Theorem aks6d1c6lem5 43043

Description: Eliminate the size hypothesis. Claim 6. (Contributed by metakunt, 15-May-2025.)

Hypotheses

Ref	Expression
aks6d1c6lem5.1	⊢ ∼ = {⟨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)))}
aks6d1c6lem5.2	⊢ 𝑃 = (chr‘𝐾)
aks6d1c6lem5.3	⊢ (𝜑 → 𝐾 ∈ Field)
aks6d1c6lem5.4	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c6lem5.5	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks6d1c6lem5.6	⊢ (𝜑 → 𝑁 ∈ ℕ)
aks6d1c6lem5.7	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks6d1c6lem5.8	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c6lem5.9	⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
aks6d1c6lem5.10	⊢ 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)))))))
aks6d1c6lem5.11	⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))
aksaks6dlem5.12	⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
aks6d1c6lem5.13	⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
aks6d1c6lem5.14	⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
aks6d1c6lem5.15	⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
aks6d1c6lem5.16	⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
aks6d1c6lem5.17	⊢ 𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))
aks6d1c6lem5.18	⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
aks6d1c6lem5.19	⊢ 𝑆 = {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}
aks6d1c6lem5.20	⊢ 𝐽 = (𝑗 ∈ ℤ ↦ (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
aks6d1c6lem5.22	⊢ 𝑈 = {𝑚 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑛 ∈ (Base‘(mulGrp‘𝐾))(𝑛(+g‘(mulGrp‘𝐾))𝑚) = (0g‘(mulGrp‘𝐾))}
aks6d1c6lem5.23	⊢ 𝑋 = (𝑏 ∈ (Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))) ↦ ∪ (𝐽 “ 𝑏))

Assertion

Ref	Expression
aks6d1c6lem5	⊢ (𝜑 → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))))

Distinct variable groups:   ∼ ,𝑎   𝐴,𝑎   𝐴,𝑏   𝐴,𝑔,𝑖,𝑥   𝐴,ℎ,𝑗   𝐴,𝑠,𝑡   𝐷,𝑠   𝑒,𝐸,𝑓,𝑦   𝑗,𝐸,𝑦   𝑥,𝐸,𝑦   𝑒,𝐺,𝑓,𝑦   𝑔,𝐺,𝑖,𝑦   ℎ,𝐺   𝑡,𝐺,𝑖,𝑦   𝐻,𝑎   𝑔,𝐻,𝑖,𝑥,𝑦   ℎ,𝐻,𝑗   𝐻,𝑠,𝑡   𝐽,𝑏   𝑦,𝐽   𝐾,𝑎   𝐾,𝑏   𝑒,𝐾,𝑓,𝑦   𝑔,𝐾,𝑖,𝑥   ℎ,𝐾,𝑗   𝐾,𝑙,𝑥,𝑦   𝑚,𝐾,𝑛   𝑡,𝐾,𝑥   ℎ,𝑀,𝑗   𝑀,𝑙,𝑦   𝑁,𝑎   𝑁,𝑏   𝑒,𝑁,𝑓   𝑗,𝑁   𝑘,𝑁,𝑙,𝑠   𝑥,𝑁,𝑘   𝑃,𝑏   𝑃,𝑒,𝑓   𝑃,𝑗   𝑃,𝑘,𝑙,𝑠   𝑥,𝑃   𝑅,𝑒,𝑓,𝑦   𝑅,𝑗   𝑅,𝑙,𝑥   𝑆,𝑎   𝑆,𝑔,𝑖,𝑥,𝑦   𝑆,ℎ,𝑗   𝑆,𝑠,𝑡   𝑈,𝑏   𝑈,𝑗   𝑈,𝑙   𝑋,𝑏   𝜑,𝑎   𝜑,𝑏   𝜑,𝑔,𝑖,𝑥,𝑦   𝜑,ℎ,𝑗   𝜑,𝑘,𝑙,𝑠   𝑦,𝑘   𝜑,𝑡

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

Proof of Theorem aks6d1c6lem5

Dummy variables 𝑐 𝑑 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		aks6d1c6lem5.1	. 2 ⊢ ∼ = {⟨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)))}
2		aks6d1c6lem5.2	. 2 ⊢ 𝑃 = (chr‘𝐾)
3		aks6d1c6lem5.3	. 2 ⊢ (𝜑 → 𝐾 ∈ Field)
4		aks6d1c6lem5.4	. 2 ⊢ (𝜑 → 𝑃 ∈ ℙ)
5		aks6d1c6lem5.5	. 2 ⊢ (𝜑 → 𝑅 ∈ ℕ)
6		aks6d1c6lem5.6	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ)
7		aks6d1c6lem5.7	. 2 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
8		aks6d1c6lem5.8	. 2 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
9		aks6d1c6lem5.9	. 2 ⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
10		aks6d1c6lem5.10	. 2 ⊢ 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)))))))
11		aks6d1c6lem5.11	. 2 ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 logb 𝑁)))
12		aksaks6dlem5.12	. 2 ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
13		aks6d1c6lem5.13	. 2 ⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
14		aks6d1c6lem5.14	. 2 ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
15		aks6d1c6lem5.15	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
16		aks6d1c6lem5.16	. 2 ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
17		aks6d1c6lem5.17	. 2 ⊢ 𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))
18		aks6d1c6lem5.18	. 2 ⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
19		aks6d1c6lem5.19	. 2 ⊢ 𝑆 = {𝑠 ∈ (ℕ0 ↑m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}
20		aks6d1c6lem5.20	. 2 ⊢ 𝐽 = (𝑗 ∈ ℤ ↦ (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
21		eqid 2760	. . . . . . . . . . 11 ⊢ (0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)) = (0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))
22	3	fldcrngd 20905	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ CRing)
23		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
24	23	crngmgp 20380	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
25	22, 24	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (mulGrp‘𝐾) ∈ CMnd)
26		aks6d1c6lem5.22	. . . . . . . . . . . 12 ⊢ 𝑈 = {𝑚 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑛 ∈ (Base‘(mulGrp‘𝐾))(𝑛(+g‘(mulGrp‘𝐾))𝑚) = (0g‘(mulGrp‘𝐾))}
27	25, 5, 26, 20, 16	aks6d1c6isolem2 43041	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (ℤring GrpHom (((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)))
