Theorem aks6d1c6lem2 42610

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 6855	. . . . . 6 ⊢ (𝜑 → (ℤRHom‘(ℤ/nℤ‘𝑅)) ∈ V)
4	2, 3	eqeltrid 2840	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ V)
5	4	imaexd 7867	. . . 4 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ V)
6		hashxrcl 14319	. . . 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 2840	. 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 12443	. . . . . . . 8 ⊢ ℕ0 ∈ V
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℕ0 ∈ V)
13	12, 12	xpexd 7705	. . . . . 6 ⊢ (𝜑 → (ℕ0 × ℕ0) ∈ V)
14	13	mptexd 7179	. . . . 5 ⊢ (𝜑 → (𝑗 ∈ (ℕ0 × ℕ0) ↦ ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀)) ∈ V)
15	10, 14	eqeltrd 2836	. . . 4 ⊢ (𝜑 → 𝐽 ∈ V)
16	15	imaexd 7867	. . 3 ⊢ (𝜑 → (𝐽 “ (ℕ0 × ℕ0)) ∈ V)
17		hashxrcl 14319	. . 3 ⊢ ((𝐽 “ (ℕ0 × ℕ0)) ∈ V → (♯‘(𝐽 “ (ℕ0 × ℕ0))) ∈ ℝ*)
18	16, 17	syl 17	. 2 ⊢ (𝜑 → (♯‘(𝐽 “ (ℕ0 × ℕ0))) ∈ ℝ*)
19		fvexd 6855	. . . . 5 ⊢ (𝜑 → ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) ∈ V)
20		cnvexg 7875	. . . . 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 7867	. . 3 ⊢ (𝜑 → (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}) ∈ V)
23		hashxrcl 14319	. . 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 5107	. 2 ⊢ (𝜑 → 𝐷 ≤ (♯‘(𝐽 “ (ℕ0 × ℕ0))))
28	22	elexd 3453	. . 3 ⊢ (𝜑 → (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}) ∈ V)
29		nfv 1916	. . . 4 ⊢ Ⅎ𝑤𝜑
30		ovexd 7402	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
31	30, 9	fmptd 7066	. . . . 5 ⊢ (𝜑 → 𝐽:(ℕ0 × ℕ0)⟶V)
32		ffun 6671	. . . . 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 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑤) → 𝑗 = 𝑤)
36	35	fveq2d 6844	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑤) → (𝐸‘𝑗) = (𝐸‘𝑤))
37	36	oveq1d 7382	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑗 = 𝑤) → ((𝐸‘𝑗)(.g‘(mulGrp‘𝐾))𝑀) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))
38		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝑤 ∈ (ℕ0 × ℕ0))
39		ovexd 7402	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ V)
40	34, 37, 38, 39	fvmptd 6955	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝐽‘𝑤) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))
41		eqid 2736	. . . . . . . . . 10 ⊢ (eval1‘𝐾) = (eval1‘𝐾)
42		eqid 2736	. . . . . . . . . 10 ⊢ (Poly1‘𝐾) = (Poly1‘𝐾)
43		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
44		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘(Poly1‘𝐾)) = (Base‘(Poly1‘𝐾))
45		aks6d1c6.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Field)
46	45	fldcrngd 20719	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ CRing)
47	46	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝐾 ∈ CRing)
48		eqid 2736	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
49	48, 43	mgpbas 20126	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾))
50		eqid 2736	. . . . . . . . . . 11 ⊢ (.g‘(mulGrp‘𝐾)) = (.g‘(mulGrp‘𝐾))
51	46	crngringd 20227	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Ring)
52	48	ringmgp 20220	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Ring → (mulGrp‘𝐾) ∈ Mnd)
53	51, 52	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ Mnd)
54	53	adantr 480	. . . . . . . . . . 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 42557	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶ℕ)
60	59	ffvelcdmda 7036	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑤) ∈ ℕ)
61	60	nnnn0d 12498	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑤) ∈ ℕ0)
62		aks6d1c6.16	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
63	48	crngmgp 20222	. . . . . . . . . . . . . . . . 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 12498	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
67	64, 66, 50	isprimroot 42532	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 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 1143	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (Base‘(mulGrp‘𝐾)))
70	69, 49	eleqtrrdi 2847	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (Base‘𝐾))
71	70	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝑀 ∈ (Base‘𝐾))
