Theorem aks6d1c2lem3 42108

Description: Lemma for aks6d1c2 42112 to simplify context. (Contributed by metakunt, 1-May-2025.)

Hypotheses

Ref	Expression
aks6d1c2.1	⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}
aks6d1c2.2	⊢ 𝑃 = (chr‘𝐾)
aks6d1c2.3	⊢ (𝜑 → 𝐾 ∈ Field)
aks6d1c2.4	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c2.5	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks6d1c2.6	⊢ (𝜑 → 𝑁 ∈ ℕ)
aks6d1c2.7	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks6d1c2.8	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c2.9	⊢ (𝜑 → 𝐹:(0...𝐴)⟶ℕ0)
aks6d1c2.10	⊢ 𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
aks6d1c2.11	⊢ (𝜑 → 𝐴 ∈ ℕ0)
aks6d1c2.12	⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
aks6d1c2.13	⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
aks6d1c2.14	⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
aks6d1c2.15	⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
aks6d1c2.16	⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
aks6d1c2.17	⊢ 𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))
aks6d1c2.18	⊢ 𝐵 = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))
aks6d1c2.19	⊢ 𝐶 = (𝐸 “ ((0...𝐵) × (0...𝐵)))
aks6d1c2.20	⊢ (𝜑 → 𝐼 ∈ 𝐶)
aks6d1c2.21	⊢ (𝜑 → 𝐽 ∈ 𝐶)
aks6d1c2.22	⊢ (𝜑 → 𝐼 < 𝐽)
aks6d1c2.23	⊢ ↑ = (.g‘(mulGrp‘(Poly1‘𝐾)))
aks6d1c2.24	⊢ 𝑋 = (var1‘𝐾)
aks6d1c2.25	⊢ 𝑆 = ((𝐽 ↑ 𝑋)(-g‘(Poly1‘𝐾))(𝐼 ↑ 𝑋))
aks6d1c2.26	⊢ (𝜑 → 𝑈 ∈ ℕ)
aks6d1c2.27	⊢ (𝜑 → 𝐽 = (𝐼 + (𝑈 · 𝑅)))
aks6d1c2p3.1	⊢ (𝜑 → 𝑠 ∈ (ℕ0 ↑m (0...𝐴)))
aks6d1c2p3.2	⊢ (𝜑 → 𝑟 ∈ (0...𝐵))
aks6d1c2p3.3	⊢ (𝜑 → 𝑜 ∈ (0...𝐵))
aks6d1c2p3.4	⊢ (𝜑 → 𝐽 = (𝑟𝐸𝑜))
aks6d1c2p3.5	⊢ (𝜑 → 𝑝 ∈ (0...𝐵))
aks6d1c2p3.6	⊢ (𝜑 → 𝑞 ∈ (0...𝐵))
aks6d1c2p3.7	⊢ (𝜑 → 𝐼 = (𝑝𝐸𝑞))
aks6d1c2p3.8	⊢ (𝜑 → 𝐼 ∈ ℕ0)

Assertion

Ref	Expression
aks6d1c2lem3	⊢ (𝜑 → (𝐽(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (𝐼(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)))

Distinct variable groups:   ∼ ,𝑎   𝐴,𝑎   𝐴,𝑔,𝑖   𝑥,𝐴   𝑒,𝐺,𝑓,𝑦   𝐾,𝑎   𝑒,𝐾,𝑓,𝑦   𝑔,𝐾,𝑖   𝑥,𝐾   𝑦,𝑀   𝑁,𝑎   𝑒,𝑁,𝑓,𝑦   𝑘,𝑁,𝑙   𝑥,𝑁   𝑃,𝑒,𝑓,𝑦   𝑃,𝑘,𝑙   𝑥,𝑃   𝑅,𝑒,𝑓,𝑦   𝑥,𝑅   𝜑,𝑎   𝑒,𝑜,𝑓,𝑦   𝑒,𝑝,𝑓,𝑦   𝑒,𝑞,𝑓,𝑦   𝑒,𝑟,𝑓,𝑦   𝑒,𝑠,𝑓,𝑦   𝜑,𝑔,𝑖   𝑔,𝑠,𝑖   𝑘,𝑜,𝑙   𝑘,𝑝,𝑙   𝜑,𝑘,𝑙   𝑘,𝑞,𝑙   𝑘,𝑟,𝑙   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑦,𝑒,𝑓,ℎ,𝑜,𝑠,𝑟,𝑞,𝑝)   𝐴(𝑦,𝑒,𝑓,ℎ,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑙)   𝐵(𝑥,𝑦,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑎,𝑙)   𝐶(𝑥,𝑦,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑎,𝑙)   𝑃(𝑔,ℎ,𝑖,𝑜,𝑠,𝑟,𝑞,𝑝,𝑎)   ∼ (𝑥,𝑦,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑙)   𝑅(𝑔,ℎ,𝑖,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑎,𝑙)   𝑆(𝑥,𝑦,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑎,𝑙)   𝑈(𝑥,𝑦,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑎,𝑙)   𝐸(𝑥,𝑦,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑎,𝑙)   ↑ (𝑥,𝑦,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑎,𝑙)   𝐹(𝑥,𝑦,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑎,𝑙)   𝐺(𝑥,𝑔,ℎ,𝑖,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑎,𝑙)   𝐻(𝑥,𝑦,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑎,𝑙)   𝐼(𝑥,𝑦,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑎,𝑙)   𝐽(𝑥,𝑦,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑎,𝑙)   𝐾(ℎ,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑙)   𝐿(𝑥,𝑦,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑎,𝑙)   𝑀(𝑥,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑎,𝑙)   𝑁(𝑔,ℎ,𝑖,𝑜,𝑠,𝑟,𝑞,𝑝)   𝑋(𝑥,𝑦,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑜,𝑠,𝑟,𝑞,𝑝,𝑎,𝑙)