28		eqid 2760	. . . . . . . . . . 11 ⊢ (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}) = (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})
29		eqid 2760	. . . . . . . . . . 11 ⊢ (ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))) = (ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))
30		aks6d1c6lem5.23	. . . . . . . . . . 11 ⊢ 𝑋 = (𝑏 ∈ (Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))) ↦ ∪ (𝐽 “ 𝑏))
31		zringbas 21666	. . . . . . . . . . 11 ⊢ ℤ = (Base‘ℤring)
32		nfcv 2922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐[𝑑](ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))
33		nfcv 2922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑑[𝑐](ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))
34		eceq1 8736	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑐 → [𝑑](ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})) = [𝑐](ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))
35	32, 33, 34	cbvmpt 5207	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ℤ ↦ [𝑑](ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))) = (𝑐 ∈ ℤ ↦ [𝑐](ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))
36	21, 27, 28, 29, 30, 31, 35	ghmquskerco 19411	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 = (𝑋 ∘ (𝑑 ∈ ℤ ↦ [𝑑](ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))))
37		eqid 2760	. . . . . . . . . . . . . . . . 17 ⊢ (RSpan‘ℤring) = (RSpan‘ℤring)
38	25, 5, 26, 20, 16, 37	aks6d1c6isolem3 43042	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((RSpan‘ℤring)‘{𝑅}) = (◡𝐽 “ {(0g‘((mulGrp‘𝐾) ↾s 𝑈))}))
39	25, 5, 26	primrootsunit 42964	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((mulGrp‘𝐾) PrimRoots 𝑅) = (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅) ∧ ((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel))
40	39	simprd 501	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel)
41	40	ablgrpd 19913	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ Grp)
42	41	grpmndd 19070	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ Mnd)
43		0zd 12627	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 ∈ ℤ)
44		simpr 490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 = 0) → 𝑤 = 0)
45	44	fveqeq2d 6886	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑤 = 0) → ((𝐽‘𝑤) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)) ↔ (𝐽‘0) = (0g‘((mulGrp‘𝐾) ↾s 𝑈))))
46	20	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐽 = (𝑗 ∈ ℤ ↦ (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
47		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 = 0) → 𝑗 = 0)
48	47	oveq1d 7428	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 = 0) → (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
49	39	simpld 500	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ((mulGrp‘𝐾) PrimRoots 𝑅) = (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅))
50	16, 49	eleqtrd 2862	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑀 ∈ (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅))
51	40	ablcmnd 19915	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ CMnd)
52	5	nnnn0d 12589	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
53		eqid 2760	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (.g‘((mulGrp‘𝐾) ↾s 𝑈)) = (.g‘((mulGrp‘𝐾) ↾s 𝑈))
54	51, 52, 53	isprimroot 42959	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑀 ∈ (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅) ↔ (𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ (𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)) → 𝑅 ∥ 𝑙))))
55	54	biimpd 232	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑀 ∈ (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅) → (𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ (𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)) → 𝑅 ∥ 𝑙))))
56	50, 55	mpd 16	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ (𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∧ ∀𝑙 ∈ ℕ0 ((𝑙(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)) → 𝑅 ∥ 𝑙)))
57	56	simp1d 1160	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
58		eqid 2760	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (Base‘((mulGrp‘𝐾) ↾s 𝑈)) = (Base‘((mulGrp‘𝐾) ↾s 𝑈))
59		eqid 2760	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0g‘((mulGrp‘𝐾) ↾s 𝑈)) = (0g‘((mulGrp‘𝐾) ↾s 𝑈))
60	58, 59, 53	mulg0 19197	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)) → (0(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
61	57, 60	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (0(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