72	49, 50, 54, 61, 71	mulgnn0cld 19071	. . . . . . . . . 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 2736	. . . . . . . . . . . . . 14 ⊢ (var1‘𝐾) = (var1‘𝐾)
77		eqid 2736	. . . . . . . . . . . . . 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 42574	. . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ 𝑆 ↔ 𝑈 ∈ {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)})
83	80, 82	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)})
84		elrabi 3630	. . . . . . . . . . . . . . 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 7037	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺‘𝑈) ∈ (Base‘(Poly1‘𝐾)))
88	87	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝐺‘𝑈) ∈ (Base‘(Poly1‘𝐾)))
89		eqidd 2737	. . . . . . . . . . 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 511	. . . . . . . . . 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 2828	. . . . . . . . . . . . . . 15 ⊢ (𝑉 ∈ 𝑆 ↔ 𝑉 ∈ {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)})
93	91, 92	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑉 ∈ {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)})
94		elrabi 3630	. . . . . . . . . . . . . . 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 7037	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺‘𝑉) ∈ (Base‘(Poly1‘𝐾)))
98	97	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝐺‘𝑉) ∈ (Base‘(Poly1‘𝐾)))
99		eqidd 2737	. . . . . . . . . . 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 511	. . . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ (-g‘(Poly1‘𝐾)) = (-g‘(Poly1‘𝐾))
102		eqid 2736	. . . . . . . . . 10 ⊢ (-g‘𝐾) = (-g‘𝐾)
103	41, 42, 43, 44, 47, 72, 90, 100, 101, 102	evl1subd 22307	. . . . . . . . 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 495	. . . . . . . 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 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑀 → (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑦) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀))
106	105	oveq2d 7383	. . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑀 → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑦) = ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀))
108	107	fveq2d 6844	. . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . 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 3433	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑘 ∈ V
111		vex 3433	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑙 ∈ V
112	110, 111	op1std 7952	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 = ⟨𝑘, 𝑙⟩ → (1st ‘𝑠) = 𝑘)
113	112	oveq2d 7383	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = ⟨𝑘, 𝑙⟩ → (𝑃↑(1st ‘𝑠)) = (𝑃↑𝑘))
114	110, 111	op2ndd 7953	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 = ⟨𝑘, 𝑙⟩ → (2nd ‘𝑠) = 𝑙)
115	114	oveq2d 7383	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 = ⟨𝑘, 𝑙⟩ → ((𝑁 / 𝑃)↑(2nd ‘𝑠)) = ((𝑁 / 𝑃)↑𝑙))
116	113, 115	oveq12d 7385	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = ⟨𝑘, 𝑙⟩ → ((𝑃↑(1st ‘𝑠)) · ((𝑁 / 𝑃)↑(2nd ‘𝑠))) = ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
117	116	mpompt 7481	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑠)) · ((𝑁 / 𝑃)↑(2nd ‘𝑠)))) = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
118	58	eqcomi 2745	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) = 𝐸
119	117, 118	eqtri 2759	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑠)) · ((𝑁 / 𝑃)↑(2nd ‘𝑠)))) = 𝐸
120	119	eqcomi 2745	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → 𝑠 = 𝑤)
123	122	fveq2d 6844	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → (1st ‘𝑠) = (1st ‘𝑤))
124	123	oveq2d 7383	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → (𝑃↑(1st ‘𝑠)) = (𝑃↑(1st ‘𝑤)))
125	122	fveq2d 6844	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → (2nd ‘𝑠) = (2nd ‘𝑤))
126	125	oveq2d 7383	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → ((𝑁 / 𝑃)↑(2nd ‘𝑠)) = ((𝑁 / 𝑃)↑(2nd ‘𝑤)))
127	124, 126	oveq12d 7385	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) ∧ 𝑠 = 𝑤) → ((𝑃↑(1st ‘𝑠)) · ((𝑁 / 𝑃)↑(2nd ‘𝑠))) = ((𝑃↑(1st ‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤))))
128		ovexd 7402	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤))) ∈ V)
