Theorem ltrniotaidvalN

Description: Value of the unique translation specified by identity value. (Contributed by NM, 25-Aug-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ltrniotaidval.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	ltrniotaidval.l	⊢ ≤ = ( le ‘ 𝐾 )
	ltrniotaidval.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	ltrniotaidval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	ltrniotaidval.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	ltrniotaidval.f	⊢ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 )
Assertion	ltrniotaidvalN	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 = ( I ↾ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ltrniotaidval.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		ltrniotaidval.l	⊢ ≤ = ( le ‘ 𝐾 )
3		ltrniotaidval.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		ltrniotaidval.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5		ltrniotaidval.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6		ltrniotaidval.f	⊢ 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = 𝑃 )
7	2 3 4 5 6	ltrniotaval	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑃 ) = 𝑃 )
8	7	3anidm23	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 ‘ 𝑃 ) = 𝑃 )
9		simpl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10	2 3 4 5 6	ltrniotacl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
11	10	3anidm23	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
12		simpr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
13	1 2 3 4 5	ltrnideq	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 = ( I ↾ 𝐵 ) ↔ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) )
14	9 11 12 13	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝐹 = ( I ↾ 𝐵 ) ↔ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) )
15	8 14	mpbird	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐹 = ( I ↾ 𝐵 ) )