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 = ( Poly1𝑅 )
deg1addle.d 𝐷 = ( deg1𝑅 )
deg1addle.r ( 𝜑𝑅 ∈ Ring )
deg1suble.b 𝐵 = ( Base ‘ 𝑌 )
deg1suble.m = ( -g𝑌 )
deg1suble.f ( 𝜑𝐹𝐵 )
deg1suble.g ( 𝜑𝐺𝐵 )
deg1sub.l ( 𝜑 → ( 𝐷𝐺 ) < ( 𝐷𝐹 ) )
Assertion deg1sub ( 𝜑 → ( 𝐷 ‘ ( 𝐹 𝐺 ) ) = ( 𝐷𝐹 ) )

Proof

Step Hyp Ref Expression
1 deg1addle.y 𝑌 = ( Poly1𝑅 )
2 deg1addle.d 𝐷 = ( deg1𝑅 )
3 deg1addle.r ( 𝜑𝑅 ∈ Ring )
4 deg1suble.b 𝐵 = ( Base ‘ 𝑌 )
5 deg1suble.m = ( -g𝑌 )
6 deg1suble.f ( 𝜑𝐹𝐵 )
7 deg1suble.g ( 𝜑𝐺𝐵 )
8 deg1sub.l ( 𝜑 → ( 𝐷𝐺 ) < ( 𝐷𝐹 ) )
9 eqid ( +g𝑌 ) = ( +g𝑌 )
10 eqid ( invg𝑌 ) = ( invg𝑌 )
11 4 9 10 5 grpsubval ( ( 𝐹𝐵𝐺𝐵 ) → ( 𝐹 𝐺 ) = ( 𝐹 ( +g𝑌 ) ( ( invg𝑌 ) ‘ 𝐺 ) ) )
12 6 7 11 syl2anc ( 𝜑 → ( 𝐹 𝐺 ) = ( 𝐹 ( +g𝑌 ) ( ( invg𝑌 ) ‘ 𝐺 ) ) )
13 12 fveq2d ( 𝜑 → ( 𝐷 ‘ ( 𝐹 𝐺 ) ) = ( 𝐷 ‘ ( 𝐹 ( +g𝑌 ) ( ( invg𝑌 ) ‘ 𝐺 ) ) ) )
14 1 ply1ring ( 𝑅 ∈ Ring → 𝑌 ∈ Ring )
15 ringgrp ( 𝑌 ∈ Ring → 𝑌 ∈ Grp )
16 3 14 15 3syl ( 𝜑𝑌 ∈ Grp )
17 4 10 grpinvcl ( ( 𝑌 ∈ Grp ∧ 𝐺𝐵 ) → ( ( invg𝑌 ) ‘ 𝐺 ) ∈ 𝐵 )
18 16 7 17 syl2anc ( 𝜑 → ( ( invg𝑌 ) ‘ 𝐺 ) ∈ 𝐵 )
19 1 2 3 4 10 7 deg1invg ( 𝜑 → ( 𝐷 ‘ ( ( invg𝑌 ) ‘ 𝐺 ) ) = ( 𝐷𝐺 ) )
20 19 8 eqbrtrd ( 𝜑 → ( 𝐷 ‘ ( ( invg𝑌 ) ‘ 𝐺 ) ) < ( 𝐷𝐹 ) )
21 1 2 3 4 9 6 18 20 deg1add ( 𝜑 → ( 𝐷 ‘ ( 𝐹 ( +g𝑌 ) ( ( invg𝑌 ) ‘ 𝐺 ) ) ) = ( 𝐷𝐹 ) )
22 13 21 eqtrd ( 𝜑 → ( 𝐷 ‘ ( 𝐹 𝐺 ) ) = ( 𝐷𝐹 ) )