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Metamath Proof Explorer

Theorem cdlemd

Description: If two translations agree at any atom not under the fiducial co-atom W , then they are equal. Lemma D in Crawley p. 113. (Contributed by NM, 2-Jun-2012)

Ref Expression
Hypotheses cdlemd.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemd.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemd.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemd.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemd ( ( ( ( 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 simpl11 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑞 ∈ 𝐴 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6 simpl12 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑞 ∈ 𝐴 ) → 𝐹 ∈ 𝑇 )
7 simpl13 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑞 ∈ 𝐴 ) → 𝐺 ∈ 𝑇 )
8 6 7 jca ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑞 ∈ 𝐴 ) → ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) )
9 simpr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑞 ∈ 𝐴 ) → 𝑞 ∈ 𝐴 )
10 simpl2 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑞 ∈ 𝐴 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
11 simpl3 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑞 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) )
12 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
13 1 12 2 3 4 cdlemd9 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑞 ∈ 𝐴 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( 𝐹 ‘ 𝑞 ) = ( 𝐺 ‘ 𝑞 ) )
14 5 8 9 10 11 13 syl311anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑞 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑞 ) = ( 𝐺 ‘ 𝑞 ) )
15 14 ralrimiva ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ∀ 𝑞 ∈ 𝐴 ( 𝐹 ‘ 𝑞 ) = ( 𝐺 ‘ 𝑞 ) )
16 2 3 4 ltrneq2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ∀ 𝑞 ∈ 𝐴 ( 𝐹 ‘ 𝑞 ) = ( 𝐺 ‘ 𝑞 ) ↔ 𝐹 = 𝐺 ) )
17 16 3ad2ant1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → ( ∀ 𝑞 ∈ 𝐴 ( 𝐹 ‘ 𝑞 ) = ( 𝐺 ‘ 𝑞 ) ↔ 𝐹 = 𝐺 ) )
18 15 17 mpbid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) → 𝐹 = 𝐺 )