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Metamath Proof Explorer

Theorem cdlemeiota

Description: A translation is uniquely determined by one of its values. (Contributed by NM, 18-Apr-2013)

Ref Expression
Hypotheses cdlemg1c.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg1c.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg1c.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg1c.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemeiota ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg1c.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemg1c.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 cdlemg1c.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 cdlemg1c.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 eqidd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) )
6 simp3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ 𝑇 )
7 1 2 3 4 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) )
8 7 3com23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) )
9 1 2 3 4 cdleme ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) ) → ∃! 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) )
10 8 9 syld3an3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ∃! 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) )
11 fveq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) )
12 11 eqeq1d ⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ↔ ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) )
13 12 riota2 ⊢ ( ( 𝐹 ∈ 𝑇 ∧ ∃! 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) → ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ↔ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) = 𝐹 ) )
14 6 10 13 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝐹 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ↔ ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) = 𝐹 ) )
15 5 14 mpbid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) = 𝐹 )
16 15 eqcomd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 = ( ℩ 𝑓 ∈ 𝑇 ( 𝑓 ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) )