Proof of Theorem aks6d1c2lem3

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		aks6d1c2p3.4	. . . 4 ⊢ (𝜑 → 𝐽 = (𝑟𝐸𝑜))
2		aks6d1c2.12	. . . . . 6 ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
3	2	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))))
4		simprl 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → 𝑘 = 𝑟)
5	4	oveq2d 7447	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → (𝑃↑𝑘) = (𝑃↑𝑟))
6		simprr 773	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → 𝑙 = 𝑜)
7	6	oveq2d 7447	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → ((𝑁 / 𝑃)↑𝑙) = ((𝑁 / 𝑃)↑𝑜))
8	5, 7	oveq12d 7449	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 = 𝑟 ∧ 𝑙 = 𝑜)) → ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)) = ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)))
9		aks6d1c2p3.2	. . . . . 6 ⊢ (𝜑 → 𝑟 ∈ (0...𝐵))
10		elfznn0 13657	. . . . . 6 ⊢ (𝑟 ∈ (0...𝐵) → 𝑟 ∈ ℕ0)
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝑟 ∈ ℕ0)
12		aks6d1c2p3.3	. . . . . 6 ⊢ (𝜑 → 𝑜 ∈ (0...𝐵))
13		elfznn0 13657	. . . . . 6 ⊢ (𝑜 ∈ (0...𝐵) → 𝑜 ∈ ℕ0)
14	12, 13	syl 17	. . . . 5 ⊢ (𝜑 → 𝑜 ∈ ℕ0)
15		ovexd 7466	. . . . 5 ⊢ (𝜑 → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ V)
16	3, 8, 11, 14, 15	ovmpod 7585	. . . 4 ⊢ (𝜑 → (𝑟𝐸𝑜) = ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)))
17	1, 16	eqtrd 2775	. . 3 ⊢ (𝜑 → 𝐽 = ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)))
18	17	oveq1d 7446	. 2 ⊢ (𝜑 → (𝐽(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)))
19		aks6d1c2p3.7	. . . . 5 ⊢ (𝜑 → 𝐼 = (𝑝𝐸𝑞))
20		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → 𝑘 = 𝑝)
21	20	oveq2d 7447	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → (𝑃↑𝑘) = (𝑃↑𝑝))
22		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → 𝑙 = 𝑞)
23	22	oveq2d 7447	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → ((𝑁 / 𝑃)↑𝑙) = ((𝑁 / 𝑃)↑𝑞))
24	21, 23	oveq12d 7449	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 = 𝑝 ∧ 𝑙 = 𝑞)) → ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)) = ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)))
25		aks6d1c2p3.5	. . . . . . 7 ⊢ (𝜑 → 𝑝 ∈ (0...𝐵))
26		elfznn0 13657	. . . . . . 7 ⊢ (𝑝 ∈ (0...𝐵) → 𝑝 ∈ ℕ0)
27	25, 26	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑝 ∈ ℕ0)
28		aks6d1c2p3.6	. . . . . . 7 ⊢ (𝜑 → 𝑞 ∈ (0...𝐵))
29		elfznn0 13657	. . . . . . 7 ⊢ (𝑞 ∈ (0...𝐵) → 𝑞 ∈ ℕ0)
30	28, 29	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑞 ∈ ℕ0)
31		ovexd 7466	. . . . . 6 ⊢ (𝜑 → ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∈ V)
32	3, 24, 27, 30, 31	ovmpod 7585	. . . . 5 ⊢ (𝜑 → (𝑝𝐸𝑞) = ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)))
33	19, 32	eqtrd 2775	. . . 4 ⊢ (𝜑 → 𝐼 = ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)))
34	33	oveq1d 7446	. . 3 ⊢ (𝜑 → (𝐼(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)))
35		fveq2 6907	. . . . . . 7 ⊢ (𝑦 = 𝑀 → (((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀))
