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Definition df-evl1 19896
Description: Define the evaluation map for the univariate polynomial algebra. The function (eval1𝑅):𝑉⟶(𝑅𝑚 𝑅) makes sense when 𝑅 is a ring, and 𝑉 is the set of polynomials in (Poly1𝑅). This function maps an element of the formal polynomial algebra (with coefficients in 𝑅) to a function from assignments to the variable from 𝑅 into an element of 𝑅 formed by evaluating the polynomial with the given assignment. (Contributed by Mario Carneiro, 12-Jun-2015.)
Assertion
Ref Expression
df-evl1 eval1 = (𝑟 ∈ V ↦ (Base‘𝑟) / 𝑏((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑟)))
Distinct variable group:   𝑟,𝑏,𝑥,𝑦

Detailed syntax breakdown of Definition df-evl1
StepHypRef Expression
1 ce1 19894 . 2 class eval1
2 vr . . 3 setvar 𝑟
3 cvv 3351 . . 3 class V
4 vb . . . 4 setvar 𝑏
52cv 1630 . . . . 5 class 𝑟
6 cbs 16064 . . . . 5 class Base
75, 6cfv 6031 . . . 4 class (Base‘𝑟)
8 vx . . . . . 6 setvar 𝑥
94cv 1630 . . . . . . 7 class 𝑏
10 c1o 7706 . . . . . . . 8 class 1𝑜
11 cmap 8009 . . . . . . . 8 class 𝑚
129, 10, 11co 6793 . . . . . . 7 class (𝑏𝑚 1𝑜)
139, 12, 11co 6793 . . . . . 6 class (𝑏𝑚 (𝑏𝑚 1𝑜))
148cv 1630 . . . . . . 7 class 𝑥
15 vy . . . . . . . 8 setvar 𝑦
1615cv 1630 . . . . . . . . . 10 class 𝑦
1716csn 4316 . . . . . . . . 9 class {𝑦}
1810, 17cxp 5247 . . . . . . . 8 class (1𝑜 × {𝑦})
1915, 9, 18cmpt 4863 . . . . . . 7 class (𝑦𝑏 ↦ (1𝑜 × {𝑦}))
2014, 19ccom 5253 . . . . . 6 class (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))
218, 13, 20cmpt 4863 . . . . 5 class (𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦}))))
22 cevl 19720 . . . . . 6 class eval
2310, 5, 22co 6793 . . . . 5 class (1𝑜 eval 𝑟)
2421, 23ccom 5253 . . . 4 class ((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑟))
254, 7, 24csb 3682 . . 3 class (Base‘𝑟) / 𝑏((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑟))
262, 3, 25cmpt 4863 . 2 class (𝑟 ∈ V ↦ (Base‘𝑟) / 𝑏((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑟)))
271, 26wceq 1631 1 wff eval1 = (𝑟 ∈ V ↦ (Base‘𝑟) / 𝑏((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑟)))
Colors of variables: wff setvar class
This definition is referenced by:  evl1fval  19907
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