Theorem aks6d1c6lem2 41774

Description: Every primitive root is root of G(u)-G(v). (Contributed by metakunt, 8-May-2025.)

Hypotheses

Ref	Expression
aks6d1c6.1	⊢ ∼ = {⟨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)))}
aks6d1c6.2	⊢ 𝑃 = (chr‘𝐾)
aks6d1c6.3	⊢ (𝜑 → 𝐾 ∈ Field)
aks6d1c6.4	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c6.5	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks6d1c6.6	⊢ (𝜑 → 𝑁 ∈ ℕ)
aks6d1c6.7	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks6d1c6.8	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c6.9	⊢ (𝜑 → 𝐴 < 𝑃)
aks6d1c6.10	⊢ 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)))))))
aks6d1c6.11	⊢ (𝜑 → 𝐴 ∈ ℕ0)
aks6d1c6.12	⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
aks6d1c6.13	⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
aks6d1c6.14	⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
aks6d1c6.15	⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
aks6d1c6.16	⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
aks6d1c6.17	⊢ 𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))
aks6d1c6.18	⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
aks6d1c6.19	⊢ 𝑆 = {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}
aks6d1c6lem2.1	⊢ (𝜑 → 𝑈 ∈ 𝑆)
aks6d1c6lem2.2	⊢ (𝜑 → 𝑉 ∈ 𝑆)
aks6d1c6lem2.3	⊢ (𝜑 → ((𝐻 ↾ 𝑆)‘𝑈) = ((𝐻 ↾ 𝑆)‘𝑉))
aks6d1c6lem2.4	⊢ (𝜑 → 𝑈 ≠ 𝑉)
aks6d1c6lem2.5	⊢ 𝐽 = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))
aks6d1c6lem2.6	⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(𝐽 “ (ℕ0 × ℕ0))))

Assertion

Ref	Expression
aks6d1c6lem2	⊢ (𝜑 → 𝐷 ≤ (♯‘(◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})))

Distinct variable groups:   ∼ ,𝑎   𝐴,𝑎   𝐴,𝑔,𝑖   𝐴,ℎ   𝐴,𝑠   𝑥,𝐴   𝑒,𝐸,𝑓,𝑦   𝑗,𝐸   𝑒,𝐺,𝑓,𝑦   ℎ,𝐺   𝐾,𝑎   𝑒,𝐾,𝑓,𝑦   𝑔,𝐾,𝑖   ℎ,𝐾   𝑗,𝐾   𝑥,𝐾   ℎ,𝑀   𝑗,𝑀   𝑦,𝑀   𝑁,𝑎   𝑒,𝑁,𝑓   𝑘,𝑁,𝑙,𝑠   𝑥,𝑁   𝑃,𝑒,𝑓   𝑃,𝑘,𝑙,𝑠   𝑥,𝑃   𝑅,𝑒,𝑓,𝑦   𝑥,𝑅   𝑆,ℎ   𝑈,𝑒,𝑓,𝑦   𝑈,𝑔,𝑖   𝑈,ℎ   𝑒,𝑉,𝑓,𝑦   𝑔,𝑉,𝑖   ℎ,𝑉   𝜑,𝑎   𝜑,𝑔,𝑖   𝜑,ℎ   𝜑,𝑗   𝜑,𝑠   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑦,𝑡,𝑒,𝑓,𝑘,𝑙)   𝐴(𝑦,𝑡,𝑒,𝑓,𝑗,𝑘,𝑙)   𝐷(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑗,𝑘,𝑠,𝑎,𝑙)   𝑃(𝑦,𝑡,𝑔,ℎ,𝑖,𝑗,𝑎)   ∼ (𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑗,𝑘,𝑠,𝑙)   𝑅(𝑡,𝑔,ℎ,𝑖,𝑗,𝑘,𝑠,𝑎,𝑙)   𝑆(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,𝑖,𝑗,𝑘,𝑠,𝑎,𝑙)   𝑈(𝑥,𝑡,𝑗,𝑘,𝑠,𝑎,𝑙)   𝐸(𝑥,𝑡,𝑔,ℎ,𝑖,𝑘,𝑠,𝑎,𝑙)   𝐺(𝑥,𝑡,𝑔,𝑖,𝑗,𝑘,𝑠,𝑎,𝑙)   𝐻(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑗,𝑘,𝑠,𝑎,𝑙)   𝐽(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑗,𝑘,𝑠,𝑎,𝑙)   𝐾(𝑡,𝑘,𝑠,𝑙)   𝐿(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑗,𝑘,𝑠,𝑎,𝑙)   𝑀(𝑥,𝑡,𝑒,𝑓,𝑔,𝑖,𝑘,𝑠,𝑎,𝑙)   𝑁(𝑦,𝑡,𝑔,ℎ,𝑖,𝑗)   𝑉(𝑥,𝑡,𝑗,𝑘,𝑠,𝑎,𝑙)

Proof of Theorem aks6d1c6lem2

Dummy variables 𝑤 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aks6d1c6.18	. . 3 ⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
2		aks6d1c6.13	. . . . . 6 ⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
3		fvexd 6911	. . . . . 6 ⊢ (𝜑 → (ℤRHom‘(ℤ/nℤ‘𝑅)) ∈ V)
4	2, 3	eqeltrid 2829	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ V)
5	4	imaexd 7924	. . . 4 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ V)
6		hashxrcl 14352	. . . 4 ⊢ ((𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ V → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℝ*)
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℝ*)
8	1, 7	eqeltrid 2829	. 2 ⊢ (𝜑 → 𝐷 ∈ ℝ*)
9		aks6d1c6lem2.5	. . . . . 6 ⊢ 𝐽 = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀))
10	9	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐽 = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)))
11		nn0ex 12511	. . . . . . . 8 ⊢ ℕ0 ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ0 ∈ V)
13	12, 12	xpexd 7754	. . . . . 6 ⊢ (𝜑 → (ℕ0 × ℕ0) ∈ V)
14	13	mptexd 7236	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) ∈ V)
15	10, 14	eqeltrd 2825	. . . 4 ⊢ (𝜑 → 𝐽 ∈ V)
16	15	imaexd 7924	. . 3 ⊢ (𝜑 → (𝐽 “ (ℕ0 × ℕ0)) ∈ V)
17		hashxrcl 14352	. . 3 ⊢ ((𝐽 “ (ℕ0 × ℕ0)) ∈ V → (♯‘(𝐽 “ (ℕ0 × ℕ0))) ∈ ℝ*)
18	16, 17	syl 17	. 2 ⊢ (𝜑 → (♯‘(𝐽 “ (ℕ0 × ℕ0))) ∈ ℝ*)
