Theorem deg1propd

Description: Property deduction for polynomial degree. (Contributed by Stefan O'Rear, 28-Mar-2015)

	Ref	Expression
Hypotheses	deg1propd.b1	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑅 ) )
	deg1propd.b2	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑆 ) )
	deg1propd.p	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) )
Assertion	deg1propd	⊢ ( 𝜑 → ( deg1 ‘ 𝑅 ) = ( deg1 ‘ 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		deg1propd.b1	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑅 ) )
2		deg1propd.b2	⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑆 ) )
3		deg1propd.p	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝑆 ) 𝑦 ) )
4	1 2 3	mdegpropd	⊢ ( 𝜑 → ( 1o mDeg 𝑅 ) = ( 1o mDeg 𝑆 ) )
5		eqid	⊢ ( deg1 ‘ 𝑅 ) = ( deg1 ‘ 𝑅 )
6	5	deg1fval	⊢ ( deg1 ‘ 𝑅 ) = ( 1o mDeg 𝑅 )
7		eqid	⊢ ( deg1 ‘ 𝑆 ) = ( deg1 ‘ 𝑆 )
8	7	deg1fval	⊢ ( deg1 ‘ 𝑆 ) = ( 1o mDeg 𝑆 )
9	4 6 8	3eqtr4g	⊢ ( 𝜑 → ( deg1 ‘ 𝑅 ) = ( deg1 ‘ 𝑆 ) )