Metamath Proof Explorer

Theorem docaclN

Description: Closure of subspace orthocomplement for DVecA partial vector space. (Contributed by NM, 6-Dec-2013) (New usage is discouraged.)

Ref Expression
Hypotheses docacl.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
docacl.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
docacl.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
docacl.n ⊢ ⊥ = ( ( ocA ‘ 𝐾 ) ‘ 𝑊 )
Assertion docaclN ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ⊥ ‘ 𝑋 ) ∈ ran 𝐼 )

Proof

Step Hyp Ref Expression
1 docacl.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 docacl.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3 docacl.i ⊢ 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
4 docacl.n ⊢ ⊥ = ( ( ocA ‘ 𝐾 ) ‘ 𝑊 )
5 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
6 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
7 eqid ⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 )
8 5 6 7 1 2 3 4 docavalN ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ⊥ ‘ 𝑋 ) = ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) )
9 1 3 diaf11N ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 )
10 f1ofun ⊢ ( 𝐼 : dom 𝐼 –1-1-onto→ ran 𝐼 → Fun 𝐼 )
11 9 10 syl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → Fun 𝐼 )
12 11 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → Fun 𝐼 )
13 hllat ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
14 13 ad2antrr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → 𝐾 ∈ Lat )
15 hlop ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP )
16 15 ad2antrr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → 𝐾 ∈ OP )
17 simpl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
18 ssrab2 ⊢ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ⊆ ran 𝐼
19 18 a1i ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ⊆ ran 𝐼 )
20 1 2 3 dia1elN ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑇 ∈ ran 𝐼 )
21 20 anim1i ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( 𝑇 ∈ ran 𝐼 ∧ 𝑋 ⊆ 𝑇 ) )
22 sseq2 ⊢ ( 𝑧 = 𝑇 → ( 𝑋 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑇 ) )
23 22 elrab ⊢ ( 𝑇 ∈ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ↔ ( 𝑇 ∈ ran 𝐼 ∧ 𝑋 ⊆ 𝑇 ) )
24 21 23 sylibr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → 𝑇 ∈ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } )
25 24 ne0d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ≠ ∅ )
26 1 3 diaintclN ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ⊆ ran 𝐼 ∧ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ≠ ∅ ) ) → ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ∈ ran 𝐼 )
27 17 19 25 26 syl12anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ∈ ran 𝐼 )
28 1 3 diacnvclN ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ∈ ran 𝐼 ) → ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ∈ dom 𝐼 )
29 27 28 syldan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ∈ dom 𝐼 )
30 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
31 30 1 3 diadmclN ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ∈ dom 𝐼 ) → ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ∈ ( Base ‘ 𝐾 ) )
32 29 31 syldan ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ∈ ( Base ‘ 𝐾 ) )
33 30 7 opoccl ⊢ ( ( 𝐾 ∈ OP ∧ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ∈ ( Base ‘ 𝐾 ) )
34 16 32 33 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ∈ ( Base ‘ 𝐾 ) )
35 30 1 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
36 35 ad2antlr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
37 30 7 opoccl ⊢ ( ( 𝐾 ∈ OP ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
38 16 36 37 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
39 30 5 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) )
40 14 34 38 39 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) )
41 30 6 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
42 14 40 36 41 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
43 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
44 30 43 6 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( le ‘ 𝐾 ) 𝑊 )
45 14 40 36 44 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( le ‘ 𝐾 ) 𝑊 )
46 30 43 1 3 diaeldm ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ dom 𝐼 ↔ ( ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) ) )
47 46 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ dom 𝐼 ↔ ( ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ( le ‘ 𝐾 ) 𝑊 ) ) )
48 42 45 47 mpbir2and ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ dom 𝐼 )
49 fvelrn ⊢ ( ( Fun 𝐼 ∧ ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ∈ dom 𝐼 ) → ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ ran 𝐼 )
50 12 48 49 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( 𝐼 ‘ ( ( ( ( oc ‘ 𝐾 ) ‘ ( ◡ 𝐼 ‘ ∩ { 𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧 } ) ) ( join ‘ 𝐾 ) ( ( oc ‘ 𝐾 ) ‘ 𝑊 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ) ∈ ran 𝐼 )
51 8 50 eqeltrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑋 ⊆ 𝑇 ) → ( ⊥ ‘ 𝑋 ) ∈ ran 𝐼 )