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Metamath Proof Explorer

Theorem fldpropd

Description: If two structures have the same group components (properties), one is a field iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015)

Ref Expression
Hypotheses drngpropd.1 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) )
drngpropd.2 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) )
drngpropd.3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) )
drngpropd.4 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) )
Assertion fldpropd ( 𝜑 → ( 𝐾 ∈ Field ↔ 𝐿 ∈ Field ) )

Proof

Step Hyp Ref Expression
1 drngpropd.1 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) )
2 drngpropd.2 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) )
3 drngpropd.3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) )
4 drngpropd.4 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( .r ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) )
5 1 2 3 4 drngpropd ⊢ ( 𝜑 → ( 𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing ) )
6 1 2 3 4 crngpropd ⊢ ( 𝜑 → ( 𝐾 ∈ CRing ↔ 𝐿 ∈ CRing ) )
7 5 6 anbi12d ⊢ ( 𝜑 → ( ( 𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing ) ↔ ( 𝐿 ∈ DivRing ∧ 𝐿 ∈ CRing ) ) )
8 isfld ⊢ ( 𝐾 ∈ Field ↔ ( 𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing ) )
9 isfld ⊢ ( 𝐿 ∈ Field ↔ ( 𝐿 ∈ DivRing ∧ 𝐿 ∈ CRing ) )
10 7 8 9 3bitr4g ⊢ ( 𝜑 → ( 𝐾 ∈ Field ↔ 𝐿 ∈ Field ) )