# 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 ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑃 ) = ( 𝐺𝑃 ) ) → 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 ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑃 ) = ( 𝐺𝑃 ) ) ∧ 𝑞𝐴 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
6 simpl12 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑃 ) = ( 𝐺𝑃 ) ) ∧ 𝑞𝐴 ) → 𝐹𝑇 )
7 simpl13 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑃 ) = ( 𝐺𝑃 ) ) ∧ 𝑞𝐴 ) → 𝐺𝑇 )
8 6 7 jca ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑃 ) = ( 𝐺𝑃 ) ) ∧ 𝑞𝐴 ) → ( 𝐹𝑇𝐺𝑇 ) )
9 simpr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑃 ) = ( 𝐺𝑃 ) ) ∧ 𝑞𝐴 ) → 𝑞𝐴 )
10 simpl2 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑃 ) = ( 𝐺𝑃 ) ) ∧ 𝑞𝐴 ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
11 simpl3 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑃 ) = ( 𝐺𝑃 ) ) ∧ 𝑞𝐴 ) → ( 𝐹𝑃 ) = ( 𝐺𝑃 ) )
12 eqid ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
13 1 12 2 3 4 cdlemd9 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ 𝑞𝐴 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑃 ) = ( 𝐺𝑃 ) ) → ( 𝐹𝑞 ) = ( 𝐺𝑞 ) )
14 5 8 9 10 11 13 syl311anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑃 ) = ( 𝐺𝑃 ) ) ∧ 𝑞𝐴 ) → ( 𝐹𝑞 ) = ( 𝐺𝑞 ) )
15 14 ralrimiva ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑃 ) = ( 𝐺𝑃 ) ) → ∀ 𝑞𝐴 ( 𝐹𝑞 ) = ( 𝐺𝑞 ) )
16 2 3 4 ltrneq2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝐺𝑇 ) → ( ∀ 𝑞𝐴 ( 𝐹𝑞 ) = ( 𝐺𝑞 ) ↔ 𝐹 = 𝐺 ) )
17 16 3ad2ant1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑃 ) = ( 𝐺𝑃 ) ) → ( ∀ 𝑞𝐴 ( 𝐹𝑞 ) = ( 𝐺𝑞 ) ↔ 𝐹 = 𝐺 ) )
18 15 17 mpbid ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝐺𝑇 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑃 ) = ( 𝐺𝑃 ) ) → 𝐹 = 𝐺 )