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 = ( Base ‘ 𝐾 )
	dihf11.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dihf11.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
	dihf11.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	dihf11.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
Assertion	dihf11	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 : 𝐵 –1-1→ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		dihf11.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		dihf11.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3		dihf11.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
4		dihf11.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5		dihf11.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
6	1 2 3 4 5	dihf11lem	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 : 𝐵 ⟶ 𝑆 )
7	1 2 3	dih11	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑥 ) = ( 𝐼 ‘ 𝑦 ) ↔ 𝑥 = 𝑦 ) )
8	7	biimpd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝐼 ‘ 𝑥 ) = ( 𝐼 ‘ 𝑦 ) → 𝑥 = 𝑦 ) )
9	8	3expb	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝐼 ‘ 𝑥 ) = ( 𝐼 ‘ 𝑦 ) → 𝑥 = 𝑦 ) )
10	9	ralrimivva	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝐼 ‘ 𝑥 ) = ( 𝐼 ‘ 𝑦 ) → 𝑥 = 𝑦 ) )
11		dff13	⊢ ( 𝐼 : 𝐵 –1-1→ 𝑆 ↔ ( 𝐼 : 𝐵 ⟶ 𝑆 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( ( 𝐼 ‘ 𝑥 ) = ( 𝐼 ‘ 𝑦 ) → 𝑥 = 𝑦 ) ) )
12	6 10 11	sylanbrc	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 : 𝐵 –1-1→ 𝑆 )