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Definition df-evls1 20454
Description: Define the evaluation map for the univariate polynomial algebra. The function (𝑆 evalSub1 𝑅):𝑉⟶(𝑆m 𝑆) makes sense when 𝑆 is a ring and 𝑅 is a subring of 𝑆, and where 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 𝑆 into an element of 𝑆 formed by evaluating the polynomial with the given assignment. (Contributed by Mario Carneiro, 12-Jun-2015.)
Assertion
Ref Expression
df-evls1 evalSub1 = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ (Base‘𝑠) / 𝑏((𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)))
Distinct variable group:   𝑟,𝑏,𝑠,𝑥,𝑦

Detailed syntax breakdown of Definition df-evls1
StepHypRef Expression
1 ces1 20452 . 2 class evalSub1
2 vs . . 3 setvar 𝑠
3 vr . . 3 setvar 𝑟
4 cvv 3473 . . 3 class V
52cv 1536 . . . . 5 class 𝑠
6 cbs 16462 . . . . 5 class Base
75, 6cfv 6331 . . . 4 class (Base‘𝑠)
87cpw 4515 . . 3 class 𝒫 (Base‘𝑠)
9 vb . . . 4 setvar 𝑏
10 vx . . . . . 6 setvar 𝑥
119cv 1536 . . . . . . 7 class 𝑏
12 c1o 8073 . . . . . . . 8 class 1o
13 cmap 8384 . . . . . . . 8 class m
1411, 12, 13co 7133 . . . . . . 7 class (𝑏m 1o)
1511, 14, 13co 7133 . . . . . 6 class (𝑏m (𝑏m 1o))
1610cv 1536 . . . . . . 7 class 𝑥
17 vy . . . . . . . 8 setvar 𝑦
1817cv 1536 . . . . . . . . . 10 class 𝑦
1918csn 4543 . . . . . . . . 9 class {𝑦}
2012, 19cxp 5529 . . . . . . . 8 class (1o × {𝑦})
2117, 11, 20cmpt 5122 . . . . . . 7 class (𝑦𝑏 ↦ (1o × {𝑦}))
2216, 21ccom 5535 . . . . . 6 class (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))
2310, 15, 22cmpt 5122 . . . . 5 class (𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦}))))
243cv 1536 . . . . . 6 class 𝑟
25 ces 20260 . . . . . . 7 class evalSub
2612, 5, 25co 7133 . . . . . 6 class (1o evalSub 𝑠)
2724, 26cfv 6331 . . . . 5 class ((1o evalSub 𝑠)‘𝑟)
2823, 27ccom 5535 . . . 4 class ((𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟))
299, 7, 28csb 3860 . . 3 class (Base‘𝑠) / 𝑏((𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟))
302, 3, 4, 8, 29cmpo 7135 . 2 class (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ (Base‘𝑠) / 𝑏((𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)))
311, 30wceq 1537 1 wff evalSub1 = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ (Base‘𝑠) / 𝑏((𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)))
Colors of variables: wff setvar class
This definition is referenced by:  reldmevls1  20456  ply1frcl  20457  evls1fval  20458
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