19		fvexd 6911	. . . . 5 ⊢ (𝜑 → ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) ∈ V)
20		cnvexg 7932	. . . . 5 ⊢ (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) ∈ V → ◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) ∈ V)
21	19, 20	syl 17	. . . 4 ⊢ (𝜑 → ◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) ∈ V)
22	21	imaexd 7924	. . 3 ⊢ (𝜑 → (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}) ∈ V)
23		hashxrcl 14352	. . 3 ⊢ ((◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}) ∈ V → (♯‘(◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})) ∈ ℝ*)
24	22, 23	syl 17	. 2 ⊢ (𝜑 → (♯‘(◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})) ∈ ℝ*)
25	1	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))
26		aks6d1c6lem2.6	. . 3 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(𝐽 “ (ℕ0 × ℕ0))))
27	25, 26	eqbrtrd 5171	. 2 ⊢ (𝜑 → 𝐷 ≤ (♯‘(𝐽 “ (ℕ0 × ℕ0))))
28	22	elexd 3483	. . 3 ⊢ (𝜑 → (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}) ∈ V)
29		nfv 1909	. . . 4 ⊢ Ⅎ𝑤𝜑
30		ovexd 7454	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
31	30, 9	fmptd 7123	. . . . 5 ⊢ (𝜑 → 𝐽:(ℕ0 × ℕ0)⟶V)
32		ffun 6726	. . . . 5 ⊢ (𝐽:(ℕ0 × ℕ0)⟶V → Fun 𝐽)
33	31, 32	syl 17	. . . 4 ⊢ (𝜑 → Fun 𝐽)
34	9	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝐽 = (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)))
35		simpr 483	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑤) → 𝑗 = 𝑤)
36	35	fveq2d 6900	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑤) → (𝐸‘𝑗) = (𝐸‘𝑤))
37	36	oveq1d 7434	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑤) → ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))
38		simpr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝑤 ∈ (ℕ0 × ℕ0))
39		ovexd 7454	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
40	34, 37, 38, 39	fvmptd 7011	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝐽‘𝑤) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))
41		eqid 2725	. . . . . . . . . 10 ⊢ (eval1‘𝐾) = (eval1‘𝐾)
42		eqid 2725	. . . . . . . . . 10 ⊢ (Poly1‘𝐾) = (Poly1‘𝐾)
43		eqid 2725	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
44		eqid 2725	. . . . . . . . . 10 ⊢ (Base‘(Poly1‘𝐾)) = (Base‘(Poly1‘𝐾))
45		aks6d1c6.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Field)
46	45	fldcrngd 20649	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ CRing)
47	46	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝐾 ∈ CRing)
48		eqid 2725	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
49	48, 43	mgpbas 20092	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
50		eqid 2725	. . . . . . . . . . 11 ⊢ (.g‘(mulGrp‘𝐾)) = (.g‘(mulGrp‘𝐾))
51	46	crngringd 20198	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Ring)
52	48	ringmgp 20191	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Ring → (mulGrp‘𝐾) ∈ Mnd)
53	51, 52	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd)
54	53	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (mulGrp‘𝐾) ∈ Mnd)
55		aks6d1c6.6	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ)
56		aks6d1c6.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ ℙ)
57		aks6d1c6.7	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
58		aks6d1c6.12	. . . . . . . . . . . . . 14 ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
59	55, 56, 57, 58	aks6d1c2p1 41721	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶ℕ)
60	59	ffvelcdmda 7093	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑤) ∈ ℕ)
61	60	nnnn0d 12565	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑤) ∈ ℕ0)
62		aks6d1c6.16	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
63	48	crngmgp 20193	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
64	46, 63	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ CMnd)
65		aks6d1c6.5	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 ∈ ℕ)
66	65	nnnn0d 12565	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