36	35	oveq2d 7447	. . . . . 6 ⊢ (𝑦 = 𝑀 → (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦)) = (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)))
37		oveq2 7439	. . . . . . 7 ⊢ (𝑦 = 𝑀 → (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑦) = (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑀))
38	37	fveq2d 6911	. . . . . 6 ⊢ (𝑦 = 𝑀 → (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑀)))
39	36, 38	eqeq12d 2751	. . . . 5 ⊢ (𝑦 = 𝑀 → ((((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑦)) ↔ (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑀))))
40		aks6d1c2.1	. . . . . . 7 ⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly1‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘𝑓)‘𝑦)) = (((eval1‘𝐾)‘𝑓)‘(𝑒(.g‘(mulGrp‘𝐾))𝑦)))}
41		aks6d1c2.2	. . . . . . 7 ⊢ 𝑃 = (chr‘𝐾)
42		aks6d1c2.3	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Field)
43		aks6d1c2.4	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℙ)
44		aks6d1c2.5	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℕ)
45		aks6d1c2.6	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
46		aks6d1c2.7	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
47		aks6d1c2.8	. . . . . . 7 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
48		aks6d1c2p3.1	. . . . . . . 8 ⊢ (𝜑 → 𝑠 ∈ (ℕ0 ↑m (0...𝐴)))
49		nn0ex 12530	. . . . . . . . . 10 ⊢ ℕ0 ∈ V
50	49	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ℕ0 ∈ V)
51		ovexd 7466	. . . . . . . . 9 ⊢ (𝜑 → (0...𝐴) ∈ V)
52		elmapg 8878	. . . . . . . . 9 ⊢ ((ℕ0 ∈ V ∧ (0...𝐴) ∈ V) → (𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ↔ 𝑠:(0...𝐴)⟶ℕ0))
53	50, 51, 52	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ↔ 𝑠:(0...𝐴)⟶ℕ0))
54	48, 53	mpbid 232	. . . . . . 7 ⊢ (𝜑 → 𝑠:(0...𝐴)⟶ℕ0)
55		aks6d1c2.10	. . . . . . 7 ⊢ 𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
56		aks6d1c2.11	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℕ0)
57		eqid 2735	. . . . . . 7 ⊢ ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) = ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))
58		aks6d1c2.14	. . . . . . 7 ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
59		aks6d1c2.15	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
60	40, 41, 42, 43, 44, 45, 46, 47, 54, 55, 56, 27, 30, 57, 58, 59	aks6d1c1rh 42107	. . . . . 6 ⊢ (𝜑 → ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∼ (𝐺‘𝑠))
61	40, 60	aks6d1c1p1rcl 42090	. . . . . . . 8 ⊢ (𝜑 → (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∈ ℕ ∧ (𝐺‘𝑠) ∈ (Base‘(Poly1‘𝐾))))
62	61	simprd 495	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑠) ∈ (Base‘(Poly1‘𝐾)))
63	61	simpld 494	. . . . . . 7 ⊢ (𝜑 → ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∈ ℕ)
64	40, 62, 63	aks6d1c1p1 42089	. . . . . 6 ⊢ (𝜑 → (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) ∼ (𝐺‘𝑠) ↔ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑦))))
65	60, 64	mpbid 232	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑦)))
66		aks6d1c2.16	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
67	39, 65, 66	rspcdva 3623	. . . 4 ⊢ (𝜑 → (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑀)))
68	33	eqcomd 2741	. . . . . . . 8 ⊢ (𝜑 → ((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞)) = 𝐼)
