Theorem trlconid

Description: The composition of two different translations is not the identity translation. (Contributed by NM, 22-Jul-2013)

	Ref	Expression
Hypotheses	trlconid.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	trlconid.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	trlconid.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	trlconid.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion	trlconid	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → ( 𝐹 ∘ 𝐺 ) ≠ ( I ↾ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		trlconid.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		trlconid.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3		trlconid.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4		trlconid.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
5		eqid	⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
6	5 2 3 4	trlcoat	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → ( 𝑅 ‘ ( 𝐹 ∘ 𝐺 ) ) ∈ ( Atoms ‘ 𝐾 ) )
7		simp1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
8		simp2l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → 𝐹 ∈ 𝑇 )
9		simp2r	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → 𝐺 ∈ 𝑇 )
10	2 3	ltrnco	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐹 ∘ 𝐺 ) ∈ 𝑇 )
11	7 8 9 10	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → ( 𝐹 ∘ 𝐺 ) ∈ 𝑇 )
12	1 5 2 3 4	trlnidatb	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∘ 𝐺 ) ∈ 𝑇 ) → ( ( 𝐹 ∘ 𝐺 ) ≠ ( I ↾ 𝐵 ) ↔ ( 𝑅 ‘ ( 𝐹 ∘ 𝐺 ) ) ∈ ( Atoms ‘ 𝐾 ) ) )
13	7 11 12	syl2anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → ( ( 𝐹 ∘ 𝐺 ) ≠ ( I ↾ 𝐵 ) ↔ ( 𝑅 ‘ ( 𝐹 ∘ 𝐺 ) ) ∈ ( Atoms ‘ 𝐾 ) ) )
14	6 13	mpbird	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → ( 𝐹 ∘ 𝐺 ) ≠ ( I ↾ 𝐵 ) )