67	64, 66, 50	isprimroot 41696	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅) ↔ (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) ∧ ∀𝑜 ∈ ℕ0 ((𝑜(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) → 𝑅 ∥ 𝑜))))
68	62, 67	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 ∈ (Base‘(mulGrp‘𝐾)) ∧ (𝑅(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) ∧ ∀𝑜 ∈ ℕ0 ((𝑜(.g‘(mulGrp‘𝐾))𝑀) = (0g‘(mulGrp‘𝐾)) → 𝑅 ∥ 𝑜)))
69	68	simp1d 1139	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (Base‘(mulGrp‘𝐾)))
70	69, 49	eleqtrrdi 2836	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (Base‘𝐾))
71	70	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝑀 ∈ (Base‘𝐾))
72	49, 50, 54, 61, 71	mulgnn0cld 19058	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ (Base‘𝐾))
73		aks6d1c6.2	. . . . . . . . . . . . . 14 ⊢ 𝑃 = (chr‘𝐾)
74		aks6d1c6.11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℕ0)
75		aks6d1c6.9	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 < 𝑃)
76		eqid 2725	. . . . . . . . . . . . . 14 ⊢ (var1‘𝐾) = (var1‘𝐾)
77		eqid 2725	. . . . . . . . . . . . . 14 ⊢ (.g‘(mulGrp‘(Poly1‘𝐾))) = (.g‘(mulGrp‘(Poly1‘𝐾)))
78		aks6d1c6.10	. . . . . . . . . . . . . 14 ⊢ 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)))))))
79	45, 56, 73, 74, 75, 76, 77, 78	aks6d1c5lem0 41738	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:(ℕ0 ↑m (0...𝐴))⟶(Base‘(Poly1‘𝐾)))
80		aks6d1c6lem2.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
81		aks6d1c6.19	. . . . . . . . . . . . . . . 16 ⊢ 𝑆 = {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}
82	81	eleq2i 2817	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ 𝑆 ↔ 𝑈 ∈ {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)})
83	80, 82	sylib 217	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)})
84		elrabi 3673	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)} → 𝑈 ∈ (ℕ0 ↑m (0...𝐴)))
85	84	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑈 ∈ {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)} → 𝑈 ∈ (ℕ0 ↑m (0...𝐴))))
86	83, 85	mpd 15	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ (ℕ0 ↑m (0...𝐴)))
87	79, 86	ffvelcdmd 7094	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺‘𝑈) ∈ (Base‘(Poly1‘𝐾)))
88	87	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝐺‘𝑈) ∈ (Base‘(Poly1‘𝐾)))
89		eqidd 2726	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
90	88, 89	jca 510	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐺‘𝑈) ∈ (Base‘(Poly1‘𝐾)) ∧ (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))))
91		aks6d1c6lem2.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉 ∈ 𝑆)
92	81	eleq2i 2817	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ 𝑆 ↔ 𝑉 ∈ {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)})
93	91, 92	sylib 217	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ∈ {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)})
94		elrabi 3673	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)} → 𝑉 ∈ (ℕ0 ↑m (0...𝐴)))
95	94	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑉 ∈ {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)} → 𝑉 ∈ (ℕ0 ↑m (0...𝐴))))
96	93, 95	mpd 15	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑉 ∈ (ℕ0 ↑m (0...𝐴)))
97	79, 96	ffvelcdmd 7094	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺‘𝑉) ∈ (Base‘(Poly1‘𝐾)))
98	97	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝐺‘𝑉) ∈ (Base‘(Poly1‘𝐾)))
99		eqidd 2726	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
100	98, 99	jca 510	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐺‘𝑉) ∈ (Base‘(Poly1‘𝐾)) ∧ (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))))
101		eqid 2725	. . . . . . . . . 10 ⊢ (-g‘(Poly1‘𝐾)) = (-g‘(Poly1‘𝐾))
102		eqid 2725	. . . . . . . . . 10 ⊢ (-g‘𝐾) = (-g‘𝐾)
103	41, 42, 43, 44, 47, 72, 90, 100, 101, 102	evl1subd 22286	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)) ∈ (Base‘(Poly1‘𝐾)) ∧ (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = ((((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))(-g‘𝐾)(((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)))))