69	68	oveq1d 7446	. . . . . . 7 ⊢ (𝜑 → (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑀) = (𝐼(.g‘(mulGrp‘𝐾))𝑀))
70	17	eqcomd 2741	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) = 𝐽)
71	70	oveq1d 7446	. . . . . . . 8 ⊢ (𝜑 → (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀) = (𝐽(.g‘(mulGrp‘𝐾))𝑀))
72		aks6d1c2.27	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 = (𝐼 + (𝑈 · 𝑅)))
73	72	oveq1d 7446	. . . . . . . . 9 ⊢ (𝜑 → (𝐽(.g‘(mulGrp‘𝐾))𝑀) = ((𝐼 + (𝑈 · 𝑅))(.g‘(mulGrp‘𝐾))𝑀))
74	42	fldcrngd 20759	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ CRing)
75		crngring 20263	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring)
76	74, 75	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Ring)
77		eqid 2735	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
78	77	ringmgp 20257	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Ring → (mulGrp‘𝐾) ∈ Mnd)
79	76, 78	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd)
80		aks6d1c2p3.8	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ ℕ0)
81		aks6d1c2.26	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ ℕ)
82	81	nnnn0d 12585	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ ℕ0)
83	44	nnnn0d 12585	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
84	82, 83	nn0mulcld 12590	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑈 · 𝑅) ∈ ℕ0)
85	77	crngmgp 20259	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
86	74, 85	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ CMnd)
87		eqid 2735	. . . . . . . . . . . . . . . 16 ⊢ (.g‘(mulGrp‘𝐾)) = (.g‘(mulGrp‘𝐾))
88	86, 83, 87	isprimroot 42075	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅) ↔ (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) ∧ ∀𝑣 ∈ ℕ0 ((𝑣(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) → 𝑅 ∥ 𝑣))))
89	88	biimpd 229	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅) → (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) ∧ ∀𝑣 ∈ ℕ0 ((𝑣(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) → 𝑅 ∥ 𝑣))))
90	66, 89	mpd 15	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) ∧ ∀𝑣 ∈ ℕ0 ((𝑣(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) → 𝑅 ∥ 𝑣)))
91	90	simp1d 1141	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (Base‘(mulGrp‘𝐾)))
92	80, 84, 91	3jca 1127	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐼 ∈ ℕ0 ∧ (𝑈 · 𝑅) ∈ ℕ0 ∧ 𝑀 ∈ (Base‘(mulGrp‘𝐾))))
93		eqid 2735	. . . . . . . . . . . 12 ⊢ (Base‘(mulGrp‘𝐾)) = (Base‘(mulGrp‘𝐾))
94		eqid 2735	. . . . . . . . . . . 12 ⊢ (+g‘(mulGrp‘𝐾)) = (+g‘(mulGrp‘𝐾))
95	93, 87, 94	mulgnn0dir 19135	. . . . . . . . . . 11 ⊢ (((mulGrp‘𝐾) ∈ Mnd ∧ (𝐼 ∈ ℕ0 ∧ (𝑈 · 𝑅) ∈ ℕ0 ∧ 𝑀 ∈ (Base‘(mulGrp‘𝐾)))) → ((𝐼 + (𝑈 · 𝑅))(.g‘(mulGrp‘𝐾))𝑀) = ((𝐼(.g‘(mulGrp‘𝐾))𝑀)(+g‘(mulGrp‘𝐾))((𝑈 · 𝑅)(.g‘(mulGrp‘𝐾))𝑀)))
96	79, 92, 95	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐼 + (𝑈 · 𝑅))(.g‘(mulGrp‘𝐾))𝑀) = ((𝐼(.g‘(mulGrp‘𝐾))𝑀)(+g‘(mulGrp‘𝐾))((𝑈 · 𝑅)(.g‘(mulGrp‘𝐾))𝑀)))
97	82, 83, 91	3jca 1127	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑈 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ∈ (Base‘(mulGrp‘𝐾))))
98	93, 87	mulgnn0ass 19141	. . . . . . . . . . . . . 14 ⊢ (((mulGrp‘𝐾) ∈ Mnd ∧ (𝑈 ∈ ℕ0 ∧ 𝑅 ∈ ℕ0 ∧ 𝑀 ∈ (Base‘(mulGrp‘𝐾)))) → ((𝑈 · 𝑅)(.g‘(mulGrp‘𝐾))𝑀) = (𝑈(.g‘(mulGrp‘𝐾))(𝑅(.g‘(mulGrp‘𝐾))𝑀)))
