Database
SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
Mathbox for Norm Megill
Construction of a vector space from a Hilbert lattice
cdleme19f
Metamath Proof Explorer
Description: Part of proof of Lemma E in Crawley p. 113, 5th paragraph on p. 114,
line 3. D , F , N , Y , G , O represent s_2,
f(s), f_s(r), t_2, f(t), f_t(r). We prove that if r <_ s \/
t, then f_t(r) = f_t(r). (Contributed by NM , 14-Nov-2012)
Ref
Expression
Hypotheses
cdleme19.l
⊢ ≤ = ( le ‘ 𝐾 )
cdleme19.j
⊢ ∨ = ( join ‘ 𝐾 )
cdleme19.m
⊢ ∧ = ( meet ‘ 𝐾 )
cdleme19.a
⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme19.h
⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme19.u
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme19.f
⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
cdleme19.g
⊢ 𝐺 = ( ( 𝑇 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 ) ) )
cdleme19.d
⊢ 𝐷 = ( ( 𝑅 ∨ 𝑆 ) ∧ 𝑊 )
cdleme19.y
⊢ 𝑌 = ( ( 𝑅 ∨ 𝑇 ) ∧ 𝑊 )
cdleme19.n
⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ 𝐷 ) )
cdleme19.o
⊢ 𝑂 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐺 ∨ 𝑌 ) )
Assertion
cdleme19f
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → 𝑁 = 𝑂 )
Proof
Step
Hyp
Ref
Expression
1
cdleme19.l
⊢ ≤ = ( le ‘ 𝐾 )
2
cdleme19.j
⊢ ∨ = ( join ‘ 𝐾 )
3
cdleme19.m
⊢ ∧ = ( meet ‘ 𝐾 )
4
cdleme19.a
⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5
cdleme19.h
⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6
cdleme19.u
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7
cdleme19.f
⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
8
cdleme19.g
⊢ 𝐺 = ( ( 𝑇 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 ) ) )
9
cdleme19.d
⊢ 𝐷 = ( ( 𝑅 ∨ 𝑆 ) ∧ 𝑊 )
10
cdleme19.y
⊢ 𝑌 = ( ( 𝑅 ∨ 𝑇 ) ∧ 𝑊 )
11
cdleme19.n
⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ 𝐷 ) )
12
cdleme19.o
⊢ 𝑂 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐺 ∨ 𝑌 ) )
13
1 2 3 4 5 6 7 8 9 10
cdleme19e
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( 𝐹 ∨ 𝐷 ) = ( 𝐺 ∨ 𝑌 ) )
14
13
oveq2d
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ 𝐷 ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐺 ∨ 𝑌 ) ) )
15
14 11 12
3eqtr4g
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → 𝑁 = 𝑂 )