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Metamath Proof Explorer

Theorem isat2

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 ) )