99	79, 97, 98	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑈 · 𝑅)(.g‘(mulGrp‘𝐾))𝑀) = (𝑈(.g‘(mulGrp‘𝐾))(𝑅(.g‘(mulGrp‘𝐾))𝑀)))
100	90	simp2d 1142	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑅(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)))
101	100	oveq2d 7447	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑈(.g‘(mulGrp‘𝐾))(𝑅(.g‘(mulGrp‘𝐾))𝑀)) = (𝑈(.g‘(mulGrp‘𝐾))(0g‘(mulGrp‘𝐾))))
102		eqid 2735	. . . . . . . . . . . . . . . 16 ⊢ (0g‘(mulGrp‘𝐾)) = (0g‘(mulGrp‘𝐾))
103	93, 87, 102	mulgnn0z 19132	. . . . . . . . . . . . . . 15 ⊢ (((mulGrp‘𝐾) ∈ Mnd ∧ 𝑈 ∈ ℕ0) → (𝑈(.g‘(mulGrp‘𝐾))(0g‘(mulGrp‘𝐾))) = (0g‘(mulGrp‘𝐾)))
104	79, 82, 103	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑈(.g‘(mulGrp‘𝐾))(0g‘(mulGrp‘𝐾))) = (0g‘(mulGrp‘𝐾)))
105	101, 104	eqtrd 2775	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑈(.g‘(mulGrp‘𝐾))(𝑅(.g‘(mulGrp‘𝐾))𝑀)) = (0g‘(mulGrp‘𝐾)))
106	99, 105	eqtrd 2775	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑈 · 𝑅)(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)))
107	106	oveq2d 7447	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐼(.g‘(mulGrp‘𝐾))𝑀)(+g‘(mulGrp‘𝐾))((𝑈 · 𝑅)(.g‘(mulGrp‘𝐾))𝑀)) = ((𝐼(.g‘(mulGrp‘𝐾))𝑀)(+g‘(mulGrp‘𝐾))(0g‘(mulGrp‘𝐾))))
108	93, 87, 79, 80, 91	mulgnn0cld 19126	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐼(.g‘(mulGrp‘𝐾))𝑀) ∈ (Base‘(mulGrp‘𝐾)))
109	93, 94, 102	mndrid 18781	. . . . . . . . . . . 12 ⊢ (((mulGrp‘𝐾) ∈ Mnd ∧ (𝐼(.g‘(mulGrp‘𝐾))𝑀) ∈ (Base‘(mulGrp‘𝐾))) → ((𝐼(.g‘(mulGrp‘𝐾))𝑀)(+g‘(mulGrp‘𝐾))(0g‘(mulGrp‘𝐾))) = (𝐼(.g‘(mulGrp‘𝐾))𝑀))
110	79, 108, 109	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐼(.g‘(mulGrp‘𝐾))𝑀)(+g‘(mulGrp‘𝐾))(0g‘(mulGrp‘𝐾))) = (𝐼(.g‘(mulGrp‘𝐾))𝑀))
111	107, 110	eqtrd 2775	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐼(.g‘(mulGrp‘𝐾))𝑀)(+g‘(mulGrp‘𝐾))((𝑈 · 𝑅)(.g‘(mulGrp‘𝐾))𝑀)) = (𝐼(.g‘(mulGrp‘𝐾))𝑀))
112	96, 111	eqtrd 2775	. . . . . . . . 9 ⊢ (𝜑 → ((𝐼 + (𝑈 · 𝑅))(.g‘(mulGrp‘𝐾))𝑀) = (𝐼(.g‘(mulGrp‘𝐾))𝑀))
113	73, 112	eqtrd 2775	. . . . . . . 8 ⊢ (𝜑 → (𝐽(.g‘(mulGrp‘𝐾))𝑀) = (𝐼(.g‘(mulGrp‘𝐾))𝑀))
114	71, 113	eqtr2d 2776	. . . . . . 7 ⊢ (𝜑 → (𝐼(.g‘(mulGrp‘𝐾))𝑀) = (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀))
115	69, 114	eqtrd 2775	. . . . . 6 ⊢ (𝜑 → (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑀) = (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀))
116	115	fveq2d 6911	. . . . 5 ⊢ (𝜑 → (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀)))
117	35	oveq2d 7447	. . . . . . . 8 ⊢ (𝑦 = 𝑀 → (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦)) = (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)))
118		oveq2 7439	. . . . . . . . 9 ⊢ (𝑦 = 𝑀 → (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑦) = (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀))
119	118	fveq2d 6911	. . . . . . . 8 ⊢ (𝑦 = 𝑀 → (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀)))
120	117, 119	eqeq12d 2751	. . . . . . 7 ⊢ (𝑦 = 𝑀 → ((((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑦)) ↔ (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀))))
121		eqid 2735	. . . . . . . . 9 ⊢ ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) = ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))
