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Metamath Proof Explorer

Theorem r1pdeglt

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

Ref Expression
Hypotheses r1pval.e ⊢ 𝐸 = ( rem1p ‘ 𝑅 )
r1pval.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
r1pval.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
r1pcl.c ⊢ 𝐶 = ( Unic1p ‘ 𝑅 )
r1pdeglt.d ⊢ 𝐷 = ( deg1 ‘ 𝑅 )
Assertion r1pdeglt ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐷 ‘ ( 𝐹 𝐸 𝐺 ) ) < ( 𝐷 ‘ 𝐺 ) )

Proof

Step Hyp Ref Expression
1 r1pval.e ⊢ 𝐸 = ( rem1p ‘ 𝑅 )
2 r1pval.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
3 r1pval.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
4 r1pcl.c ⊢ 𝐶 = ( Unic1p ‘ 𝑅 )
5 r1pdeglt.d ⊢ 𝐷 = ( deg1 ‘ 𝑅 )
6 simp2 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → 𝐹 ∈ 𝐵 )
7 2 3 4 uc1pcl ⊢ ( 𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐵 )
8 7 3ad2ant3 ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → 𝐺 ∈ 𝐵 )
9 eqid ⊢ ( quot1p ‘ 𝑅 ) = ( quot1p ‘ 𝑅 )
10 eqid ⊢ ( .r ‘ 𝑃 ) = ( .r ‘ 𝑃 )
11 eqid ⊢ ( -g ‘ 𝑃 ) = ( -g ‘ 𝑃 )
12 1 2 3 9 10 11 r1pval ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → ( 𝐹 𝐸 𝐺 ) = ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) )
13 6 8 12 syl2anc ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐹 𝐸 𝐺 ) = ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) )
14 13 fveq2d ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐷 ‘ ( 𝐹 𝐸 𝐺 ) ) = ( 𝐷 ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ) )
15 eqid ⊢ ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) = ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 )
16 9 2 3 5 11 10 4 q1peqb ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ∈ 𝐵 ∧ ( 𝐷 ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ) < ( 𝐷 ‘ 𝐺 ) ) ↔ ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) = ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ) )
17 15 16 mpbiri ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ∈ 𝐵 ∧ ( 𝐷 ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ) < ( 𝐷 ‘ 𝐺 ) ) )
18 17 simprd ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐷 ‘ ( 𝐹 ( -g ‘ 𝑃 ) ( ( 𝐹 ( quot1p ‘ 𝑅 ) 𝐺 ) ( .r ‘ 𝑃 ) 𝐺 ) ) ) < ( 𝐷 ‘ 𝐺 ) )
19 14 18 eqbrtrd ⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐶 ) → ( 𝐷 ‘ ( 𝐹 𝐸 𝐺 ) ) < ( 𝐷 ‘ 𝐺 ) )