129	121, 127, 38, 128	fvmptd 6955	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝐾 ∈ Field)
132	56	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝑃 ∈ ℙ)
133	65	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝑅 ∈ ℕ)
134	55	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝑁 ∈ ℕ)
135	57	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝑃 ∥ 𝑁)
136		aks6d1c6.8	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
137	136	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝑁 gcd 𝑅) = 1)
138		ovexd 7402	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (0...𝐴) ∈ V)
139	12, 138	elmapd 8787	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑈 ∈ (ℕ0 ↑m (0...𝐴)) ↔ 𝑈:(0...𝐴)⟶ℕ0))
140	86, 139	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑈:(0...𝐴)⟶ℕ0)
141	140	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝑈:(0...𝐴)⟶ℕ0)
142	74	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝐴 ∈ ℕ0)
143		xp1st 7974	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (ℕ0 × ℕ0) → (1st ‘𝑤) ∈ ℕ0)
144	143	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (1st ‘𝑤) ∈ ℕ0)
145		xp2nd 7975	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑤) ∈ ℕ0)
146	145	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (2nd ‘𝑤) ∈ ℕ0)
147		eqid 2736	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 42564	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤))) ∼ (𝐺‘𝑈))
153	129, 152	eqbrtrd 5107	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑤) ∼ (𝐺‘𝑈))
154	130, 88, 60	aks6d1c1p1 42546	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
157	109, 155, 156	rspcdva 3565	. . . . . . . . . . . 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 2742	. . . . . . . . . . 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 5943	. . . . . . . . . . . . . . . . 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 4020	. . . . . . . . . . . . . . . . . . . 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 3956	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑆 ⊆ (ℕ0 ↑m (0...𝐴)))
166	165	resmptd 6005	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀)) ↾ 𝑆) = (ℎ ∈ 𝑆 ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀)))
167	161, 166	eqtrd 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐻 ↾ 𝑆) = (ℎ ∈ 𝑆 ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀)))
168		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ℎ = 𝑈) → ℎ = 𝑈)
169	168	fveq2d 6844	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ℎ = 𝑈) → (𝐺‘ℎ) = (𝐺‘𝑈))
170	169	fveq2d 6844	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ℎ = 𝑈) → ((eval1‘𝐾)‘(𝐺‘ℎ)) = ((eval1‘𝐾)‘(𝐺‘𝑈)))
171	170	fveq1d 6842	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ℎ = 𝑈) → (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀))
172		fvexd 6855	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀) ∈ V)
173	167, 171, 80, 172	fvmptd 6955	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐻 ↾ 𝑆)‘𝑈) = (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀))
174	173	eqcomd 2742	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀) = ((𝐻 ↾ 𝑆)‘𝑈))
175		aks6d1c6lem2.3	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐻 ↾ 𝑆)‘𝑈) = ((𝐻 ↾ 𝑆)‘𝑉))
176		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ℎ = 𝑉) → ℎ = 𝑉)
177	176	fveq2d 6844	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ℎ = 𝑉) → (𝐺‘ℎ) = (𝐺‘𝑉))
178	177	fveq2d 6844	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ℎ = 𝑉) → ((eval1‘𝐾)‘(𝐺‘ℎ)) = ((eval1‘𝐾)‘(𝐺‘𝑉)))
179	178	fveq1d 6842	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ℎ = 𝑉) → (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀))
180		fvexd 6855	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀) ∈ V)
181	167, 179, 91, 180	fvmptd 6955	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐻 ↾ 𝑆)‘𝑉) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀))
182	174, 175, 181	3eqtrd 2775	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀))
183	182	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑈))‘𝑀) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀))
