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 ∧ 𝑊𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ ( ( ( 𝑋 ) ( 𝑊 ) ) 𝑊 ) ) = ( 𝑁 ‘ ( 𝐼𝑋 ) ) )

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 ∧ 𝑊𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
9 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
10 9 4 6 diadmclN ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → 𝑋 ∈ ( Base ‘ 𝐾 ) )
11 eqid ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
12 11 4 6 diadmleN ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → 𝑋 ( le ‘ 𝐾 ) 𝑊 )
13 9 11 4 5 6 diass ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋 ∈ ( Base ‘ 𝐾 ) ∧ 𝑋 ( le ‘ 𝐾 ) 𝑊 ) ) → ( 𝐼𝑋 ) ⊆ 𝑇 )
14 8 10 12 13 syl12anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼𝑋 ) ⊆ 𝑇 )
15 1 2 3 4 5 6 7 docavalN ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐼𝑋 ) ⊆ 𝑇 ) → ( 𝑁 ‘ ( 𝐼𝑋 ) ) = ( 𝐼 ‘ ( ( ( ‘ ( 𝐼 { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼𝑋 ) ⊆ 𝑧 } ) ) ( 𝑊 ) ) 𝑊 ) ) )
16 14 15 syldan ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝑁 ‘ ( 𝐼𝑋 ) ) = ( 𝐼 ‘ ( ( ( ‘ ( 𝐼 { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼𝑋 ) ⊆ 𝑧 } ) ) ( 𝑊 ) ) 𝑊 ) ) )
17 4 6 diaclN ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼𝑋 ) ∈ ran 𝐼 )
18 intmin ( ( 𝐼𝑋 ) ∈ ran 𝐼 { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼𝑋 ) ⊆ 𝑧 } = ( 𝐼𝑋 ) )
19 17 18 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼𝑋 ) ⊆ 𝑧 } = ( 𝐼𝑋 ) )
20 19 fveq2d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼𝑋 ) ⊆ 𝑧 } ) = ( 𝐼 ‘ ( 𝐼𝑋 ) ) )
21 4 6 diaf11N ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝐼 : dom 𝐼1-1-onto→ ran 𝐼 )
22 f1ocnvfv1 ( ( 𝐼 : dom 𝐼1-1-onto→ ran 𝐼𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ ( 𝐼𝑋 ) ) = 𝑋 )
23 21 22 sylan ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ ( 𝐼𝑋 ) ) = 𝑋 )
24 20 23 eqtrd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼𝑋 ) ⊆ 𝑧 } ) = 𝑋 )
25 24 fveq2d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( ‘ ( 𝐼 { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼𝑋 ) ⊆ 𝑧 } ) ) = ( 𝑋 ) )
26 25 oveq1d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( ( ‘ ( 𝐼 { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼𝑋 ) ⊆ 𝑧 } ) ) ( 𝑊 ) ) = ( ( 𝑋 ) ( 𝑊 ) ) )
27 26 fvoveq1d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ ( ( ( ‘ ( 𝐼 { 𝑧 ∈ ran 𝐼 ∣ ( 𝐼𝑋 ) ⊆ 𝑧 } ) ) ( 𝑊 ) ) 𝑊 ) ) = ( 𝐼 ‘ ( ( ( 𝑋 ) ( 𝑊 ) ) 𝑊 ) ) )
28 16 27 eqtr2d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ ( ( ( 𝑋 ) ( 𝑊 ) ) 𝑊 ) ) = ( 𝑁 ‘ ( 𝐼𝑋 ) ) )