104	103	simprd 494	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = ((((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))(-g‘𝐾)(((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))))
105		fveq2 6896	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑀 → (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑦) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀))
106	105	oveq2d 7435	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑀 → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑦)) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀)))
107		oveq2 7427	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑀 → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))
108	107	fveq2d 6900	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑀 → (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
109	106, 108	eqeq12d 2741	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑀 → (((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦)) ↔ ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))))
110		vex 3465	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑘 ∈ V
111		vex 3465	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑙 ∈ V
112	110, 111	op1std 8004	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 = ⟨𝑘, 𝑙⟩ → (1st ‘𝑠) = 𝑘)
113	112	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = ⟨𝑘, 𝑙⟩ → (𝑃↑(1st ‘𝑠)) = (𝑃↑𝑘))
114	110, 111	op2ndd 8005	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 = ⟨𝑘, 𝑙⟩ → (2nd ‘𝑠) = 𝑙)
115	114	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = ⟨𝑘, 𝑙⟩ → ((𝑁 / 𝑃)↑(2nd ‘𝑠)) = ((𝑁 / 𝑃)↑𝑙))
116	113, 115	oveq12d 7437	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = ⟨𝑘, 𝑙⟩ → ((𝑃↑(1st ‘𝑠)) · ((𝑁 / 𝑃)↑(2nd ‘𝑠))) = ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
117	116	mpompt 7534	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑠)) · ((𝑁 / 𝑃)↑(2nd ‘𝑠)))) = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
118	58	eqcomi 2734	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) = 𝐸
119	117, 118	eqtri 2753	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑠)) · ((𝑁 / 𝑃)↑(2nd ‘𝑠)))) = 𝐸
120	119	eqcomi 2734	. . . . . . . . . . . . . . . . 17 ⊢ 𝐸 = (𝑠 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑠)) · ((𝑁 / 𝑃)↑(2nd ‘𝑠))))
121	120	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝐸 = (𝑠 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑠)) · ((𝑁 / 𝑃)↑(2nd ‘𝑠)))))
122		simpr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → 𝑠 = 𝑤)
123	122	fveq2d 6900	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → (1st ‘𝑠) = (1st ‘𝑤))
124	123	oveq2d 7435	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → (𝑃↑(1st ‘𝑠)) = (𝑃↑(1st ‘𝑤)))
125	122	fveq2d 6900	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → (2nd ‘𝑠) = (2nd ‘𝑤))
126	125	oveq2d 7435	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → ((𝑁 / 𝑃)↑(2nd ‘𝑠)) = ((𝑁 / 𝑃)↑(2nd ‘𝑤)))
127	124, 126	oveq12d 7437	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → ((𝑃↑(1st ‘𝑠)) · ((𝑁 / 𝑃)↑(2nd ‘𝑠))) = ((𝑃↑(1st ‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤))))
128		ovexd 7454	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤))) ∈ V)
129	121, 127, 38, 128	fvmptd 7011	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑤) = ((𝑃↑(1st ‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤))))
130		aks6d1c6.1	. . . . . . . . . . . . . . . 16 ⊢ ∼ = {⟨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)))}
131	45	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝐾 ∈ Field)
132	56	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝑃 ∈ ℙ)
133	65	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝑅 ∈ ℕ)
134	55	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝑁 ∈ ℕ)
135	57	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝑃 ∥ 𝑁)
136		aks6d1c6.8	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
137	136	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝑁 gcd 𝑅) = 1)