184	183	oveq2d 7383	. . . . . . . . . . 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 2771	. . . . . . . . . 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 6840	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑀 → (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑦) = (((eval1‘𝐾)‘(𝐺‘𝑉))‘𝑀))
187	186	oveq2d 7383	. . . . . . . . . . . 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 6844	. . . . . . . . . . . 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 2752	. . . . . . . . . . 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 8787	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑉 ∈ (ℕ0 ↑m (0...𝐴)) ↔ 𝑉:(0...𝐴)⟶ℕ0))
191	96, 190	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑉:(0...𝐴)⟶ℕ0)
192	191	adantr 480	. . . . . . . . . . . . . 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 42564	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑤)) · ((𝑁 / 𝑃)↑(2nd ‘𝑤))) ∼ (𝐺‘𝑉))
194	129, 193	eqbrtrd 5107	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑤) ∼ (𝐺‘𝑉))
195	130, 98, 60	aks6d1c1p1 42546	. . . . . . . . . . . 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 232	. . . . . . . . . . 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 3565	. . . . . . . . . 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 2771	. . . . . . . . 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 20228	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ Grp)
200	199	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → 𝐾 ∈ Grp)
201	41, 42, 43, 44, 47, 72, 88	fveval1fvcl 22298	. . . . . . . . . 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 22298	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1‘𝐾)‘(𝐺‘𝑉))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ (Base‘𝐾))
203		eqid 2736	. . . . . . . . . . 11 ⊢ (0g‘𝐾) = (0g‘𝐾)
204	43, 203, 102	grpsubeq0 19002	. . . . . . . . . 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 1374	. . . . . . . . 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 257	. . . . . . . 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 2771	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) = (0g‘𝐾))
208		fvexd 6855	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ V)
209		elsng 4581	. . . . . . . 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 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)))‘((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀)) ∈ {(0g‘𝐾)})
212		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ↑s (Base‘𝐾)) = (𝐾 ↑s (Base‘𝐾))
213	41, 42, 212, 43	evl1rhm 22297	. . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 14 ⊢ (Base‘(𝐾 ↑s (Base‘𝐾))) = (Base‘(𝐾 ↑s (Base‘𝐾)))
216	44, 215	rhmf 20464	. . . . . . . . . . . . 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 6855	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Base‘𝐾) ∈ V)
219	212, 43	pwsbas 17450	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Field ∧ (Base‘𝐾) ∈ V) → ((Base‘𝐾) ↑m (Base‘𝐾)) = (Base‘(𝐾 ↑s (Base‘𝐾))))
220	45, 218, 219	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((Base‘𝐾) ↑m (Base‘𝐾)) = (Base‘(𝐾 ↑s (Base‘𝐾))))
221	220	feq3d 6653	. . . . . . . . . . . 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 257	. . . . . . . . . . 11 ⊢ (𝜑 → (eval1‘𝐾):(Base‘(Poly1‘𝐾))⟶((Base‘𝐾) ↑m (Base‘𝐾)))
223	42	ply1ring 22211	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Ring → (Poly1‘𝐾) ∈ Ring)
224	51, 223	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Poly1‘𝐾) ∈ Ring)
225		ringgrp 20219	. . . . . . . . . . . . 13 ⊢ ((Poly1‘𝐾) ∈ Ring → (Poly1‘𝐾) ∈ Grp)
226	224, 225	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (Poly1‘𝐾) ∈ Grp)
227	44, 101	grpsubcl 18996	. . . . . . . . . . . 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 1374	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉)) ∈ (Base‘(Poly1‘𝐾)))
229	222, 228	ffvelcdmd 7037	. . . . . . . . . 10 ⊢ (𝜑 → ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) ∈ ((Base‘𝐾) ↑m (Base‘𝐾)))
230	218, 218	elmapd 8787	. . . . . . . . . 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 232	. . . . . . . . 9 ⊢ (𝜑 → ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))):(Base‘𝐾)⟶(Base‘𝐾))