62	61	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 = 0) → (0(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
63	48, 62	eqtrd 2795	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 = 0) → (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
64		fvexd 6893	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∈ V)
65	46, 63, 43, 64	fvmptd 6994	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐽‘0) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
66	43, 45, 65	rspcedvd 3578	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ∃𝑤 ∈ ℤ (𝐽‘𝑤) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
67	41	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → ((mulGrp‘𝐾) ↾s 𝑈) ∈ Grp)
68		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℤ)
69	57	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
70	58, 53, 67, 68, 69	mulgcld 19219	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
71	70, 20	fmptd 7107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐽:ℤ⟶(Base‘((mulGrp‘𝐾) ↾s 𝑈)))
72	71	ffnd 6703	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐽 Fn ℤ)
73		fvelrnb 6938	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐽 Fn ℤ → ((0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∈ ran 𝐽 ↔ ∃𝑤 ∈ ℤ (𝐽‘𝑤) = (0g‘((mulGrp‘𝐾) ↾s 𝑈))))
74	72, 73	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∈ ran 𝐽 ↔ ∃𝑤 ∈ ℤ (𝐽‘𝑤) = (0g‘((mulGrp‘𝐾) ↾s 𝑈))))
75	66, 74	mpbird 260	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∈ ran 𝐽)
76	71	frnd 6711	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ran 𝐽 ⊆ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
77		eqid 2760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽) = (((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)
78	77, 58, 59	ress0g 18867	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((mulGrp‘𝐾) ↾s 𝑈) ∈ Mnd ∧ (0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∈ ran 𝐽 ∧ ran 𝐽 ⊆ (Base‘((mulGrp‘𝐾) ↾s 𝑈))) → (0g‘((mulGrp‘𝐾) ↾s 𝑈)) = (0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)))
79	42, 75, 76, 78	syl3anc 1398	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0g‘((mulGrp‘𝐾) ↾s 𝑈)) = (0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)))
80	79	sneqd 4596	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → {(0g‘((mulGrp‘𝐾) ↾s 𝑈))} = {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})
81	80	imaeq2d 6056	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (◡𝐽 “ {(0g‘((mulGrp‘𝐾) ↾s 𝑈))}) = (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))
82	38, 81	eqtr2d 2796	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}) = ((RSpan‘ℤring)‘{𝑅}))
83	82	oveq2d 7429	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})) = (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})))
84	83	eceq2d 8740	. . . . . . . . . . . . 13 ⊢ (𝜑 → [𝑑](ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})) = [𝑑](ℤring ~QG ((RSpan‘ℤring)‘{𝑅})))
85	84	mpteq2dv 5199	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑑 ∈ ℤ ↦ [𝑑](ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))) = (𝑑 ∈ ℤ ↦ [𝑑](ℤring ~QG ((RSpan‘ℤring)‘{𝑅}))))
86		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})) = (ℤring ~QG ((RSpan‘ℤring)‘{𝑅}))
87		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
88	37, 86, 87, 13	znzrh2 21758	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℕ0 → 𝐿 = (𝑑 ∈ ℤ ↦ [𝑑](ℤring ~QG ((RSpan‘ℤring)‘{𝑅}))))
89	52, 88	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 = (𝑑 ∈ ℤ ↦ [𝑑](ℤring ~QG ((RSpan‘ℤring)‘{𝑅}))))
90	89	eqcomd 2766	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑑 ∈ ℤ ↦ [𝑑](ℤring ~QG ((RSpan‘ℤring)‘{𝑅}))) = 𝐿)
91	85, 90	eqtrd 2795	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑑 ∈ ℤ ↦ [𝑑](ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))) = 𝐿)
92	91	coeq2d 5842	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∘ (𝑑 ∈ ℤ ↦ [𝑑](ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))) = (𝑋 ∘ 𝐿))
93	36, 92	eqtrd 2795	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 = (𝑋 ∘ 𝐿))
94	93	coeq2d 5842	. . . . . . . 8 ⊢ (𝜑 → (◡𝑋 ∘ 𝐽) = (◡𝑋 ∘ (𝑋 ∘ 𝐿)))
95		coass 6262	. . . . . . . . 9 ⊢ ((◡𝑋 ∘ 𝑋) ∘ 𝐿) = (◡𝑋 ∘ (𝑋 ∘ 𝐿))
96	95	eqcomi 2769	. . . . . . . 8 ⊢ (◡𝑋 ∘ (𝑋 ∘ 𝐿)) = ((◡𝑋 ∘ 𝑋) ∘ 𝐿)
97	94, 96	eqtrdi 2811	. . . . . . 7 ⊢ (𝜑 → (◡𝑋 ∘ 𝐽) = ((◡𝑋 ∘ 𝑋) ∘ 𝐿))
98	77, 58	ressbas2 17330	. . . . . . . . . . . . 13 ⊢ (ran 𝐽 ⊆ (Base‘((mulGrp‘𝐾) ↾s 𝑈)) → ran 𝐽 = (Base‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)))
99	76, 98	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐽 = (Base‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)))
