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Metamath Proof Explorer

Theorem diaocN

Description: Value of partial isomorphism A at lattice orthocomplement (using a Sasaki projection to get orthocomplement relative to the fiducial co-atom W ). (Contributed by NM, 6-Dec-2013) (New usage is discouraged.)

Ref Expression
Hypotheses diaoc.j ⊢ ∨ = ( join ‘ 𝐾 )
diaoc.m ⊢ ∧ = ( meet ‘ 𝐾 )
diaoc.o ⊢ ⊥ = ( oc ‘ 𝐾 )
diaoc.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
diaoc.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
diaoc.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
diaoc.n ⊢ 𝑁 = ( ( ocA ‘ 𝐾 ) ‘ 𝑊 )
Assertion diaocN ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ ( ( ( ⊥ ‘ 𝑋 ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) ) = ( 𝑁 ‘ ( 𝐼 ‘ 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 diaoc.j ⊢ ∨ = ( join ‘ 𝐾 )
2 diaoc.m ⊢ ∧ = ( meet ‘ 𝐾 )
3 diaoc.o ⊢ ⊥ = ( oc ‘ 𝐾 )
4 diaoc.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5 diaoc.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 diaoc.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
7 diaoc.n ⊢ 𝑁 = ( ( ocA ‘ 𝐾 ) ‘ 𝑊 )
8 simpl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
10 9 4 6 diadmclN ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → 𝑋 ∈ ( Base ‘ 𝐾 ) )
11 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
12 11 4 6 diadmleN ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → 𝑋 ( le ‘ 𝐾 ) 𝑊 )
13 9 11 4 5 6 diass ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ ( Base ‘ 𝐾 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼 ‘ 𝑋 ) ⊆ 𝑇 )
14 8 10 12 13 syl12anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) ⊆ 𝑇 )
15 1 2 3 4 5 6 7 docavalN ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑇 ) → ( 𝑁 ‘ ( 𝐼 ‘ 𝑋 ) ) = ( 𝐼 ‘ ( ( ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) ) )
16 14 15 syldan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝑁 ‘ ( 𝐼 ‘ 𝑋 ) ) = ( 𝐼 ‘ ( ( ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) ) )
17 4 6 diaclN ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) ∈ ran 𝐼 )
18 intmin ⊢ ( ( 𝐼 ‘ 𝑋 ) ∈ ran 𝐼 → ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } = ( 𝐼 ‘ 𝑋 ) )
19 17 18 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } = ( 𝐼 ‘ 𝑋 ) )
20 19 fveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } ) = ( ◡ 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) )
21 4 6 diaf11N ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 )
22 f1ocnvfv1 ⊢ ( ( 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 ∧ 𝑋 ∈ dom 𝐼 ) → ( ◡ 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) = 𝑋 )
23 21 22 sylan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( ◡ 𝐼 ‘ ( 𝐼 ‘ 𝑋 ) ) = 𝑋 )
24 20 23 eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } ) = 𝑋 )
25 24 fveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } ) ) = ( ⊥ ‘ 𝑋 ) )
26 25 oveq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑊 ) ) = ( ( ⊥ ‘ 𝑋 ) ∨ ( ⊥ ‘ 𝑊 ) ) )
27 26 fvoveq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ ( ( ( ⊥ ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼 ‘ 𝑋 ) ⊆ 𝑧 } ) ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) ) = ( 𝐼 ‘ ( ( ( ⊥ ‘ 𝑋 ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) ) )
28 16 27 eqtr2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ ( ( ( ⊥ ‘ 𝑋 ) ∨ ( ⊥ ‘ 𝑊 ) ) ∧ 𝑊 ) ) = ( 𝑁 ‘ ( 𝐼 ‘ 𝑋 ) ) )