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Definition df-sfl 31244
Description: Define the splitting field of a finite collection of polynomials, given a total ordered base field. The output is a tuple 𝑆, 𝐹 where 𝑆 is the totally ordered splitting field and 𝐹 is an injective homomorphism from the original field 𝑟. (Contributed by Mario Carneiro, 2-Dec-2014.)
Assertion
Ref Expression
df-sfl splitFld = (𝑟 ∈ V, 𝑝 ∈ V ↦ (℩𝑥𝑓(𝑓 Isom < , (lt‘𝑟)((1...(#‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(#‘𝑝)))))
Distinct variable group:   𝑒,𝑓,𝑔,𝑝,𝑟,𝑥

Detailed syntax breakdown of Definition df-sfl
StepHypRef Expression
1 csf 31236 . 2 class splitFld
2 vr . . 3 setvar 𝑟
3 vp . . 3 setvar 𝑝
4 cvv 3186 . . 3 class V
5 c1 9881 . . . . . . . 8 class 1
63cv 1479 . . . . . . . . 9 class 𝑝
7 chash 13057 . . . . . . . . 9 class #
86, 7cfv 5847 . . . . . . . 8 class (#‘𝑝)
9 cfz 12268 . . . . . . . 8 class ...
105, 8, 9co 6604 . . . . . . 7 class (1...(#‘𝑝))
11 clt 10018 . . . . . . 7 class <
122cv 1479 . . . . . . . 8 class 𝑟
13 cplt 16862 . . . . . . . 8 class lt
1412, 13cfv 5847 . . . . . . 7 class (lt‘𝑟)
15 vf . . . . . . . 8 setvar 𝑓
1615cv 1479 . . . . . . 7 class 𝑓
1710, 6, 11, 14, 16wiso 5848 . . . . . 6 wff 𝑓 Isom < , (lt‘𝑟)((1...(#‘𝑝)), 𝑝)
18 vx . . . . . . . 8 setvar 𝑥
1918cv 1479 . . . . . . 7 class 𝑥
20 ve . . . . . . . . . 10 setvar 𝑒
21 vg . . . . . . . . . 10 setvar 𝑔
2221cv 1479 . . . . . . . . . . 11 class 𝑔
2320cv 1479 . . . . . . . . . . . 12 class 𝑒
24 csf1 31235 . . . . . . . . . . . 12 class splitFld1
2512, 23, 24co 6604 . . . . . . . . . . 11 class (𝑟 splitFld1 𝑒)
2622, 25cfv 5847 . . . . . . . . . 10 class ((𝑟 splitFld1 𝑒)‘𝑔)
2720, 21, 4, 4, 26cmpt2 6606 . . . . . . . . 9 class (𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔))
28 cc0 9880 . . . . . . . . . . . 12 class 0
29 cid 4984 . . . . . . . . . . . . . 14 class I
30 cbs 15781 . . . . . . . . . . . . . . 15 class Base
3112, 30cfv 5847 . . . . . . . . . . . . . 14 class (Base‘𝑟)
3229, 31cres 5076 . . . . . . . . . . . . 13 class ( I ↾ (Base‘𝑟))
3312, 32cop 4154 . . . . . . . . . . . 12 class 𝑟, ( I ↾ (Base‘𝑟))⟩
3428, 33cop 4154 . . . . . . . . . . 11 class ⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩
3534csn 4148 . . . . . . . . . 10 class {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}
3616, 35cun 3553 . . . . . . . . 9 class (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩})
3727, 36, 28cseq 12741 . . . . . . . 8 class seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))
388, 37cfv 5847 . . . . . . 7 class (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(#‘𝑝))
3919, 38wceq 1480 . . . . . 6 wff 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(#‘𝑝))
4017, 39wa 384 . . . . 5 wff (𝑓 Isom < , (lt‘𝑟)((1...(#‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(#‘𝑝)))
4140, 15wex 1701 . . . 4 wff 𝑓(𝑓 Isom < , (lt‘𝑟)((1...(#‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(#‘𝑝)))
4241, 18cio 5808 . . 3 class (℩𝑥𝑓(𝑓 Isom < , (lt‘𝑟)((1...(#‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(#‘𝑝))))
432, 3, 4, 4, 42cmpt2 6606 . 2 class (𝑟 ∈ V, 𝑝 ∈ V ↦ (℩𝑥𝑓(𝑓 Isom < , (lt‘𝑟)((1...(#‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(#‘𝑝)))))
441, 43wceq 1480 1 wff splitFld = (𝑟 ∈ V, 𝑝 ∈ V ↦ (℩𝑥𝑓(𝑓 Isom < , (lt‘𝑟)((1...(#‘𝑝)), 𝑝) ∧ 𝑥 = (seq0((𝑒 ∈ V, 𝑔 ∈ V ↦ ((𝑟 splitFld1 𝑒)‘𝑔)), (𝑓 ∪ {⟨0, ⟨𝑟, ( I ↾ (Base‘𝑟))⟩⟩}))‘(#‘𝑝)))))
Colors of variables: wff setvar class
This definition is referenced by: (None)
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