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 )