100	21, 27, 28, 29, 30, 99	ghmqusker 19414	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ ((ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))) GrpIso (((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)))
101		eqid 2760	. . . . . . . . . . . 12 ⊢ (Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))) = (Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))))
102		eqid 2760	. . . . . . . . . . . 12 ⊢ (Base‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)) = (Base‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))
103	101, 102	gimf1o 19390	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ((ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))) GrpIso (((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)) → 𝑋:(Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))))–1-1-onto→(Base‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)))
104	100, 103	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋:(Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))))–1-1-onto→(Base‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)))
105		f1ococnv1 6847	. . . . . . . . . 10 ⊢ (𝑋:(Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))))–1-1-onto→(Base‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)) → (◡𝑋 ∘ 𝑋) = ( I ↾ (Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))))))
106	104, 105	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (◡𝑋 ∘ 𝑋) = ( I ↾ (Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))))))
107	106	coeq1d 5841	. . . . . . . 8 ⊢ (𝜑 → ((◡𝑋 ∘ 𝑋) ∘ 𝐿) = (( I ↾ (Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))))) ∘ 𝐿))
108	87	zncrng 21757	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ ℕ0 → (ℤ/nℤ‘𝑅) ∈ CRing)
109	52, 108	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℤ/nℤ‘𝑅) ∈ CRing)
110		crngring 20384	. . . . . . . . . . . 12 ⊢ ((ℤ/nℤ‘𝑅) ∈ CRing → (ℤ/nℤ‘𝑅) ∈ Ring)
111	13	zrhrhm 21724	. . . . . . . . . . . 12 ⊢ ((ℤ/nℤ‘𝑅) ∈ Ring → 𝐿 ∈ (ℤring RingHom (ℤ/nℤ‘𝑅)))
112		eqid 2760	. . . . . . . . . . . . 13 ⊢ (Base‘(ℤ/nℤ‘𝑅)) = (Base‘(ℤ/nℤ‘𝑅))
113	31, 112	rhmf 20626	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ (ℤring RingHom (ℤ/nℤ‘𝑅)) → 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅)))
114	109, 110, 111, 113	4syl 20	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅)))
115		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑅}))) = (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})))
116	37, 115, 87	znbas2 21752	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ ℕ0 → (Base‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})))) = (Base‘(ℤ/nℤ‘𝑅)))
117	52, 116	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})))) = (Base‘(ℤ/nℤ‘𝑅)))
118	117	feq3d 6687	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿:ℤ⟶(Base‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})))) ↔ 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅))))
119	114, 118	mpbird 260	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})))))
120	82	eqcomd 2766	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((RSpan‘ℤring)‘{𝑅}) = (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))
121	120	oveq2d 7429	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})) = (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))
122	121	oveq2d 7429	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑅}))) = (ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))))
123	122	fveq2d 6882	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})))) = (Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))))
124	123	feq3d 6687	. . . . . . . . . 10 ⊢ (𝜑 → (𝐿:ℤ⟶(Base‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})))) ↔ 𝐿:ℤ⟶(Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))))))
125	119, 124	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))))
126		fcoi2 6750	. . . . . . . . 9 ⊢ (𝐿:ℤ⟶(Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))) → (( I ↾ (Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))))) ∘ 𝐿) = 𝐿)
127	125, 126	syl 18	. . . . . . . 8 ⊢ (𝜑 → (( I ↾ (Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))))) ∘ 𝐿) = 𝐿)
128	107, 127	eqtrd 2795	. . . . . . 7 ⊢ (𝜑 → ((◡𝑋 ∘ 𝑋) ∘ 𝐿) = 𝐿)
129	97, 128	eqtr2d 2796	. . . . . 6 ⊢ (𝜑 → 𝐿 = (◡𝑋 ∘ 𝐽))
130	129	imaeq1d 6055	. . . . 5 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) = ((◡𝑋 ∘ 𝐽) “ (𝐸 “ (ℕ0 × ℕ0))))
131		imaco 6247	. . . . . 6 ⊢ ((◡𝑋 ∘ 𝐽) “ (𝐸 “ (ℕ0 × ℕ0))) = (◡𝑋 “ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))))
132	131	a1i 11	. . . . 5 ⊢ (𝜑 → ((◡𝑋 ∘ 𝐽) “ (𝐸 “ (ℕ0 × ℕ0))) = (◡𝑋 “ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))))
