Definition df-sfl1 35958

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 35945	. 2 class splitFld1
2		vr	. . 3 setvar 𝑟
3		vj	. . 3 setvar 𝑗
4		cvv 3453	. . 3 class V
5		vp	. . . 4 setvar 𝑝
6	2	cv 1558	. . . . 5 class 𝑟
7		cpl1 22219	. . . . 5 class Poly1
8	6, 7	cfv 6517	. . . 4 class (Poly1‘𝑟)
9		c1 11071	. . . . . . 7 class 1
10	5	cv 1558	. . . . . . . 8 class 𝑝
11		cdg1 26094	. . . . . . . 8 class deg1
12	6, 10, 11	co 7392	. . . . . . 7 class (𝑟deg1𝑝)
13		cfz 13509	. . . . . . 7 class ...
14	9, 12, 13	co 7392	. . . . . 6 class (1...(𝑟deg1𝑝))
15		ccrd 9890	. . . . . 6 class card
16	14, 15	cfv 6517	. . . . 5 class (card‘(1...(𝑟deg1𝑝)))
17		vs	. . . . . . 7 setvar 𝑠
18		vf	. . . . . . 7 setvar 𝑓
19		vm	. . . . . . . 8 setvar 𝑚
20	17	cv 1558	. . . . . . . . 9 class 𝑠
21	20, 7	cfv 6517	. . . . . . . 8 class (Poly1‘𝑠)
22		vb	. . . . . . . . 9 setvar 𝑏
23		vg	. . . . . . . . . . . . 13 setvar 𝑔
24	23	cv 1558	. . . . . . . . . . . 12 class 𝑔
25	18	cv 1558	. . . . . . . . . . . . 13 class 𝑓
26	10, 25	ccom 5649	. . . . . . . . . . . 12 class (𝑝 ∘ 𝑓)
27	19	cv 1558	. . . . . . . . . . . . 13 class 𝑚
28		cdsr 20382	. . . . . . . . . . . . 13 class ∥r
29	27, 28	cfv 6517	. . . . . . . . . . . 12 class (∥r‘𝑚)
30	24, 26, 29	wbr 5099	. . . . . . . . . . 11 wff 𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓)
31	20, 24, 11	co 7392	. . . . . . . . . . . 12 class (𝑠deg1𝑔)
32		clt 11213	. . . . . . . . . . . 12 class <
33	9, 31, 32	wbr 5099	. . . . . . . . . . 11 wff 1 < (𝑠deg1𝑔)
34	30, 33	wa 399	. . . . . . . . . 10 wff (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))
35		cmn1 26166	. . . . . . . . . . . 12 class Monic1p
36	20, 35	cfv 6517	. . . . . . . . . . 11 class (Monic1p‘𝑠)
37		cir 20384	. . . . . . . . . . . 12 class Irred
38	27, 37	cfv 6517	. . . . . . . . . . 11 class (Irred‘𝑚)
39	36, 38	cin 3903	. . . . . . . . . 10 class ((Monic1p‘𝑠) ∩ (Irred‘𝑚))
40	34, 23, 39	crab 3413	. . . . . . . . 9 class {𝑔 ∈ ((Monic1p‘𝑠) ∩ (Irred‘𝑚)) ∣ (𝑔(∥r‘𝑚)(𝑝 ∘ 𝑓) ∧ 1 < (𝑠deg1𝑔))}
41		c0g 17451	. . . . . . . . . . . . 13 class 0g
42	27, 41	cfv 6517	. . . . . . . . . . . 12 class (0g‘𝑚)
43	26, 42	wceq 1559	. . . . . . . . . . 11 wff (𝑝 ∘ 𝑓) = (0g‘𝑚)
44	22	cv 1558	. . . . . . . . . . . 12 class 𝑏
45		c0 4285	. . . . . . . . . . . 12 class ∅
46	44, 45	wceq 1559	. . . . . . . . . . 11 wff 𝑏 = ∅
47	43, 46	wo 858	. . . . . . . . . 10 wff ((𝑝 ∘ 𝑓) = (0g‘𝑚) ∨ 𝑏 = ∅)
48	20, 25	cop 4587	. . . . . . . . . 10 class ⟨𝑠, 𝑓⟩
49		vh	. . . . . . . . . . 11 setvar ℎ
50		cglb 18325	. . . . . . . . . . . 12 class glb
51	44, 50	cfv 6517	. . . . . . . . . . 11 class (glb‘𝑏)
52		vt	. . . . . . . . . . . 12 setvar 𝑡
53	49	cv 1558	. . . . . . . . . . . . 13 class ℎ
54		cpfl 35944	. . . . . . . . . . . . 13 class polyFld
55	20, 53, 54	co 7392	. . . . . . . . . . . 12 class (𝑠 polyFld ℎ)
56	52	cv 1558	. . . . . . . . . . . . . 14 class 𝑡
57		c1st 7964	. . . . . . . . . . . . . 14 class 1st
58	56, 57	cfv 6517	. . . . . . . . . . . . 13 class (1st ‘𝑡)
59		c2nd 7965	. . . . . . . . . . . . . . 15 class 2nd
60	56, 59	cfv 6517	. . . . . . . . . . . . . 14 class (2nd ‘𝑡)
61	25, 60	ccom 5649	. . . . . . . . . . . . 13 class (𝑓 ∘ (2nd ‘𝑡))
62	58, 61	cop 4587	. . . . . . . . . . . 12 class ⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩
63	52, 55, 62	csb 3852	. . . . . . . . . . 11 class ⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩
64	49, 51, 63	csb 3852	. . . . . . . . . 10 class ⦋(glb‘𝑏) / ℎ⦌⦋(𝑠 polyFld ℎ) / 𝑡⦌⟨(1st ‘𝑡), (𝑓 ∘ (2nd ‘𝑡))⟩
65	47, 48, 64	cif 4479	. . . . . . . . 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 3852	. . . . . . . 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 3852	. . . . . . 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 7394	. . . . . 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 1558	. . . . . 6 class 𝑗
70	68, 69	crdg 8375	. . . . 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 6517	. . . 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 5180	. . 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 7394	. 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 1559	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)