Metamath Proof Explorer


Theorem dgradd

Description: The degree of a sum of polynomials is at most the maximum of the degrees. (Contributed by Mario Carneiro, 24-Jul-2014)

Ref Expression
Hypotheses dgradd.1 𝑀 = ( deg ‘ 𝐹 )
dgradd.2 𝑁 = ( deg ‘ 𝐺 )
Assertion dgradd ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( deg ‘ ( 𝐹f + 𝐺 ) ) ≤ if ( 𝑀𝑁 , 𝑁 , 𝑀 ) )

Proof

Step Hyp Ref Expression
1 dgradd.1 𝑀 = ( deg ‘ 𝐹 )
2 dgradd.2 𝑁 = ( deg ‘ 𝐺 )
3 eqid ( coeff ‘ 𝐹 ) = ( coeff ‘ 𝐹 )
4 eqid ( coeff ‘ 𝐺 ) = ( coeff ‘ 𝐺 )
5 3 4 1 2 coeaddlem ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( ( coeff ‘ ( 𝐹f + 𝐺 ) ) = ( ( coeff ‘ 𝐹 ) ∘f + ( coeff ‘ 𝐺 ) ) ∧ ( deg ‘ ( 𝐹f + 𝐺 ) ) ≤ if ( 𝑀𝑁 , 𝑁 , 𝑀 ) ) )
6 5 simprd ( ( 𝐹 ∈ ( Poly ‘ 𝑆 ) ∧ 𝐺 ∈ ( Poly ‘ 𝑆 ) ) → ( deg ‘ ( 𝐹f + 𝐺 ) ) ≤ if ( 𝑀𝑁 , 𝑁 , 𝑀 ) )