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Metamath Proof Explorer

Theorem ltrncom

Description: Composition is commutative for translations. Part of proof of Lemma G of Crawley p. 116. (Contributed by NM, 5-Jun-2013)

Ref Expression
Hypotheses ltrncom.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ltrncom.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion ltrncom ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐹 ∘ 𝐺 ) = ( 𝐺 ∘ 𝐹 ) )

Proof

Step Hyp Ref Expression
1 ltrncom.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 ltrncom.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ ( Base ‘ 𝐾 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
4 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ ( Base ‘ 𝐾 ) ) ) → 𝐹 ∈ 𝑇 )
5 simpl3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ ( Base ‘ 𝐾 ) ) ) → 𝐺 ∈ 𝑇 )
6 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ ( Base ‘ 𝐾 ) ) ) → 𝐹 = ( I ↾ ( Base ‘ 𝐾 ) ) )
7 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
8 7 1 2 cdlemg47a ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ ( Base ‘ 𝐾 ) ) ) → ( 𝐹 ∘ 𝐺 ) = ( 𝐺 ∘ 𝐹 ) )
9 3 4 5 6 8 syl121anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 = ( I ↾ ( Base ‘ 𝐾 ) ) ) → ( 𝐹 ∘ 𝐺 ) = ( 𝐺 ∘ 𝐹 ) )
10 simpll1 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) = ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11 simpll2 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) = ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) → 𝐹 ∈ 𝑇 )
12 simpll3 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) = ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) → 𝐺 ∈ 𝑇 )
13 simplr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) = ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) → 𝐹 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) )
14 simpr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) = ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) = ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) )
15 eqid ⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
16 7 1 2 15 cdlemg48 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝐹 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) = ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) → ( 𝐹 ∘ 𝐺 ) = ( 𝐺 ∘ 𝐹 ) )
17 10 11 12 13 14 16 syl122anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) = ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) → ( 𝐹 ∘ 𝐺 ) = ( 𝐺 ∘ 𝐹 ) )
18 simpll1 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) ≠ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
19 simpll2 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) ≠ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) → 𝐹 ∈ 𝑇 )
20 simpll3 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) ≠ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) → 𝐺 ∈ 𝑇 )
21 simpr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) ≠ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) → ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) ≠ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) )
22 1 2 15 cdlemg44 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) ≠ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) → ( 𝐹 ∘ 𝐺 ) = ( 𝐺 ∘ 𝐹 ) )
23 18 19 20 21 22 syl121anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ) ∧ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐹 ) ≠ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) → ( 𝐹 ∘ 𝐺 ) = ( 𝐺 ∘ 𝐹 ) )
24 17 23 pm2.61dane ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝐹 ≠ ( I ↾ ( Base ‘ 𝐾 ) ) ) → ( 𝐹 ∘ 𝐺 ) = ( 𝐺 ∘ 𝐹 ) )
25 9 24 pm2.61dane ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐹 ∘ 𝐺 ) = ( 𝐺 ∘ 𝐹 ) )