138		ovexd 7454	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (0...𝐴) ∈ V)
139	12, 138	elmapd 8859	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑈 ∈ (ℕ0 ↑m (0...𝐴)) ↔ 𝑈:(0...𝐴)⟶ℕ0))
140	86, 139	mpbid 231	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑈:(0...𝐴)⟶ℕ0)
141	140	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝑈:(0...𝐴)⟶ℕ0)
142	74	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝐴 ∈ ℕ0)
143		xp1st 8026	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (ℕ0 × ℕ0) → (1st ‘𝑤) ∈ ℕ0)
144	143	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (1st ‘𝑤) ∈ ℕ0)
145		xp2nd 8027	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑤) ∈ ℕ0)
146	145	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (2nd ‘𝑤) ∈ ℕ0)
147		eqid 2725	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃↑(1st ‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤))) = ((𝑃↑(1st ‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤)))
148		aks6d1c6.14	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
149	148	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
150		aks6d1c6.15	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
151	150	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
152	130, 73, 131, 132, 133, 134, 135, 137, 141, 78, 142, 144, 146, 147, 149, 151	aks6d1c1rh 41728	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤))) ∼ (𝐺‘𝑈))
153	129, 152	eqbrtrd 5171	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑤) ∼ (𝐺‘𝑈))
154	130, 88, 60	aks6d1c1p1 41710	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑤) ∼ (𝐺‘𝑈) ↔ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦))))
155	153, 154	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦)))
156	62	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
157	109, 155, 156	rspcdva 3607	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
158	157	eqcomd 2731	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀)))
159		aks6d1c6.17	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))
160	159	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀)))
161	160	reseq1d 5984	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐻 ↾ 𝑆) = ((ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀)) ↾ 𝑆))
162	81	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑆 = {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)})
163		ssrab2 4073	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)} ⊆ (ℕ0 ↑m (0...𝐴))
164	163	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)} ⊆ (ℕ0 ↑m (0...𝐴)))
165	162, 164	eqsstrd 4015	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑆 ⊆ (ℕ0 ↑m (0...𝐴)))
166	165	resmptd 6045	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀)) ↾ 𝑆) = (ℎ ∈ 𝑆 ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀)))
167	161, 166	eqtrd 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐻 ↾ 𝑆) = (ℎ ∈ 𝑆 ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀)))
168		simpr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ℎ = 𝑈) → ℎ = 𝑈)
169	168	fveq2d 6900	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ℎ = 𝑈) → (𝐺‘ℎ) = (𝐺‘𝑈))
170	169	fveq2d 6900	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ℎ = 𝑈) → ((eval1‘𝐾)‘(𝐺‘ℎ)) = ((eval1‘𝐾)‘(𝐺‘𝑈)))
171	170	fveq1d 6898	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ℎ = 𝑈) → (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀))
172		fvexd 6911	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀) ∈ V)
173	167, 171, 80, 172	fvmptd 7011	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐻 ↾ 𝑆)‘𝑈) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀))
174	173	eqcomd 2731	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀) = ((𝐻 ↾ 𝑆)‘𝑈))
175		aks6d1c6lem2.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐻 ↾ 𝑆)‘𝑈) = ((𝐻 ↾ 𝑆)‘𝑉))
176		simpr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ℎ = 𝑉) → ℎ = 𝑉)
177	176	fveq2d 6900	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ℎ = 𝑉) → (𝐺‘ℎ) = (𝐺‘𝑉))
178	177	fveq2d 6900	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ℎ = 𝑉) → ((eval1‘𝐾)‘(𝐺‘ℎ)) = ((eval1‘𝐾)‘(𝐺‘𝑉)))
