Theorem dia2dimlem4

Description: Lemma for dia2dim . Show that the composition (sum) of translations (vectors) G and D equals F . Part of proof of Lemma M in Crawley p. 121 line 5. (Contributed by NM, 8-Sep-2014)

	Ref	Expression
Hypotheses	dia2dimlem4.l	⊢ ≤ = ( le ‘ 𝐾 )
	dia2dimlem4.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	dia2dimlem4.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	dia2dimlem4.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	dia2dimlem4.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	dia2dimlem4.p	⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
	dia2dimlem4.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑇 )
	dia2dimlem4.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑇 )
	dia2dimlem4.gv	⊢ ( 𝜑 → ( 𝐺 ‘ 𝑃 ) = 𝑄 )
	dia2dimlem4.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑇 )
	dia2dimlem4.dv	⊢ ( 𝜑 → ( 𝐷 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) )
Assertion	dia2dimlem4	⊢ ( 𝜑 → ( 𝐷 ∘ 𝐺 ) = 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		dia2dimlem4.l	⊢ ≤ = ( le ‘ 𝐾 )
2		dia2dimlem4.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		dia2dimlem4.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		dia2dimlem4.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		dia2dimlem4.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6		dia2dimlem4.p	⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
7		dia2dimlem4.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑇 )
8		dia2dimlem4.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑇 )
9		dia2dimlem4.gv	⊢ ( 𝜑 → ( 𝐺 ‘ 𝑃 ) = 𝑄 )
10		dia2dimlem4.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑇 )
11		dia2dimlem4.dv	⊢ ( 𝜑 → ( 𝐷 ‘ 𝑄 ) = ( 𝐹 ‘ 𝑃 ) )
12	3 4	ltrnco	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐷 ∘ 𝐺 ) ∈ 𝑇 )
13	5 10 8 12	syl3anc	⊢ ( 𝜑 → ( 𝐷 ∘ 𝐺 ) ∈ 𝑇 )
14	6	simpld	⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
15	1 2 3 4	ltrncoval	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → ( ( 𝐷 ∘ 𝐺 ) ‘ 𝑃 ) = ( 𝐷 ‘ ( 𝐺 ‘ 𝑃 ) ) )
16	5 10 8 14 15	syl121anc	⊢ ( 𝜑 → ( ( 𝐷 ∘ 𝐺 ) ‘ 𝑃 ) = ( 𝐷 ‘ ( 𝐺 ‘ 𝑃 ) ) )
17	9	fveq2d	⊢ ( 𝜑 → ( 𝐷 ‘ ( 𝐺 ‘ 𝑃 ) ) = ( 𝐷 ‘ 𝑄 ) )
18	16 17 11	3eqtrd	⊢ ( 𝜑 → ( ( 𝐷 ∘ 𝐺 ) ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) )
19	1 2 3 4	cdlemd	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐷 ∘ 𝐺 ) ∈ 𝑇 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( ( 𝐷 ∘ 𝐺 ) ‘ 𝑃 ) = ( 𝐹 ‘ 𝑃 ) ) → ( 𝐷 ∘ 𝐺 ) = 𝐹 )
20	5 13 7 6 18 19	syl311anc	⊢ ( 𝜑 → ( 𝐷 ∘ 𝐺 ) = 𝐹 )