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Metamath Proof Explorer

Theorem idltrn

Description: The identity function is a lattice translation. Remark below Lemma B in Crawley p. 112. (Contributed by NM, 18-May-2012)

Ref Expression
Hypotheses idltrn.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
idltrn.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
idltrn.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion idltrn ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( I ↾ 𝐵 ) ∈ 𝑇 )

Proof

Step Hyp Ref Expression
1 idltrn.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 idltrn.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 idltrn.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4 eqid ⊢ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) = ( ( LDil ‘ 𝐾 ) ‘ 𝑊 )
5 1 2 4 idldil ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( I ↾ 𝐵 ) ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) )
6 simpll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7 simplrr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑞 ∈ ( Atoms ‘ 𝐾 ) )
8 simprr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 )
9 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
10 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
11 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
12 eqid ⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
13 9 10 11 12 2 lhpmat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑞 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑞 ( meet ‘ 𝐾 ) 𝑊 ) = ( 0. ‘ 𝐾 ) )
14 6 7 8 13 syl12anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑞 ( meet ‘ 𝐾 ) 𝑊 ) = ( 0. ‘ 𝐾 ) )
15 1 12 atbase ⊢ ( 𝑞 ∈ ( Atoms ‘ 𝐾 ) → 𝑞 ∈ 𝐵 )
16 fvresi ⊢ ( 𝑞 ∈ 𝐵 → ( ( I ↾ 𝐵 ) ‘ 𝑞 ) = 𝑞 )
17 7 15 16 3syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( I ↾ 𝐵 ) ‘ 𝑞 ) = 𝑞 )
18 17 oveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑞 ( join ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑞 ) ) = ( 𝑞 ( join ‘ 𝐾 ) 𝑞 ) )
19 simplll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝐾 ∈ HL )
20 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
21 20 12 hlatjidm ⊢ ( ( 𝐾 ∈ HL ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) → ( 𝑞 ( join ‘ 𝐾 ) 𝑞 ) = 𝑞 )
22 19 7 21 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑞 ( join ‘ 𝐾 ) 𝑞 ) = 𝑞 )
23 18 22 eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑞 ( join ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑞 ) ) = 𝑞 )
24 23 oveq1d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( 𝑞 ( join ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( 𝑞 ( meet ‘ 𝐾 ) 𝑊 ) )
25 simplrl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → 𝑝 ∈ ( Atoms ‘ 𝐾 ) )
26 1 12 atbase ⊢ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) → 𝑝 ∈ 𝐵 )
27 fvresi ⊢ ( 𝑝 ∈ 𝐵 → ( ( I ↾ 𝐵 ) ‘ 𝑝 ) = 𝑝 )
28 25 26 27 3syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( I ↾ 𝐵 ) ‘ 𝑝 ) = 𝑝 )
29 28 oveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑝 ( join ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑝 ) ) = ( 𝑝 ( join ‘ 𝐾 ) 𝑝 ) )
30 20 12 hlatjidm ⊢ ( ( 𝐾 ∈ HL ∧ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ) → ( 𝑝 ( join ‘ 𝐾 ) 𝑝 ) = 𝑝 )
31 19 25 30 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑝 ( join ‘ 𝐾 ) 𝑝 ) = 𝑝 )
32 29 31 eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑝 ( join ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑝 ) ) = 𝑝 )
33 32 oveq1d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( 𝑝 ( meet ‘ 𝐾 ) 𝑊 ) )
34 simprl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 )
35 9 10 11 12 2 lhpmat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑝 ( meet ‘ 𝐾 ) 𝑊 ) = ( 0. ‘ 𝐾 ) )
36 6 25 34 35 syl12anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝑝 ( meet ‘ 𝐾 ) 𝑊 ) = ( 0. ‘ 𝐾 ) )
37 33 36 eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( 0. ‘ 𝐾 ) )
38 14 24 37 3eqtr4rd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) ∧ ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
39 38 ex ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∧ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ) ) → ( ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) )
40 39 ralrimivva ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∀ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∀ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) )
41 9 20 10 12 2 4 3 isltrn ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( I ↾ 𝐵 ) ∈ 𝑇 ↔ ( ( I ↾ 𝐵 ) ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ∧ ∀ 𝑝 ∈ ( Atoms ‘ 𝐾 ) ∀ 𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑝 ( le ‘ 𝐾 ) 𝑊 ∧ ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ) → ( ( 𝑝 ( join ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑝 ) ) ( meet ‘ 𝐾 ) 𝑊 ) = ( ( 𝑞 ( join ‘ 𝐾 ) ( ( I ↾ 𝐵 ) ‘ 𝑞 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) )
42 5 40 41 mpbir2and ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( I ↾ 𝐵 ) ∈ 𝑇 )