Metamath Proof Explorer
Description: Part of proof of Lemma E in Crawley p. 113. C represents s_1,
which we prove is an atom. (Contributed by NM, 10-Jun-2012)
|
|
Ref |
Expression |
|
Hypotheses |
cdleme8.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
|
|
cdleme8.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
|
|
cdleme8.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
|
|
cdleme8.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
|
|
cdleme8.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
|
|
cdleme8.4 |
⊢ 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) |
|
Assertion |
cdleme9a |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑆 ) ) → 𝐶 ∈ 𝐴 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme8.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
cdleme8.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
cdleme8.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
4 |
|
cdleme8.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
cdleme8.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
6 |
|
cdleme8.4 |
⊢ 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) |
7 |
1 2 3 4 5 6
|
lhpat2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑆 ) ) → 𝐶 ∈ 𝐴 ) |