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Metamath Proof Explorer

Theorem ltrneq3

Description: Two translations agree at any atom not under the fiducial co-atom W iff they are equal. (Contributed by NM, 25-Jul-2013)

Ref Expression
Hypotheses cdlemd.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemd.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemd.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemd.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion ltrneq3 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ↔ 𝐹 = 𝐺 ) )

Proof

Step Hyp Ref Expression
1 cdlemd.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemd.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 cdlemd.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 cdlemd.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6 simpl2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → 𝐹 ∈ 𝑇 )
7 simpl2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → 𝐺 ∈ 𝑇 )
8 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
9 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) )
10 1 2 3 4 cdlemd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → 𝐹 = 𝐺 )
11 5 6 7 8 9 10 syl311anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → 𝐹 = 𝐺 )
12 fveq1 ⊢ ( 𝐹 = 𝐺 → ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) )
13 12 adantl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) ∧ 𝐹 = 𝐺 ) → ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) )
14 11 13 impbida ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ↔ 𝐹 = 𝐺 ) )