Definition df-sfl1 35512

Description: Temporary construction for the splitting field of a polynomial. The inputs are a field 𝑟 and a polynomial 𝑝 that we want to split, along with a tuple 𝑗 in the same format as the output. The output is a tuple ⟨𝑆, 𝐹⟩ where 𝑆 is the splitting field and 𝐹 is an injective homomorphism from the original field 𝑟.

The function works by repeatedly finding the smallest monic irreducible factor, and extending the field by that factor using the polyFld construction. We keep track of a total order in each of the splitting fields so that we can pick an element definably without needing global choice. (Contributed by Mario Carneiro, 2-Dec-2014.)

Assertion

Ref	Expression
df-sfl1	⊢ splitFld1 = (𝑟 ∈ V, 𝑗 ∈ V ↦ (𝑝 ∈ (Poly1‘𝑟) ↦ (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)), 𝑗)‘(card‘(1...(𝑟deg1𝑝))))))

Distinct variable group:   𝑓,𝑏,𝑔,ℎ,𝑗,𝑚,𝑝,𝑟,𝑠,𝑡

Detailed syntax breakdown of Definition df-sfl1

Step	Hyp	Ref	Expression
1		csf1 35499	. 2 class splitFld1
2		vr	. . 3 setvar 𝑟
3		vj	. . 3 setvar 𝑗
4		cvv 3472	. . 3 class V
5		vp	. . . 4 setvar 𝑝
6	2	cv 1533	. . . . 5 class 𝑟
7		cpl1 22165	. . . . 5 class Poly1
8	6, 7	cfv 6556	. . . 4 class (Poly1‘𝑟)
9		c1 11160	. . . . . . 7 class 1
10	5	cv 1533	. . . . . . . 8 class 𝑝
11		cdg1 26078	. . . . . . . 8 class deg1
12	6, 10, 11	co 7427	. . . . . . 7 class (𝑟deg1𝑝)
13		cfz 13542	. . . . . . 7 class ...
14	9, 12, 13	co 7427	. . . . . 6 class (1...(𝑟deg1𝑝))
15		ccrd 9979	. . . . . 6 class card
16	14, 15	cfv 6556	. . . . 5 class (card‘(1...(𝑟deg1𝑝)))
17		vs	. . . . . . 7 setvar 𝑠
18		vf	. . . . . . 7 setvar 𝑓
19		vm	. . . . . . . 8 setvar 𝑚
20	17	cv 1533	. . . . . . . . 9 class 𝑠
21	20, 7	cfv 6556	. . . . . . . 8 class (Poly1‘𝑠)
22		vb	. . . . . . . . 9 setvar 𝑏
23		vg	. . . . . . . . . . . . 13 setvar 𝑔
24	23	cv 1533	. . . . . . . . . . . 12 class 𝑔
25	18	cv 1533	. . . . . . . . . . . . 13 class 𝑓
26	10, 25	ccom 5688	. . . . . . . . . . . 12 class (𝑝 ∘ 𝑓)
27	19	cv 1533	. . . . . . . . . . . . 13 class 𝑚
28		cdsr 20348	. . . . . . . . . . . . 13 class ∥r
29	27, 28	cfv 6556	. . . . . . . . . . . 12 class (∥r‘𝑚)
30	24, 26, 29	wbr 5154	. . . . . . . . . . 11 wff 𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓)
31	20, 24, 11	co 7427	. . . . . . . . . . . 12 class (𝑠deg1𝑔)
32		clt 11299	. . . . . . . . . . . 12 class <
33	9, 31, 32	wbr 5154	. . . . . . . . . . 11 wff 1 < (𝑠deg1𝑔)
34	30, 33	wa 394	. . . . . . . . . 10 wff (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))
35		cmn1 26150	. . . . . . . . . . . 12 class Monic1p
36	20, 35	cfv 6556	. . . . . . . . . . 11 class (Monic1p‘𝑠)
37		cir 20350	. . . . . . . . . . . 12 class Irred
38	27, 37	cfv 6556	. . . . . . . . . . 11 class (Irred‘𝑚)
39	36, 38	cin 3957	. . . . . . . . . 10 class ((Monic1p‘𝑠) ∩ (Irred‘𝑚))
40	34, 23, 39	crab 3428	. . . . . . . . 9 class {𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))}