232	231	ffund 6672	. . . . . . . 8 ⊢ (𝜑 → Fun ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))))
233	232	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → Fun ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))))
234	231	ffnd 6669	. . . . . . . . . . 11 ⊢ (𝜑 → ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) Fn (Base‘𝐾))
235	234	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) Fn (Base‘𝐾))
236	235	fndmd 6603	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → dom ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) = (Base‘𝐾))
237	236	eqcomd 2742	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (Base‘𝐾) = dom ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))))
238	72, 237	eleqtrd 2838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑤)(.g‘(mulGrp‘𝐾))𝑀) ∈ dom ((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))))
239		fvimacnv 7005	. . . . . . 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 585	. . . . . 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 232	. . . . 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 2836	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ (ℕ0 × ℕ0)) → (𝐽‘𝑤) ∈ (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}))
243	29, 33, 242	funimassd 6906	. . 3 ⊢ (𝜑 → (𝐽 “ (ℕ0 × ℕ0)) ⊆ (◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)}))
244		hashss 14371	. . 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 585	. 2 ⊢ (𝜑 → (♯‘(𝐽 “ (ℕ0 × ℕ0))) ≤ (♯‘(◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})))
246	8, 18, 24, 27, 245	xrletrd 13113	1 ⊢ (𝜑 → 𝐷 ≤ (♯‘(◡((eval1‘𝐾)‘((𝐺‘𝑈)(-g‘(Poly1‘𝐾))(𝐺‘𝑉))) “ {(0g‘𝐾)})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  {crab 3389  Vcvv 3429   ⊆ wss 3889  {csn 4567  ⟨cop 4573   class class class wbr 5085  {copab 5147   ↦ cmpt 5166   × cxp 5629  ^◡ccnv 5630  dom cdm 5631   ↾ cres 5633   “ cima 5634  Fun wfun 6492   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  1^st c1st 7940  2^nd c2nd 7941   ↑_m cmap 8773  0cc0 11038  1c1 11039   · cmul 11043  ℝ^*cxr 11178   < clt 11179   ≤ cle 11180   − cmin 11377   / cdiv 11807  ℕcn 12174  ℕ₀cn0 12437  ...cfz 13461  ↑cexp 14023  ♯chash 14292  Σcsu 15648   ∥ cdvds 16221   gcd cgcd 16463  ℙcprime 16640  Basecbs 17179  +_gcplusg 17220  0_gc0g 17402   Σ_g cgsu 17403   ↑_s cpws 17409  Mndcmnd 18702  Grpcgrp 18909  -_gcsg 18911  ._gcmg 19043  CMndccmn 19755  mulGrpcmgp 20121  Ringcrg 20214  CRingccrg 20215   RingHom crh 20449   RingIso crs 20450  Fieldcfield 20707  ℤRHomczrh 21479  chrcchr 21481  ℤ/nℤczn 21482  algSccascl 21832  var₁cv1 22139  Poly₁cpl1 22140  eval₁ce1 22279   PrimRoots cprimroots 42530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116  ax-addf 11117  ax-mulf 11118

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-ofr 7632  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-tpos 8176  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8643  df-map 8775  df-pm 8776  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-sup 9355  df-inf 9356  df-oi 9425  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-xnn0 12511  df-z 12525  df-dec 12645  df-uz 12789  df-rp 12943  df-fz 13462  df-fzo 13609  df-fl 13751  df-mod 13829  df-seq 13964  df-exp 14024  df-fac 14236  df-bc 14265  df-hash 14293  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-dvds 16222  df-gcd 16464  df-prm 16641  df-phi 16736  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-mulg 19044  df-subg 19099  df-ghm 19188  df-cntz 19292  df-od 19503  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-srg 20168  df-ring 20216  df-cring 20217  df-oppr 20317  df-dvdsr 20337  df-unit 20338  df-invr 20368  df-dvr 20381  df-rhm 20452  df-rim 20453  df-subrng 20523  df-subrg 20547  df-drng 20708  df-field 20709  df-lmod 20857  df-lss 20927  df-lsp 20967  df-cnfld 21353  df-zring 21427  df-zrh 21483  df-chr 21485  df-assa 21833  df-asp 21834  df-ascl 21835  df-psr 21889  df-mvr 21890  df-mpl 21891  df-opsr 21893  df-evls 22052  df-evl 22053  df-psr1 22143  df-vr1 22144  df-ply1 22145  df-coe1 22146  df-evl1 22281  df-primroots 42531

This theorem is referenced by:  aks6d1c6lem3  42611