isomorphism H for a lattice K is a one-to-one function. Part of proof after Lemma N of Crawley p. 122 line 6. (Contributed by NM, 7-Mar-2014)

	Ref	Expression
Hypotheses	dihf11.b	$$⊢B={\mathrm{Base}}_K$$
	dihf11.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	dihf11.i	$$⊢I=\mathrm{DIsoH}\left(K\right)\left(W\right)$$
	dihf11.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	dihf11.s	$$⊢S=\mathrm{LSubSp}\left(U\right)$$
Assertion	dihf11	$$⊢\left(K\in \mathrm{HL}\wedge W\in H\right)\to I:B\underset{1-1}{⟶}S$$

Proof

Step	Hyp	Ref	Expression
1		dihf11.b	$$⊢B={\mathrm{Base}}_K$$
2		dihf11.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
3		dihf11.i	$$⊢I=\mathrm{DIsoH}\left(K\right)\left(W\right)$$
4		dihf11.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
5		dihf11.s	$$⊢S=\mathrm{LSubSp}\left(U\right)$$
6	1 2 3 4 5	dihf11lem	$$⊢\left(K\in \mathrm{HL}\wedge W\in H\right)\to I:B⟶S$$
7	1 2 3	dih11	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge x\in B\wedge y\in B\right)\to \left(I\left(x\right)=I\left(y\right)\leftrightarrow x=y\right)$$
8	7	biimpd	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge x\in B\wedge y\in B\right)\to \left(I\left(x\right)=I\left(y\right)\to x=y\right)$$
9	8	3expb	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(x\in B\wedge y\in B\right)\right)\to \left(I\left(x\right)=I\left(y\right)\to x=y\right)$$
10	9	ralrimivva	$$⊢\left(K\in \mathrm{HL}\wedge W\in H\right)\to \forall x\in B\forall y\in B\left(I\left(x\right)=I\left(y\right)\to x=y\right)$$
11		dff13	$$⊢I:B\underset{1-1}{⟶}S\leftrightarrow \left(I:B⟶S\wedge \forall x\in B\forall y\in B\left(I\left(x\right)=I\left(y\right)\to x=y\right)\right)$$
12	6 10 11	sylanbrc	$$⊢\left(K\in \mathrm{HL}\wedge W\in H\right)\to I:B\underset{1-1}{⟶}S$$