Theorem ltrn11at

Description: Frequently used one-to-one property of lattice translation atoms. (Contributed by NM, 5-May-2013)

	Ref	Expression
Hypotheses	ltrneq2.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	ltrneq2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	ltrneq2.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion	ltrn11at	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑄 ) )

Proof

Step	Hyp	Ref	Expression
1		ltrneq2.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2		ltrneq2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3		ltrneq2.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4		simp33	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑃 ≠ 𝑄 )
5		simp1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6		simp2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝐹 ∈ 𝑇 )
7		simp31	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑃 ∈ 𝐴 )
8		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
9	8 1	atbase	⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
10	7 9	syl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
11		simp32	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑄 ∈ 𝐴 )
12	8 1	atbase	⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
13	11 12	syl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
14	8 2 3	ltrn11	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑄 ) ↔ 𝑃 = 𝑄 ) )
15	5 6 10 13 14	syl112anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑄 ) ↔ 𝑃 = 𝑄 ) )
16	15	necon3bid	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ( ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑄 ) ↔ 𝑃 ≠ 𝑄 ) )
17	4 16	mpbird	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑄 ) )