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Metamath Proof Explorer

Theorem deg1sub

Description: Exact degree of a difference of two polynomials of unequal degree. (Contributed by Stefan O'Rear, 28-Mar-2015)

Ref Expression
Hypotheses deg1addle.y ⊢ 𝑌 = ( Poly1 ‘ 𝑅 )
deg1addle.d ⊢ 𝐷 = ( deg1 ‘ 𝑅 )
deg1addle.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
deg1suble.b ⊢ 𝐵 = ( Base ‘ 𝑌 )
deg1suble.m ⊢ − = ( -g ‘ 𝑌 )
deg1suble.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
deg1suble.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐵 )
deg1sub.l ⊢ ( 𝜑 → ( 𝐷 ‘ 𝐺 ) < ( 𝐷 ‘ 𝐹 ) )
Assertion deg1sub ( 𝜑 → ( 𝐷 ‘ ( 𝐹 − 𝐺 ) ) = ( 𝐷 ‘ 𝐹 ) )

Proof

Step Hyp Ref Expression
1 deg1addle.y ⊢ 𝑌 = ( Poly1 ‘ 𝑅 )
2 deg1addle.d ⊢ 𝐷 = ( deg1 ‘ 𝑅 )
3 deg1addle.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
4 deg1suble.b ⊢ 𝐵 = ( Base ‘ 𝑌 )
5 deg1suble.m ⊢ − = ( -g ‘ 𝑌 )
6 deg1suble.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
7 deg1suble.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐵 )
8 deg1sub.l ⊢ ( 𝜑 → ( 𝐷 ‘ 𝐺 ) < ( 𝐷 ‘ 𝐹 ) )
9 eqid ⊢ ( +g ‘ 𝑌 ) = ( +g ‘ 𝑌 )
10 eqid ⊢ ( invg ‘ 𝑌 ) = ( invg ‘ 𝑌 )
11 4 9 10 5 grpsubval ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → ( 𝐹 − 𝐺 ) = ( 𝐹 ( +g ‘ 𝑌 ) ( ( invg ‘ 𝑌 ) ‘ 𝐺 ) ) )
12 6 7 11 syl2anc ⊢ ( 𝜑 → ( 𝐹 − 𝐺 ) = ( 𝐹 ( +g ‘ 𝑌 ) ( ( invg ‘ 𝑌 ) ‘ 𝐺 ) ) )
13 12 fveq2d ⊢ ( 𝜑 → ( 𝐷 ‘ ( 𝐹 − 𝐺 ) ) = ( 𝐷 ‘ ( 𝐹 ( +g ‘ 𝑌 ) ( ( invg ‘ 𝑌 ) ‘ 𝐺 ) ) ) )
14 1 ply1ring ⊢ ( 𝑅 ∈ Ring → 𝑌 ∈ Ring )
15 ringgrp ⊢ ( 𝑌 ∈ Ring → 𝑌 ∈ Grp )
16 3 14 15 3syl ⊢ ( 𝜑 → 𝑌 ∈ Grp )
17 4 10 grpinvcl ⊢ ( ( 𝑌 ∈ Grp ∧ 𝐺 ∈ 𝐵 ) → ( ( invg ‘ 𝑌 ) ‘ 𝐺 ) ∈ 𝐵 )
18 16 7 17 syl2anc ⊢ ( 𝜑 → ( ( invg ‘ 𝑌 ) ‘ 𝐺 ) ∈ 𝐵 )
19 1 2 3 4 10 7 deg1invg ⊢ ( 𝜑 → ( 𝐷 ‘ ( ( invg ‘ 𝑌 ) ‘ 𝐺 ) ) = ( 𝐷 ‘ 𝐺 ) )
20 19 8 eqbrtrd ⊢ ( 𝜑 → ( 𝐷 ‘ ( ( invg ‘ 𝑌 ) ‘ 𝐺 ) ) < ( 𝐷 ‘ 𝐹 ) )
21 1 2 3 4 9 6 18 20 deg1add ⊢ ( 𝜑 → ( 𝐷 ‘ ( 𝐹 ( +g ‘ 𝑌 ) ( ( invg ‘ 𝑌 ) ‘ 𝐺 ) ) ) = ( 𝐷 ‘ 𝐹 ) )
22 13 21 eqtrd ⊢ ( 𝜑 → ( 𝐷 ‘ ( 𝐹 − 𝐺 ) ) = ( 𝐷 ‘ 𝐹 ) )