133	130, 132	eqtrd 2795	. . . 4 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) = (◡𝑋 “ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))))
134	133	fveq2d 6882	. . 3 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) = (♯‘(◡𝑋 “ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))))))
135		simplll 787	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) ∧ 𝑢 ∈ ℤ) ∧ (𝐽‘𝑢) = 𝑤) → 𝜑)
136		simplr 781	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) ∧ 𝑢 ∈ ℤ) ∧ (𝐽‘𝑢) = 𝑤) → 𝑢 ∈ ℤ)
137	135, 136	jca 521	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) ∧ 𝑢 ∈ ℤ) ∧ (𝐽‘𝑢) = 𝑤) → (𝜑 ∧ 𝑢 ∈ ℤ))
138		simplr 781	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → 𝑧 ∈ (0...(𝑅 − 1)))
139		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) ∧ 𝑣 = 𝑧) → 𝑣 = 𝑧)
140	139	fveqeq2d 6886	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) ∧ 𝑣 = 𝑧) → ((𝐽‘𝑣) = (𝐽‘𝑢) ↔ (𝐽‘𝑧) = (𝐽‘𝑢)))
141	20	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → 𝐽 = (𝑗 ∈ ℤ ↦ (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
142		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) ∧ 𝑗 = 𝑧) → 𝑗 = 𝑧)
143	142	oveq1d 7428	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) ∧ 𝑗 = 𝑧) → (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (𝑧(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
144		fzssz 13580	. . . . . . . . . . . . . . . . . . 19 ⊢ (0...(𝑅 − 1)) ⊆ ℤ
145	144, 138	sselid 3929	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → 𝑧 ∈ ℤ)
146		ovexd 7448	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → (𝑧(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) ∈ V)
147	141, 143, 145, 146	fvmptd 6994	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → (𝐽‘𝑧) = (𝑧(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
148		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) ∧ 𝑗 = 𝑢) → 𝑗 = 𝑢)
149	148	oveq1d 7428	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) ∧ 𝑗 = 𝑢) → (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (𝑢(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
150		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ ℤ) → 𝑢 ∈ ℤ)
151	150	ad3antrrr 743	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → 𝑢 ∈ ℤ)
152		ovexd 7448	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → (𝑢(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) ∈ V)
153	141, 149, 151, 152	fvmptd 6994	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → (𝐽‘𝑢) = (𝑢(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
154		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → 𝑢 = ((𝑦 · 𝑅) + 𝑧))
155	154	oveq1d 7428	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → (𝑢(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (((𝑦 · 𝑅) + 𝑧)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
156	41	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → ((mulGrp‘𝐾) ↾s 𝑈) ∈ Grp)
157		simplr 781	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → 𝑦 ∈ ℤ)
158	5	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑢 ∈ ℤ) → 𝑅 ∈ ℕ)
159	158	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → 𝑅 ∈ ℕ)
160	159	nnzd 12641	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → 𝑅 ∈ ℤ)
161	157, 160	zmulcld 12731	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → (𝑦 · 𝑅) ∈ ℤ)
162	144	sseli 3927	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ (0...(𝑅 − 1)) → 𝑧 ∈ ℤ)
163	162	adantl 487	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → 𝑧 ∈ ℤ)
164	57	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → 𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
165	161, 163, 164	3jca 1146	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → ((𝑦 · 𝑅) ∈ ℤ ∧ 𝑧 ∈ ℤ ∧ 𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈))))
166		eqid 2760	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (+g‘((mulGrp‘𝐾) ↾s 𝑈)) = (+g‘((mulGrp‘𝐾) ↾s 𝑈))
167	58, 53, 166	mulgdir 19229	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((mulGrp‘𝐾) ↾s 𝑈) ∈ Grp ∧ ((𝑦 · 𝑅) ∈ ℤ ∧ 𝑧 ∈ ℤ ∧ 𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))) → (((𝑦 · 𝑅) + 𝑧)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (((𝑦 · 𝑅)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)(+g‘((mulGrp‘𝐾) ↾s 𝑈))(𝑧(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
168	156, 165, 167	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → (((𝑦 · 𝑅) + 𝑧)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (((𝑦 · 𝑅)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)(+g‘((mulGrp‘𝐾) ↾s 𝑈))(𝑧(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
169	157, 160, 164	3jca 1146	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → (𝑦 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈))))
