Metamath Proof Explorer
Description: The predicate "is an atom". ( elatcv0 analog.) (Contributed by NM, 18-Jun-2012)
|
|
Ref |
Expression |
|
Hypotheses |
isatom.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
|
|
isatom.z |
⊢ 0 = ( 0. ‘ 𝐾 ) |
|
|
isatom.c |
⊢ 𝐶 = ( ⋖ ‘ 𝐾 ) |
|
|
isatom.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
|
Assertion |
isat2 |
⊢ ( ( 𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐵 ) → ( 𝑃 ∈ 𝐴 ↔ 0 𝐶 𝑃 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
isatom.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
isatom.z |
⊢ 0 = ( 0. ‘ 𝐾 ) |
3 |
|
isatom.c |
⊢ 𝐶 = ( ⋖ ‘ 𝐾 ) |
4 |
|
isatom.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
1 2 3 4
|
isat |
⊢ ( 𝐾 ∈ 𝐷 → ( 𝑃 ∈ 𝐴 ↔ ( 𝑃 ∈ 𝐵 ∧ 0 𝐶 𝑃 ) ) ) |
6 |
5
|
baibd |
⊢ ( ( 𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐵 ) → ( 𝑃 ∈ 𝐴 ↔ 0 𝐶 𝑃 ) ) |