122	40, 41, 42, 43, 44, 45, 46, 47, 54, 55, 56, 11, 14, 121, 58, 59	aks6d1c1rh 42107	. . . . . . . 8 ⊢ (𝜑 → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∼ (𝐺‘𝑠))
123	40, 122	aks6d1c1p1rcl 42090	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ ℕ ∧ (𝐺‘𝑠) ∈ (Base‘(Poly1‘𝐾))))
124	123	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∈ ℕ)
125	40, 62, 124	aks6d1c1p1 42089	. . . . . . . 8 ⊢ (𝜑 → (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜)) ∼ (𝐺‘𝑠) ↔ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑦))))
126	122, 125	mpbid 232	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑦)))
127	120, 126, 66	rspcdva 3623	. . . . . 6 ⊢ (𝜑 → (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀)))
128	127	eqcomd 2741	. . . . 5 ⊢ (𝜑 → (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))𝑀)) = (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)))
129	116, 128	eqtrd 2775	. . . 4 ⊢ (𝜑 → (((eval1‘𝐾)‘(𝐺‘𝑠))‘(((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))𝑀)) = (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)))
130	67, 129	eqtrd 2775	. . 3 ⊢ (𝜑 → (((𝑃↑𝑝) · ((𝑁 / 𝑃)↑𝑞))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)))
131	34, 130	eqtr2d 2776	. 2 ⊢ (𝜑 → (((𝑃↑𝑟) · ((𝑁 / 𝑃)↑𝑜))(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (𝐼(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)))
132	18, 131	eqtrd 2775	1 ⊢ (𝜑 → (𝐽(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)) = (𝐼(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑠))‘𝑀)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   class class class wbr 5148  {copab 5210   ↦ cmpt 5231   × cxp 5687   “ cima 5692  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433   ↑_m cmap 8865  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158   < clt 11293   / cdiv 11918  ℕcn 12264  ℕ₀cn0 12524  ...cfz 13544  ⌊cfl 13827  ↑cexp 14099  ♯chash 14366  √csqrt 15269   ∥ cdvds 16287   gcd cgcd 16528  ℙcprime 16705  Basecbs 17245  +_gcplusg 17298  0_gc0g 17486   Σ_g cgsu 17487  Mndcmnd 18760  -_gcsg 18966  ._gcmg 19098  CMndccmn 19813  mulGrpcmgp 20152  Ringcrg 20251  CRingccrg 20252   RingIso crs 20487  Fieldcfield 20747  ℤRHomczrh 21528  chrcchr 21530  ℤ/nℤczn 21531  algSccascl 21890  var₁cv1 22193  Poly₁cpl1 22194  eval₁ce1 22334   PrimRoots cprimroots 42073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232  ax-mulf 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-xnn0 12598  df-z 12612  df-dec 12732  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-fac 14310  df-bc 14339  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-dvds 16288  df-gcd 16529  df-prm 16706  df-phi 16800  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-od 19561  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-srg 20205  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-dvr 20418  df-rhm 20489  df-rim 20490  df-subrng 20563  df-subrg 20587  df-drng 20748  df-field 20749  df-lmod 20877  df-lss 20948  df-lsp 20988  df-cnfld 21383  df-zring 21476  df-zrh 21532  df-chr 21534  df-assa 21891  df-asp 21892  df-ascl 21893  df-psr 21947  df-mvr 21948  df-mpl 21949  df-opsr 21951  df-evls 22116  df-evl 22117  df-psr1 22197  df-vr1 22198  df-ply1 22199  df-coe1 22200  df-evl1 22336  df-primroots 42074

This theorem is referenced by:  aks6d1c2lem4  42109