Definition df-evl1 19502

Description: Define the evaluation map for the univariate polynomial algebra. The function (eval₁‘𝑅):𝑉⟶(𝑅 ↑_𝑚 𝑅) makes sense when 𝑅 is a ring, and 𝑉 is the set of polynomials in (Poly₁‘𝑅). 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 ∈ (𝑏 ↑𝑚 (𝑏 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑟)))

Distinct variable group:   𝑟,𝑏,𝑥,𝑦

Detailed syntax breakdown of Definition df-evl1

Step	Hyp	Ref	Expression
1		ce1 19500	. 2 class eval1
2		vr	. . 3 setvar 𝑟
3		cvv 3173	. . 3 class V
4		vb	. . . 4 setvar 𝑏
5	2	cv 1474	. . . . 5 class 𝑟
6		cbs 15695	. . . . 5 class Base
7	5, 6	cfv 5804	. . . 4 class (Base‘𝑟)
8		vx	. . . . . 6 setvar 𝑥
9	4	cv 1474	. . . . . . 7 class 𝑏
10		c1o 7440	. . . . . . . 8 class 1𝑜
11		cmap 7744	. . . . . . . 8 class ↑𝑚
12	9, 10, 11	co 6549	. . . . . . 7 class (𝑏 ↑𝑚 1𝑜)
13	9, 12, 11	co 6549	. . . . . 6 class (𝑏 ↑𝑚 (𝑏 ↑𝑚 1𝑜))
14	8	cv 1474	. . . . . . 7 class 𝑥
15		vy	. . . . . . . 8 setvar 𝑦
16	15	cv 1474	. . . . . . . . . 10 class 𝑦
17	16	csn 4125	. . . . . . . . 9 class {𝑦}
18	10, 17	cxp 5036	. . . . . . . 8 class (1𝑜 × {𝑦})
19	15, 9, 18	cmpt 4643	. . . . . . 7 class (𝑦 ∈ 𝑏 ↦ (1𝑜 × {𝑦}))
20	14, 19	ccom 5042	. . . . . 6 class (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 × {𝑦})))
21	8, 13, 20	cmpt 4643	. . . . 5 class (𝑥 ∈ (𝑏 ↑𝑚 (𝑏 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 × {𝑦}))))
22		cevl 19326	. . . . . 6 class eval
23	10, 5, 22	co 6549	. . . . 5 class (1𝑜 eval 𝑟)
24	21, 23	ccom 5042	. . . 4 class ((𝑥 ∈ (𝑏 ↑𝑚 (𝑏 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑟))
25	4, 7, 24	csb 3499	. . 3 class ⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑𝑚 (𝑏 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑟))
26	2, 3, 25	cmpt 4643	. 2 class (𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑𝑚 (𝑏 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑟)))
27	1, 26	wceq 1475	1 wff eval1 = (𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑𝑚 (𝑏 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑟)))

This definition is referenced by:  evl1fval  19513