179	178	fveq1d 6898	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ℎ = 𝑉) → (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀))
180		fvexd 6911	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀) ∈ V)
181	167, 179, 91, 180	fvmptd 7011	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐻 ↾ 𝑆)‘𝑉) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀))
182	174, 175, 181	3eqtrd 2769	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀))
183	182	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀))
184	183	oveq2d 7435	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀)) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀)))
185	158, 184	eqtrd 2765	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀)))
186		fveq2 6896	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑀 → (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑦) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀))
187	186	oveq2d 7435	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑀 → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑦)) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀)))
188	107	fveq2d 6900	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑀 → (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
189	187, 188	eqeq12d 2741	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑀 → (((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦)) ↔ ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))))
190	12, 138	elmapd 8859	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑉 ∈ (ℕ0 ↑m (0...𝐴)) ↔ 𝑉:(0...𝐴)⟶ℕ0))
191	96, 190	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉:(0...𝐴)⟶ℕ0)
192	191	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝑉:(0...𝐴)⟶ℕ0)
193	130, 73, 131, 132, 133, 134, 135, 137, 192, 78, 142, 144, 146, 147, 149, 151	aks6d1c1rh 41728	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤))) ∼ (𝐺‘𝑉))
194	129, 193	eqbrtrd 5171	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑤) ∼ (𝐺‘𝑉))
195	130, 98, 60	aks6d1c1p1 41710	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑤) ∼ (𝐺‘𝑉) ↔ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦))))
196	194, 195	mpbid 231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑦)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦)))
197	189, 196, 156	rspcdva 3607	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))(((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
198	185, 197	eqtrd 2765	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)))
199	46	crnggrpd 20199	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ Grp)
200	199	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝐾 ∈ Grp)
201	41, 42, 43, 44, 47, 72, 88	fveval1fvcl 22277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ (Base‘𝐾))
202	41, 42, 43, 44, 47, 72, 98	fveval1fvcl 22277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ (Base‘𝐾))
203		eqid 2725	. . . . . . . . . . 11 ⊢ (0g‘𝐾) = (0g‘𝐾)
204	43, 203, 102	grpsubeq0 18990	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Grp ∧ (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ (Base‘𝐾) ∧ (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ (Base‘𝐾)) → (((((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))(-g‘𝐾)(((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))) = (0g‘𝐾) ↔ (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))))
205	200, 201, 202, 204	syl3anc 1368	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))(-g‘𝐾)(((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))) = (0g‘𝐾) ↔ (((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))))
206	198, 205	mpbird 256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((((eval1‘𝐾)‘(𝐺‘𝑈))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))(-g‘𝐾)(((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))) = (0g‘𝐾))
207	104, 206	eqtrd 2765	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (0g‘𝐾))
208		fvexd 6911	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ V)
