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 ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝐹𝑇 ) → 𝐹 = ( 𝑓𝑇 ( 𝑓𝑃 ) = ( 𝐹𝑃 ) ) )

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 ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝐹𝑇 ) → ( 𝐹𝑃 ) = ( 𝐹𝑃 ) )
6 simp3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝐹𝑇 ) → 𝐹𝑇 )
7 1 2 3 4 ltrnel ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( ( 𝐹𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹𝑃 ) 𝑊 ) )
8 7 3com23 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝐹𝑇 ) → ( ( 𝐹𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹𝑃 ) 𝑊 ) )
9 1 2 3 4 cdleme ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( ( 𝐹𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹𝑃 ) 𝑊 ) ) → ∃! 𝑓𝑇 ( 𝑓𝑃 ) = ( 𝐹𝑃 ) )
10 8 9 syld3an3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝐹𝑇 ) → ∃! 𝑓𝑇 ( 𝑓𝑃 ) = ( 𝐹𝑃 ) )
11 fveq1 ( 𝑓 = 𝐹 → ( 𝑓𝑃 ) = ( 𝐹𝑃 ) )
12 11 eqeq1d ( 𝑓 = 𝐹 → ( ( 𝑓𝑃 ) = ( 𝐹𝑃 ) ↔ ( 𝐹𝑃 ) = ( 𝐹𝑃 ) ) )
13 12 riota2 ( ( 𝐹𝑇 ∧ ∃! 𝑓𝑇 ( 𝑓𝑃 ) = ( 𝐹𝑃 ) ) → ( ( 𝐹𝑃 ) = ( 𝐹𝑃 ) ↔ ( 𝑓𝑇 ( 𝑓𝑃 ) = ( 𝐹𝑃 ) ) = 𝐹 ) )
14 6 10 13 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝐹𝑇 ) → ( ( 𝐹𝑃 ) = ( 𝐹𝑃 ) ↔ ( 𝑓𝑇 ( 𝑓𝑃 ) = ( 𝐹𝑃 ) ) = 𝐹 ) )
15 5 14 mpbid ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝐹𝑇 ) → ( 𝑓𝑇 ( 𝑓𝑃 ) = ( 𝐹𝑃 ) ) = 𝐹 )
16 15 eqcomd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝐹𝑇 ) → 𝐹 = ( 𝑓𝑇 ( 𝑓𝑃 ) = ( 𝐹𝑃 ) ) )