170	58, 53	mulgass 19234	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((mulGrp‘𝐾) ↾s 𝑈) ∈ Grp ∧ (𝑦 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))) → ((𝑦 · 𝑅)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (𝑦(.g‘((mulGrp‘𝐾) ↾s 𝑈))(𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
171	156, 169, 170	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → ((𝑦 · 𝑅)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (𝑦(.g‘((mulGrp‘𝐾) ↾s 𝑈))(𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
172	56	simp2d 1161	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
173	172	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑢 ∈ ℤ) → (𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
174	173	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) → (𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
175	174	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → (𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
176	175	oveq2d 7429	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → (𝑦(.g‘((mulGrp‘𝐾) ↾s 𝑈))(𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑦(.g‘((mulGrp‘𝐾) ↾s 𝑈))(0g‘((mulGrp‘𝐾) ↾s 𝑈))))
177	58, 53, 59	mulgz 19225	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((mulGrp‘𝐾) ↾s 𝑈) ∈ Grp ∧ 𝑦 ∈ ℤ) → (𝑦(.g‘((mulGrp‘𝐾) ↾s 𝑈))(0g‘((mulGrp‘𝐾) ↾s 𝑈))) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
178	156, 157, 177	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → (𝑦(.g‘((mulGrp‘𝐾) ↾s 𝑈))(0g‘((mulGrp‘𝐾) ↾s 𝑈))) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
179	176, 178	eqtrd 2795	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → (𝑦(.g‘((mulGrp‘𝐾) ↾s 𝑈))(𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
180	171, 179	eqtrd 2795	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → ((𝑦 · 𝑅)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
181	180	oveq1d 7428	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → (((𝑦 · 𝑅)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)(+g‘((mulGrp‘𝐾) ↾s 𝑈))(𝑧(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = ((0g‘((mulGrp‘𝐾) ↾s 𝑈))(+g‘((mulGrp‘𝐾) ↾s 𝑈))(𝑧(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)))
182	58, 53, 156, 163, 164	mulgcld 19219	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → (𝑧(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
183	58, 166, 59, 156, 182	grplidd 19093	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → ((0g‘((mulGrp‘𝐾) ↾s 𝑈))(+g‘((mulGrp‘𝐾) ↾s 𝑈))(𝑧(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑧(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
184	181, 183	eqtrd 2795	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → (((𝑦 · 𝑅)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)(+g‘((mulGrp‘𝐾) ↾s 𝑈))(𝑧(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀)) = (𝑧(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
185	168, 184	eqtrd 2795	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → (((𝑦 · 𝑅) + 𝑧)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (𝑧(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
186	185	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → (((𝑦 · 𝑅) + 𝑧)(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (𝑧(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
187	155, 186	eqtrd 2795	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → (𝑢(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (𝑧(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
188	153, 187	eqtr2d 2796	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → (𝑧(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (𝐽‘𝑢))
189	147, 188	eqtrd 2795	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → (𝐽‘𝑧) = (𝐽‘𝑢))
190	138, 140, 189	rspcedvd 3578	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = (𝐽‘𝑢))
191	150, 158	remexz 42970	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ ℤ) → ∃𝑦 ∈ ℤ ∃𝑧 ∈ (0...(𝑅 − 1))𝑢 = ((𝑦 · 𝑅) + 𝑧))
192	190, 191	r19.29vva 3222	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ ℤ) → ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = (𝐽‘𝑢))
193	137, 192	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) ∧ 𝑢 ∈ ℤ) ∧ (𝐽‘𝑢) = 𝑤) → ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = (𝐽‘𝑢))
194		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) ∧ 𝑢 ∈ ℤ) ∧ (𝐽‘𝑢) = 𝑤) → (𝐽‘𝑢) = 𝑤)
195	194	eqcomd 2766	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) ∧ 𝑢 ∈ ℤ) ∧ (𝐽‘𝑢) = 𝑤) → 𝑤 = (𝐽‘𝑢))
196	195	eqeq2d 2771	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) ∧ 𝑢 ∈ ℤ) ∧ (𝐽‘𝑢) = 𝑤) → ((𝐽‘𝑣) = 𝑤 ↔ (𝐽‘𝑣) = (𝐽‘𝑢)))
197	196	rexbidv 3186	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) ∧ 𝑢 ∈ ℤ) ∧ (𝐽‘𝑢) = 𝑤) → (∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = 𝑤 ↔ ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = (𝐽‘𝑢)))
