Metamath Proof Explorer


Theorem r1pdeglt

Description: The remainder has a degree smaller than the divisor. (Contributed by Stefan O'Rear, 28-Mar-2015)

Ref Expression
Hypotheses r1pval.e 𝐸 = ( rem1p𝑅 )
r1pval.p 𝑃 = ( Poly1𝑅 )
r1pval.b 𝐵 = ( Base ‘ 𝑃 )
r1pcl.c 𝐶 = ( Unic1p𝑅 )
r1pdeglt.d 𝐷 = ( deg1𝑅 )
Assertion r1pdeglt ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → ( 𝐷 ‘ ( 𝐹 𝐸 𝐺 ) ) < ( 𝐷𝐺 ) )

Proof

Step Hyp Ref Expression
1 r1pval.e 𝐸 = ( rem1p𝑅 )
2 r1pval.p 𝑃 = ( Poly1𝑅 )
3 r1pval.b 𝐵 = ( Base ‘ 𝑃 )
4 r1pcl.c 𝐶 = ( Unic1p𝑅 )
5 r1pdeglt.d 𝐷 = ( deg1𝑅 )
6 simp2 ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → 𝐹𝐵 )
7 2 3 4 uc1pcl ( 𝐺𝐶𝐺𝐵 )
8 7 3ad2ant3 ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → 𝐺𝐵 )
9 eqid ( quot1p𝑅 ) = ( quot1p𝑅 )
10 eqid ( .r𝑃 ) = ( .r𝑃 )
11 eqid ( -g𝑃 ) = ( -g𝑃 )
12 1 2 3 9 10 11 r1pval ( ( 𝐹𝐵𝐺𝐵 ) → ( 𝐹 𝐸 𝐺 ) = ( 𝐹 ( -g𝑃 ) ( ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ( .r𝑃 ) 𝐺 ) ) )
13 6 8 12 syl2anc ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → ( 𝐹 𝐸 𝐺 ) = ( 𝐹 ( -g𝑃 ) ( ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ( .r𝑃 ) 𝐺 ) ) )
14 13 fveq2d ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → ( 𝐷 ‘ ( 𝐹 𝐸 𝐺 ) ) = ( 𝐷 ‘ ( 𝐹 ( -g𝑃 ) ( ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ( .r𝑃 ) 𝐺 ) ) ) )
15 eqid ( 𝐹 ( quot1p𝑅 ) 𝐺 ) = ( 𝐹 ( quot1p𝑅 ) 𝐺 )
16 9 2 3 5 11 10 4 q1peqb ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → ( ( ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ∈ 𝐵 ∧ ( 𝐷 ‘ ( 𝐹 ( -g𝑃 ) ( ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ( .r𝑃 ) 𝐺 ) ) ) < ( 𝐷𝐺 ) ) ↔ ( 𝐹 ( quot1p𝑅 ) 𝐺 ) = ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ) )
17 15 16 mpbiri ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → ( ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ∈ 𝐵 ∧ ( 𝐷 ‘ ( 𝐹 ( -g𝑃 ) ( ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ( .r𝑃 ) 𝐺 ) ) ) < ( 𝐷𝐺 ) ) )
18 17 simprd ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → ( 𝐷 ‘ ( 𝐹 ( -g𝑃 ) ( ( 𝐹 ( quot1p𝑅 ) 𝐺 ) ( .r𝑃 ) 𝐺 ) ) ) < ( 𝐷𝐺 ) )
19 14 18 eqbrtrd ( ( 𝑅 ∈ Ring ∧ 𝐹𝐵𝐺𝐶 ) → ( 𝐷 ‘ ( 𝐹 𝐸 𝐺 ) ) < ( 𝐷𝐺 ) )