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Metamath Proof Explorer

Theorem trlco

Description: The trace of a composition of translations is less than or equal to the join of their traces. Part of proof of Lemma G of Crawley p. 116, second paragraph on p. 117. (Contributed by NM, 2-Jun-2013)

Ref Expression
Hypotheses trlco.l ⊢ ≤ = ( le ‘ 𝐾 )
trlco.j ⊢ ∨ = ( join ‘ 𝐾 )
trlco.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
trlco.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
trlco.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion trlco ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝐹 ∘ 𝐺 ) ) ≤ ( ( 𝑅 ‘ 𝐹 ) ∨ ( 𝑅 ‘ 𝐺 ) ) )

Proof

Step Hyp Ref Expression
1 trlco.l ⊢ ≤ = ( le ‘ 𝐾 )
2 trlco.j ⊢ ∨ = ( join ‘ 𝐾 )
3 trlco.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 trlco.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 trlco.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6 eqid ⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
7 1 6 3 lhpexnle ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ¬ 𝑝 ≤ 𝑊 )
8 7 3ad2ant1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ∃ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ¬ 𝑝 ≤ 𝑊 )
9 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ≤ 𝑊 ) ) → 𝐹 ∈ 𝑇 )
11 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ≤ 𝑊 ) ) → 𝐺 ∈ 𝑇 )
12 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ≤ 𝑊 ) ) → ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ≤ 𝑊 ) )
13 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
14 1 2 3 4 5 13 6 trlcolem ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ≤ 𝑊 ) ) → ( 𝑅 ‘ ( 𝐹 ∘ 𝐺 ) ) ≤ ( ( 𝑅 ‘ 𝐹 ) ∨ ( 𝑅 ‘ 𝐺 ) ) )
15 9 10 11 12 14 syl121anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ≤ 𝑊 ) ) → ( 𝑅 ‘ ( 𝐹 ∘ 𝐺 ) ) ≤ ( ( 𝑅 ‘ 𝐹 ) ∨ ( 𝑅 ‘ 𝐺 ) ) )
16 8 15 rexlimddv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝐹 ∘ 𝐺 ) ) ≤ ( ( 𝑅 ‘ 𝐹 ) ∨ ( 𝑅 ‘ 𝐺 ) ) )