Metamath Proof Explorer

Theorem r1pval

Description: Value of the polynomial remainder function. (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 )
r1pval.q ⊢ 𝑄 = ( quot1p ‘ 𝑅 )
r1pval.t ⊢ · = ( .r ‘ 𝑃 )
r1pval.m ⊢ − = ( -g ‘ 𝑃 )
Assertion r1pval ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → ( 𝐹 𝐸 𝐺 ) = ( 𝐹 − ( ( 𝐹 𝑄 𝐺 ) · 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 r1pval.q ⊢ 𝑄 = ( quot1p ‘ 𝑅 )
5 r1pval.t ⊢ · = ( .r ‘ 𝑃 )
6 r1pval.m ⊢ − = ( -g ‘ 𝑃 )
7 2 3 elbasfv ⊢ ( 𝐹 ∈ 𝐵 → 𝑅 ∈ V )
8 7 adantr ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → 𝑅 ∈ V )
9 fveq2 ⊢ ( 𝑟 = 𝑅 → ( Poly1 ‘ 𝑟 ) = ( Poly1 ‘ 𝑅 ) )
10 9 2 eqtr4di ⊢ ( 𝑟 = 𝑅 → ( Poly1 ‘ 𝑟 ) = 𝑃 )
11 10 fveq2d ⊢ ( 𝑟 = 𝑅 → ( Base ‘ ( Poly1 ‘ 𝑟 ) ) = ( Base ‘ 𝑃 ) )
12 11 3 eqtr4di ⊢ ( 𝑟 = 𝑅 → ( Base ‘ ( Poly1 ‘ 𝑟 ) ) = 𝐵 )
13 12 csbeq1d ⊢ ( 𝑟 = 𝑅 → ⦋ ( Base ‘ ( Poly1 ‘ 𝑟 ) ) / 𝑏 ⦌ ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑓 ( -g ‘ ( Poly1 ‘ 𝑟 ) ) ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) ) ) = ⦋ 𝐵 / 𝑏 ⦌ ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑓 ( -g ‘ ( Poly1 ‘ 𝑟 ) ) ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) ) ) )
14 3 fvexi ⊢ 𝐵 ∈ V
15 14 a1i ⊢ ( 𝑟 = 𝑅 → 𝐵 ∈ V )
16 simpr ⊢ ( ( 𝑟 = 𝑅 ∧ 𝑏 = 𝐵 ) → 𝑏 = 𝐵 )
17 10 fveq2d ⊢ ( 𝑟 = 𝑅 → ( -g ‘ ( Poly1 ‘ 𝑟 ) ) = ( -g ‘ 𝑃 ) )
18 17 6 eqtr4di ⊢ ( 𝑟 = 𝑅 → ( -g ‘ ( Poly1 ‘ 𝑟 ) ) = − )
19 eqidd ⊢ ( 𝑟 = 𝑅 → 𝑓 = 𝑓 )
20 10 fveq2d ⊢ ( 𝑟 = 𝑅 → ( .r ‘ ( Poly1 ‘ 𝑟 ) ) = ( .r ‘ 𝑃 ) )
21 20 5 eqtr4di ⊢ ( 𝑟 = 𝑅 → ( .r ‘ ( Poly1 ‘ 𝑟 ) ) = · )
22 fveq2 ⊢ ( 𝑟 = 𝑅 → ( quot1p ‘ 𝑟 ) = ( quot1p ‘ 𝑅 ) )
23 22 4 eqtr4di ⊢ ( 𝑟 = 𝑅 → ( quot1p ‘ 𝑟 ) = 𝑄 )
24 23 oveqd ⊢ ( 𝑟 = 𝑅 → ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) = ( 𝑓 𝑄 𝑔 ) )
25 eqidd ⊢ ( 𝑟 = 𝑅 → 𝑔 = 𝑔 )
26 21 24 25 oveq123d ⊢ ( 𝑟 = 𝑅 → ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) = ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) )
27 18 19 26 oveq123d ⊢ ( 𝑟 = 𝑅 → ( 𝑓 ( -g ‘ ( Poly1 ‘ 𝑟 ) ) ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) ) = ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) )
28 27 adantr ⊢ ( ( 𝑟 = 𝑅 ∧ 𝑏 = 𝐵 ) → ( 𝑓 ( -g ‘ ( Poly1 ‘ 𝑟 ) ) ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) ) = ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) )
29 16 16 28 mpoeq123dv ⊢ ( ( 𝑟 = 𝑅 ∧ 𝑏 = 𝐵 ) → ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑓 ( -g ‘ ( Poly1 ‘ 𝑟 ) ) ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) ) )
30 15 29 csbied ⊢ ( 𝑟 = 𝑅 → ⦋ 𝐵 / 𝑏 ⦌ ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑓 ( -g ‘ ( Poly1 ‘ 𝑟 ) ) ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) ) )
31 13 30 eqtrd ⊢ ( 𝑟 = 𝑅 → ⦋ ( Base ‘ ( Poly1 ‘ 𝑟 ) ) / 𝑏 ⦌ ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑓 ( -g ‘ ( Poly1 ‘ 𝑟 ) ) ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) ) ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) ) )
32 df-r1p ⊢ rem1p = ( 𝑟 ∈ V ↦ ⦋ ( Base ‘ ( Poly1 ‘ 𝑟 ) ) / 𝑏 ⦌ ( 𝑓 ∈ 𝑏 , 𝑔 ∈ 𝑏 ↦ ( 𝑓 ( -g ‘ ( Poly1 ‘ 𝑟 ) ) ( ( 𝑓 ( quot1p ‘ 𝑟 ) 𝑔 ) ( .r ‘ ( Poly1 ‘ 𝑟 ) ) 𝑔 ) ) ) )
33 14 14 mpoex ⊢ ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) ) ∈ V
34 31 32 33 fvmpt ⊢ ( 𝑅 ∈ V → ( rem1p ‘ 𝑅 ) = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) ) )
35 1 34 syl5eq ⊢ ( 𝑅 ∈ V → 𝐸 = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) ) )
36 8 35 syl ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → 𝐸 = ( 𝑓 ∈ 𝐵 , 𝑔 ∈ 𝐵 ↦ ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) ) )
37 simpl ⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → 𝑓 = 𝐹 )
38 oveq12 ⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → ( 𝑓 𝑄 𝑔 ) = ( 𝐹 𝑄 𝐺 ) )
39 simpr ⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → 𝑔 = 𝐺 )
40 38 39 oveq12d ⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) = ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) )
41 37 40 oveq12d ⊢ ( ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) → ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) = ( 𝐹 − ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) ) )
42 41 adantl ⊢ ( ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) ∧ ( 𝑓 = 𝐹 ∧ 𝑔 = 𝐺 ) ) → ( 𝑓 − ( ( 𝑓 𝑄 𝑔 ) · 𝑔 ) ) = ( 𝐹 − ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) ) )
43 simpl ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → 𝐹 ∈ 𝐵 )
44 simpr ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → 𝐺 ∈ 𝐵 )
45 ovexd ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → ( 𝐹 − ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) ) ∈ V )
46 36 42 43 44 45 ovmpod ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵 ) → ( 𝐹 𝐸 𝐺 ) = ( 𝐹 − ( ( 𝐹 𝑄 𝐺 ) · 𝐺 ) ) )