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Metamath Proof Explorer

Theorem deg1fvi

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

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

Proof

Step Hyp Ref Expression
1 fvi ⊢ ( 𝑅 ∈ V → ( I ‘ 𝑅 ) = 𝑅 )
2 1 fveq2d ⊢ ( 𝑅 ∈ V → ( deg1 ‘ ( I ‘ 𝑅 ) ) = ( deg1 ‘ 𝑅 ) )
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 → ( deg1 ‘ 𝑅 ) = ∅ )
14 10 12 13 3eqtr4a ⊢ ( ¬ 𝑅 ∈ V → ( deg1 ‘ ( I ‘ 𝑅 ) ) = ( deg1 ‘ 𝑅 ) )
15 2 14 pm2.61i ⊢ ( deg1 ‘ ( I ‘ 𝑅 ) ) = ( deg1 ‘ 𝑅 )
16 15 eqcomi ⊢ ( deg1 ‘ 𝑅 ) = ( deg1 ‘ ( I ‘ 𝑅 ) )