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 ) )