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 ) ) )

Metamath Proof Explorer

Theorem diaocN

Proof