Metamath Proof Explorer


Theorem deg1fvi

Description: Univariate polynomial degree respects protection. (Contributed by Stefan O'Rear, 28-Mar-2015)

Ref Expression
Assertion deg1fvi ( deg1𝑅 ) = ( deg1 ‘ ( I ‘ 𝑅 ) )

Proof

Step Hyp Ref Expression
1 fvi ( 𝑅 ∈ V → ( I ‘ 𝑅 ) = 𝑅 )
2 1 fveq2d ( 𝑅 ∈ V → ( deg1 ‘ ( I ‘ 𝑅 ) ) = ( deg1𝑅 ) )
3 eqid ( deg1 ‘ ∅ ) = ( deg1 ‘ ∅ )
4 eqid ( Poly1 ‘ ∅ ) = ( Poly1 ‘ ∅ )
5 00ply1bas ∅ = ( Base ‘ ( Poly1 ‘ ∅ ) )
6 3 4 5 deg1xrf ( deg1 ‘ ∅ ) : ∅ ⟶ ℝ*
7 ffn ( ( deg1 ‘ ∅ ) : ∅ ⟶ ℝ* → ( deg1 ‘ ∅ ) Fn ∅ )
8 6 7 ax-mp ( deg1 ‘ ∅ ) Fn ∅
9 fn0 ( ( deg1 ‘ ∅ ) Fn ∅ ↔ ( deg1 ‘ ∅ ) = ∅ )
10 8 9 mpbi ( deg1 ‘ ∅ ) = ∅
11 fvprc ( ¬ 𝑅 ∈ V → ( I ‘ 𝑅 ) = ∅ )
12 11 fveq2d ( ¬ 𝑅 ∈ V → ( deg1 ‘ ( I ‘ 𝑅 ) ) = ( deg1 ‘ ∅ ) )
13 fvprc ( ¬ 𝑅 ∈ V → ( deg1𝑅 ) = ∅ )
14 10 12 13 3eqtr4a ( ¬ 𝑅 ∈ V → ( deg1 ‘ ( I ‘ 𝑅 ) ) = ( deg1𝑅 ) )
15 2 14 pm2.61i ( deg1 ‘ ( I ‘ 𝑅 ) ) = ( deg1𝑅 )
16 15 eqcomi ( deg1𝑅 ) = ( deg1 ‘ ( I ‘ 𝑅 ) )