209		elsng 4644	. . . . . . . 8 ⊢ ((((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ V → ((((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ {(0g‘𝐾)} ↔ (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (0g‘𝐾)))
210	208, 209	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ {(0g‘𝐾)} ↔ (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (0g‘𝐾)))
211	207, 210	mpbird 256	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ {(0g‘𝐾)})
212		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ↑s (Base‘𝐾)) = (𝐾 ↑s (Base‘𝐾))
213	41, 42, 212, 43	evl1rhm 22276	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ CRing → (eval1‘𝐾) ∈ ((Poly1‘𝐾) RingHom (𝐾 ↑s (Base‘𝐾))))
214	46, 213	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (eval1‘𝐾) ∈ ((Poly1‘𝐾) RingHom (𝐾 ↑s (Base‘𝐾))))
215		eqid 2725	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝐾 ↑s (Base‘𝐾))) = (Base‘(𝐾 ↑s (Base‘𝐾)))
216	44, 215	rhmf 20436	. . . . . . . . . . . . 13 ⊢ ((eval1‘𝐾) ∈ ((Poly1‘𝐾) RingHom (𝐾 ↑s (Base‘𝐾))) → (eval1‘𝐾):(Base‘(Poly1‘𝐾))⟶(Base‘(𝐾 ↑s (Base‘𝐾))))
217	214, 216	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (eval1‘𝐾):(Base‘(Poly1‘𝐾))⟶(Base‘(𝐾 ↑s (Base‘𝐾))))
218		fvexd 6911	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Base‘𝐾) ∈ V)
219	212, 43	pwsbas 17472	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Field ∧ (Base‘𝐾) ∈ V) → ((Base‘𝐾) ↑m (Base‘𝐾)) = (Base‘(𝐾 ↑s (Base‘𝐾))))
220	45, 218, 219	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((Base‘𝐾) ↑m (Base‘𝐾)) = (Base‘(𝐾 ↑s (Base‘𝐾))))
221	220	feq3d 6710	. . . . . . . . . . . 12 ⊢ (𝜑 → ((eval1‘𝐾):(Base‘(Poly1‘𝐾))⟶((Base‘𝐾) ↑m (Base‘𝐾)) ↔ (eval1‘𝐾):(Base‘(Poly1‘𝐾))⟶(Base‘(𝐾 ↑s (Base‘𝐾)))))
222	217, 221	mpbird 256	. . . . . . . . . . 11 ⊢ (𝜑 → (eval1‘𝐾):(Base‘(Poly1‘𝐾))⟶((Base‘𝐾) ↑m (Base‘𝐾)))
223	42	ply1ring 22190	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Ring → (Poly1‘𝐾) ∈ Ring)
224	51, 223	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Poly1‘𝐾) ∈ Ring)
225		ringgrp 20190	. . . . . . . . . . . . 13 ⊢ ((Poly1‘𝐾) ∈ Ring → (Poly1‘𝐾) ∈ Grp)
226	224, 225	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (Poly1‘𝐾) ∈ Grp)
227	44, 101	grpsubcl 18984	. . . . . . . . . . . 12 ⊢ (((Poly1‘𝐾) ∈ Grp ∧ (𝐺‘𝑈) ∈ (Base‘(Poly1‘𝐾)) ∧ (𝐺‘𝑉) ∈ (Base‘(Poly1‘𝐾))) → ((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)) ∈ (Base‘(Poly1‘𝐾)))
228	226, 87, 97, 227	syl3anc 1368	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)) ∈ (Base‘(Poly1‘𝐾)))
229	222, 228	ffvelcdmd 7094	. . . . . . . . . 10 ⊢ (𝜑 → ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) ∈ ((Base‘𝐾) ↑m (Base‘𝐾)))
230	218, 218	elmapd 8859	. . . . . . . . . 10 ⊢ (𝜑 → (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) ∈ ((Base‘𝐾) ↑m (Base‘𝐾)) ↔ ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))):(Base‘𝐾)⟶(Base‘𝐾)))
231	229, 230	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))):(Base‘𝐾)⟶(Base‘𝐾))
232	231	ffund 6727	. . . . . . . 8 ⊢ (𝜑 → Fun ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))))
233	232	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → Fun ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))))
234	231	ffnd 6724	. . . . . . . . . . 11 ⊢ (𝜑 → ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) Fn (Base‘𝐾))
235	234	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) Fn (Base‘𝐾))
236	235	fndmd 6660	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → dom ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) = (Base‘𝐾))
237	236	eqcomd 2731	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (Base‘𝐾) = dom ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))))
238	72, 237	eleqtrd 2827	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ dom ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))))
239		fvimacnv 7061	. . . . . . 7 ⊢ ((Fun ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) ∧ ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ dom ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))) → ((((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ {(0g‘𝐾)} ↔ ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})))