198	193, 197	mpbird 260	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) ∧ 𝑢 ∈ ℤ) ∧ (𝐽‘𝑢) = 𝑤) → ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = 𝑤)
199		ssidd 3954	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ℤ ⊆ ℤ)
200		fvelimab 6950	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 Fn ℤ ∧ ℤ ⊆ ℤ) → (𝑤 ∈ (𝐽 “ ℤ) ↔ ∃𝑢 ∈ ℤ (𝐽‘𝑢) = 𝑤))
201	72, 199, 200	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑤 ∈ (𝐽 “ ℤ) ↔ ∃𝑢 ∈ ℤ (𝐽‘𝑢) = 𝑤))
202	201	biimpd 232	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑤 ∈ (𝐽 “ ℤ) → ∃𝑢 ∈ ℤ (𝐽‘𝑢) = 𝑤))
203	202	imp 412	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) → ∃𝑢 ∈ ℤ (𝐽‘𝑢) = 𝑤)
204	198, 203	r19.29a 3170	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) → ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = 𝑤)
205	144	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...(𝑅 − 1)) ⊆ ℤ)
206		fvelimab 6950	. . . . . . . . . . . . 13 ⊢ ((𝐽 Fn ℤ ∧ (0...(𝑅 − 1)) ⊆ ℤ) → (𝑤 ∈ (𝐽 “ (0...(𝑅 − 1))) ↔ ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = 𝑤))
207	72, 205, 206	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑤 ∈ (𝐽 “ (0...(𝑅 − 1))) ↔ ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = 𝑤))
208	207	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) → (𝑤 ∈ (𝐽 “ (0...(𝑅 − 1))) ↔ ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = 𝑤))
209	204, 208	mpbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) → 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1))))
210	209	ex 418	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ (𝐽 “ ℤ) → 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))))
211	210	ssrdv 3937	. . . . . . . 8 ⊢ (𝜑 → (𝐽 “ ℤ) ⊆ (𝐽 “ (0...(𝑅 − 1))))
212	207	biimpd 232	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑤 ∈ (𝐽 “ (0...(𝑅 − 1))) → ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = 𝑤))
213	212	imp 412	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))) → ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = 𝑤)
214	144	sseli 3927	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (0...(𝑅 − 1)) → 𝑣 ∈ ℤ)
215	214	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ (0...(𝑅 − 1)) ∧ (𝐽‘𝑣) = 𝑤) → 𝑣 ∈ ℤ)
216	215	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))) ∧ (𝑣 ∈ (0...(𝑅 − 1)) ∧ (𝐽‘𝑣) = 𝑤)) → 𝑣 ∈ ℤ)
217		simprr 785	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))) ∧ (𝑣 ∈ (0...(𝑅 − 1)) ∧ (𝐽‘𝑣) = 𝑤)) → (𝐽‘𝑣) = 𝑤)
218	213, 216, 217	reximssdv 3180	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))) → ∃𝑣 ∈ ℤ (𝐽‘𝑣) = 𝑤)
219	72	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))) → 𝐽 Fn ℤ)
220		ssidd 3954	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))) → ℤ ⊆ ℤ)
221		fvelimab 6950	. . . . . . . . . . . 12 ⊢ ((𝐽 Fn ℤ ∧ ℤ ⊆ ℤ) → (𝑤 ∈ (𝐽 “ ℤ) ↔ ∃𝑣 ∈ ℤ (𝐽‘𝑣) = 𝑤))
222	219, 220, 221	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))) → (𝑤 ∈ (𝐽 “ ℤ) ↔ ∃𝑣 ∈ ℤ (𝐽‘𝑣) = 𝑤))
223	218, 222	mpbird 260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))) → 𝑤 ∈ (𝐽 “ ℤ))
224	223	ex 418	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ (𝐽 “ (0...(𝑅 − 1))) → 𝑤 ∈ (𝐽 “ ℤ)))
225	224	ssrdv 3937	. . . . . . . 8 ⊢ (𝜑 → (𝐽 “ (0...(𝑅 − 1))) ⊆ (𝐽 “ ℤ))
226	211, 225	eqssd 3948	. . . . . . 7 ⊢ (𝜑 → (𝐽 “ ℤ) = (𝐽 “ (0...(𝑅 − 1))))
227	72	fnfund 6633	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐽)
228		fzfid 14037	. . . . . . . 8 ⊢ (𝜑 → (0...(𝑅 − 1)) ∈ Fin)
229		imafi 9285	. . . . . . . 8 ⊢ ((Fun 𝐽 ∧ (0...(𝑅 − 1)) ∈ Fin) → (𝐽 “ (0...(𝑅 − 1))) ∈ Fin)
230	227, 228, 229	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝐽 “ (0...(𝑅 − 1))) ∈ Fin)
231	226, 230	eqeltrd 2860	. . . . . 6 ⊢ (𝜑 → (𝐽 “ ℤ) ∈ Fin)
232	6, 4, 7, 12	aks6d1c2p1 42984	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶ℕ)
233		nnssz 12637	. . . . . . . . . . . 12 ⊢ ℕ ⊆ ℤ
234	233	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ℕ ⊆ ℤ)
235	232, 234	jca 521	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸:(ℕ0 × ℕ0)⟶ℕ ∧ ℕ ⊆ ℤ))
236		fss 6719	. . . . . . . . . 10 ⊢ ((𝐸:(ℕ0 × ℕ0)⟶ℕ ∧ ℕ ⊆ ℤ) → 𝐸:(ℕ0 × ℕ0)⟶ℤ)
237	235, 236	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶ℤ)
238	237	frnd 6711	. . . . . . . 8 ⊢ (𝜑 → ran 𝐸 ⊆ ℤ)
239	232	ffnd 6703	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 Fn (ℕ0 × ℕ0))
240		fnima 6662	. . . . . . . . . 10 ⊢ (𝐸 Fn (ℕ0 × ℕ0) → (𝐸 “ (ℕ0 × ℕ0)) = ran 𝐸)
241	239, 240	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 “ (ℕ0 × ℕ0)) = ran 𝐸)
242	241	sseq1d 3962	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 “ (ℕ0 × ℕ0)) ⊆ ℤ ↔ ran 𝐸 ⊆ ℤ))
