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Metamath Proof Explorer

Theorem ltrnco

Description: The composition of two translations is a translation. Part of proof of Lemma G of Crawley p. 116, line 15 on p. 117. (Contributed by NM, 31-May-2013)

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

Proof

Step Hyp Ref Expression
1 ltrnco.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 ltrnco.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
4 eqid ⊢ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
5 1 4 2 ltrnldil ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) )
6 5 3adant3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) )
7 1 4 2 ltrnldil ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → 𝐺 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) )
8 7 3adant2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → 𝐺 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) )
9 1 4 ldilco ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ∧ 𝐺 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ) → ( 𝐹 ∘ 𝐺 ) ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) )
10 3 6 8 9 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐹 ∘ 𝐺 ) ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) )
11 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12 simp2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑝 ∈ ( Atoms ‘ 𝐾 ) )
13 simp3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 )
14 12 13 jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) )
15 simp2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑞 ∈ ( Atoms ‘ 𝐾 ) )
16 simp3r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 )
17 15 16 jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑞 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) )
18 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝐹 ∈ 𝑇 )
19 simp13 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝐺 ∈ 𝑇 )
20 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
21 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
22 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
23 eqid ⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
24 20 21 22 23 1 2 cdlemg41 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ∧ ( 𝑞 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
25 11 14 17 18 19 24 syl122anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
26 25 3exp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) → ( ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) )
27 26 ralrimivv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ∀ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∀ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) )
28 20 21 22 23 1 4 2 isltrn ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( 𝐹 ∘ 𝐺 ) ∈ 𝑇 ↔ ( ( 𝐹 ∘ 𝐺 ) ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∀ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) )
29 28 3ad2ant1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ( 𝐹 ∘ 𝐺 ) ∈ 𝑇 ↔ ( ( 𝐹 ∘ 𝐺 ) ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∀ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ( 𝐹 ∘ 𝐺 ) ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) )
30 10 27 29 mpbir2and ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝐹 ∘ 𝐺 ) ∈ 𝑇 )