41		c0g 17467	. . . . . . . . . . . . 13 class 0g
42	27, 41	cfv 6556	. . . . . . . . . . . 12 class (0g‘𝑚)
43	26, 42	wceq 1534	. . . . . . . . . . 11 wff (𝑝 ∘ 𝑓) = (0g‘𝑚)
44	22	cv 1533	. . . . . . . . . . . 12 class 𝑏
45		c0 4333	. . . . . . . . . . . 12 class ∅
46	44, 45	wceq 1534	. . . . . . . . . . 11 wff 𝑏 = ∅
47	43, 46	wo 845	. . . . . . . . . 10 wff ((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅)
48	20, 25	cop 4640	. . . . . . . . . 10 class ⟨𝑠, 𝑓⟩
49		vh	. . . . . . . . . . 11 setvar ℎ
50		cglb 18348	. . . . . . . . . . . 12 class glb
51	44, 50	cfv 6556	. . . . . . . . . . 11 class (glb‘𝑏)
52		vt	. . . . . . . . . . . 12 setvar 𝑡
53	49	cv 1533	. . . . . . . . . . . . 13 class ℎ
54		cpfl 35498	. . . . . . . . . . . . 13 class polyFld
55	20, 53, 54	co 7427	. . . . . . . . . . . 12 class (𝑠 polyFld ℎ)
56	52	cv 1533	. . . . . . . . . . . . . 14 class 𝑡
57		c1st 8006	. . . . . . . . . . . . . 14 class 1st
58	56, 57	cfv 6556	. . . . . . . . . . . . 13 class (1st ‘𝑡)
59		c2nd 8007	. . . . . . . . . . . . . . 15 class 2nd
60	56, 59	cfv 6556	. . . . . . . . . . . . . 14 class (2nd ‘𝑡)
61	25, 60	ccom 5688	. . . . . . . . . . . . 13 class (𝑓 ∘ (2nd ‘𝑡))
62	58, 61	cop 4640	. . . . . . . . . . . 12 class ⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩
63	52, 55, 62	csb 3903	. . . . . . . . . . 11 class ⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩
64	49, 51, 63	csb 3903	. . . . . . . . . 10 class ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩
65	47, 48, 64	cif 4534	. . . . . . . . 9 class if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)
66	22, 40, 65	csb 3903	. . . . . . . 8 class ⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)
67	19, 21, 66	csb 3903	. . . . . . 7 class ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)
68	17, 18, 4, 4, 67	cmpo 7429	. . . . . 6 class (𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩))
69	3	cv 1533	. . . . . 6 class 𝑗
70	68, 69	crdg 8443	. . . . 5 class rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)), 𝑗)
71	16, 70	cfv 6556	. . . 4 class (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)), 𝑗)‘(card‘(1...(𝑟deg1𝑝))))
72	5, 8, 71	cmpt 5237	. . 3 class (𝑝 ∈ (Poly1‘𝑟) ↦ (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)), 𝑗)‘(card‘(1...(𝑟deg1𝑝)))))
73	2, 3, 4, 4, 72	cmpo 7429	. 2 class (𝑟 ∈ V, 𝑗 ∈ V ↦ (𝑝 ∈ (Poly1‘𝑟) ↦ (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)), 𝑗)‘(card‘(1...(𝑟deg1𝑝))))))
74	1, 73	wceq 1534	1 wff splitFld1 = (𝑟 ∈ V, 𝑗 ∈ V ↦ (𝑝 ∈ (Poly1‘𝑟) ↦ (rec((𝑠 ∈ V, 𝑓 ∈ V ↦ ⦋(Poly1‘𝑠) / 𝑚⦌⦋{𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))} / 𝑏⦌if(((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅), ⟨𝑠, 𝑓⟩, ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩)), 𝑗)‘(card‘(1...(𝑟deg1𝑝))))))

This definition is referenced by: (None)