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Metamath Proof Explorer

Theorem trlcocnvat

Description: Commonly used special case of trlcoat . (Contributed by NM, 1-Jul-2013)

Ref Expression
Hypotheses trlcoat.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
trlcoat.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
trlcoat.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
trlcoat.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion trlcocnvat ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → ( 𝑅 ‘ ( 𝐹 ∘ ◡ 𝐺 ) ) ∈ 𝐴 )

Proof

Step Hyp Ref Expression
1 trlcoat.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2 trlcoat.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 trlcoat.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4 trlcoat.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
5 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → 𝐹 ∈ 𝑇 )
7 simp2r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → 𝐺 ∈ 𝑇 )
8 2 3 ltrncnv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → ◡ 𝐺 ∈ 𝑇 )
9 5 7 8 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → ◡ 𝐺 ∈ 𝑇 )
10 simp3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) )
11 2 3 4 trlcnv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → ( 𝑅 ‘ ◡ 𝐺 ) = ( 𝑅 ‘ 𝐺 ) )
12 5 7 11 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → ( 𝑅 ‘ ◡ 𝐺 ) = ( 𝑅 ‘ 𝐺 ) )
13 10 12 neeqtrrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ ◡ 𝐺 ) )
14 1 2 3 4 trlcoat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ◡ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ ◡ 𝐺 ) ) → ( 𝑅 ‘ ( 𝐹 ∘ ◡ 𝐺 ) ) ∈ 𝐴 )
15 5 6 9 13 14 syl121anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) → ( 𝑅 ‘ ( 𝐹 ∘ ◡ 𝐺 ) ) ∈ 𝐴 )