240	233, 238, 239	syl2anc 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ {(0g‘𝐾)} ↔ ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})))
241	211, 240	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}))
242	40, 241	eqeltrd 2825	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝐽‘𝑤) ∈ (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}))
243	29, 33, 242	funimassd 6964	. . 3 ⊢ (𝜑 → (𝐽 “ (ℕ0 × ℕ0)) ⊆ (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}))
244		hashss 14404	. . 3 ⊢ (((◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}) ∈ V ∧ (𝐽 “ (ℕ0 × ℕ0)) ⊆ (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})) → (♯‘(𝐽 “ (ℕ0 × ℕ0))) ≤ (♯‘(◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})))
245	28, 243, 244	syl2anc 582	. 2 ⊢ (𝜑 → (♯‘(𝐽 “ (ℕ0 × ℕ0))) ≤ (♯‘(◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})))
246	8, 18, 24, 27, 245	xrletrd 13176	1 ⊢ (𝜑 → 𝐷 ≤ (♯‘(◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2929  ∀wral 3050  {crab 3418  Vcvv 3461   ⊆ wss 3944  {csn 4630  ⟨cop 4636   class class class wbr 5149  {copab 5211   ↦ cmpt 5232   × cxp 5676  ^◡ccnv 5677  dom cdm 5678   ↾ cres 5680   “ cima 5681  Fun wfun 6543   Fn wfn 6544  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419   ∈ cmpo 7421  1^st c1st 7992  2^nd c2nd 7993   ↑_m cmap 8845  0cc0 11140  1c1 11141   · cmul 11145  ℝ^*cxr 11279   < clt 11280   ≤ cle 11281   − cmin 11476   / cdiv 11903  ℕcn 12245  ℕ₀cn0 12505  ...cfz 13519  ↑cexp 14062  ♯chash 14325  Σcsu 15668   ∥ cdvds 16234   gcd cgcd 16472  ℙcprime 16645  Basecbs 17183  +_gcplusg 17236  0_gc0g 17424   Σ_g cgsu 17425   ↑_s cpws 17431  Mndcmnd 18697  Grpcgrp 18898  -_gcsg 18900  ._gcmg 19031  CMndccmn 19747  mulGrpcmgp 20086  Ringcrg 20185  CRingccrg 20186   RingHom crh 20420   RingIso crs 20421  Fieldcfield 20637  ℤRHomczrh 21442  chrcchr 21444  ℤ/nℤczn 21445  algSccascl 21803  var₁cv1 22118  Poly₁cpl1 22119  eval₁ce1 22258   PrimRoots cprimroots 41694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218  ax-addf 11219  ax-mulf 11220

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-ofr 7686  df-om 7872  df-1st 7994  df-2nd 7995  df-supp 8166  df-tpos 8232  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-oadd 8491  df-er 8725  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9388  df-sup 9467  df-inf 9468  df-oi 9535  df-dju 9926  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-xnn0 12578  df-z 12592  df-dec 12711  df-uz 12856  df-rp 13010  df-fz 13520  df-fzo 13663  df-fl 13793  df-mod 13871  df-seq 14003  df-exp 14063  df-fac 14269  df-bc 14298  df-hash 14326  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-dvds 16235  df-gcd 16473  df-prm 16646  df-phi 16738  df-struct 17119  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-starv 17251  df-sca 17252  df-vsca 17253  df-ip 17254  df-tset 17255  df-ple 17256  df-ds 17258  df-unif 17259  df-hom 17260  df-cco 17261  df-0g 17426  df-gsum 17427  df-prds 17432  df-pws 17434  df-mre 17569  df-mrc 17570  df-acs 17572  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-mhm 18743  df-submnd 18744  df-grp 18901  df-minusg 18902  df-sbg 18903  df-mulg 19032  df-subg 19086  df-ghm 19176  df-cntz 19280  df-od 19495  df-cmn 19749  df-abl 19750  df-mgp 20087  df-rng 20105  df-ur 20134  df-srg 20139  df-ring 20187  df-cring 20188  df-oppr 20285  df-dvdsr 20308  df-unit 20309  df-invr 20339  df-dvr 20352  df-rhm 20423  df-rim 20424  df-subrng 20495  df-subrg 20520  df-drng 20638  df-field 20639  df-lmod 20757  df-lss 20828  df-lsp 20868  df-cnfld 21297  df-zring 21390  df-zrh 21446  df-chr 21448  df-assa 21804  df-asp 21805  df-ascl 21806  df-psr 21859  df-mvr 21860  df-mpl 21861  df-opsr 21863  df-evls 22040  df-evl 22041  df-psr1 22122  df-vr1 22123  df-ply1 22124  df-coe1 22125  df-evl1 22260  df-primroots 41695

This theorem is referenced by:  aks6d1c6lem3  41775