243	238, 242	mpbird 260	. . . . . . 7 ⊢ (𝜑 → (𝐸 “ (ℕ0 × ℕ0)) ⊆ ℤ)
244		imass2 6098	. . . . . . 7 ⊢ ((𝐸 “ (ℕ0 × ℕ0)) ⊆ ℤ → (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) ⊆ (𝐽 “ ℤ))
245	243, 244	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) ⊆ (𝐽 “ ℤ))
246	231, 245	ssfid 9239	. . . . 5 ⊢ (𝜑 → (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ Fin)
247		dff1o2 6823	. . . . . . . 8 ⊢ (𝑋:(Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))))–1-1-onto→(Base‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)) ↔ (𝑋 Fn (Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))) ∧ Fun ◡𝑋 ∧ ran 𝑋 = (Base‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))))
248	247	biimpi 219	. . . . . . 7 ⊢ (𝑋:(Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))))–1-1-onto→(Base‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)) → (𝑋 Fn (Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))) ∧ Fun ◡𝑋 ∧ ran 𝑋 = (Base‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))))
249	248	simp2d 1161	. . . . . 6 ⊢ (𝑋:(Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))))–1-1-onto→(Base‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)) → Fun ◡𝑋)
250	104, 249	syl 18	. . . . 5 ⊢ (𝜑 → Fun ◡𝑋)
251		imadomfi 42868	. . . . 5 ⊢ (((𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ Fin ∧ Fun ◡𝑋) → (◡𝑋 “ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))) ≼ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))))
252	246, 250, 251	syl2anc 596	. . . 4 ⊢ (𝜑 → (◡𝑋 “ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))) ≼ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))))
253		hashdomi 14444	. . . 4 ⊢ ((◡𝑋 “ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))) ≼ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) → (♯‘(◡𝑋 “ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))))) ≤ (♯‘(𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))))
254	252, 253	syl 18	. . 3 ⊢ (𝜑 → (♯‘(◡𝑋 “ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))))) ≤ (♯‘(𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))))
255	134, 254	eqbrtrd 5127	. 2 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))))
256	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 255, 26	aks6d1c6lem4 43039	1 ⊢ (𝜑 → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3899  {csn 4584  ∪ cuni 4867   class class class wbr 5103  {copab 5167   ↦ cmpt 5186   I cid 5549   × cxp 5653  ^◡ccnv 5654  ran crn 5656   ↾ cres 5657   “ cima 5658   ∘ ccom 5659  Fun wfun 6527   Fn wfn 6528  ⟶wf 6529  –1-1-onto→wf1o 6532  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415  [cec 8694   ↑_m cmap 8826   ≼ cdom 8950  Fincfn 8952  0cc0 11124  1c1 11125   + caddc 11127   · cmul 11129   ≤ cle 11268   − cmin 11465   / cdiv 11895  ℕcn 12257  2c2 12319  ℕ₀cn0 12528  ℤcz 12615  ...cfz 13561  ⌊cfl 13851  ↑cexp 14125  Ccbc 14366  ♯chash 14394  √csqrt 15320  Σcsu 15773   ∥ cdvds 16342   gcd cgcd 16584  ℙcprime 16761  ϕcphi 16855  Basecbs 17301   ↾_s cress 17322  +_gcplusg 17342  0_gc0g 17524   Σ_g cgsu 17525   /_s cqus 17591  Mndcmnd 18836  Grpcgrp 19057  ._gcmg 19190   ~_QG cqg 19245   GrpIso cgim 19384  CMndccmn 19907  Abelcabl 19908  mulGrpcmgp 20273  Ringcrg 20372  CRingccrg 20373   RingHom crh 20610   RingIso crs 20611  Fieldcfield 20891  RSpancrsp 21394  ℤ_ringczring 21659  ℤRHomczrh 21712  chrcchr 21714  ℤ/nℤczn 21715  algSccascl 22067  var₁cv1 22401  Poly₁cpl1 22402  eval₁ce1 22539   log_b clogb 27001   PrimRoots cprimroots 42957

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-inf2 9620  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202  ax-addf 11203  ax-mulf 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678  df-ofr 7679  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8159  df-tpos 8224  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-oadd 8459  df-er 8696  df-ec 8698  df-qs 8702  df-map 8828  df-pm 8829  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-fsupp 9332  df-fi 9381  df-sup 9412  df-inf 9413  df-oi 9482  df-dju 9906  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  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 12529  df-xnn0 12602  df-z 12616  df-dec 12737  df-uz 12888  df-q 12998  df-rp 13043  df-xneg 13163  df-xadd 13164  df-xmul 13165  df-ioo 13402  df-ioc 13403  df-ico 13404  df-icc 13405  df-fz 13562  df-fzo 13710  df-fl 13853  df-mod 13931  df-seq 14066  df-exp 14126  df-fac 14338  df-bc 14367  df-hash 14395  df-shft 15140  df-cj 15186  df-re 15187  df-im 15188  df-sqrt 15322  df-abs 15323  df-limsup 15558  df-clim 15575  df-rlim 15576  df-sum 15774  df-ef 16153  df-sin 16155  df-cos 16156  df-pi 16158  df-dvds 16343  df-gcd 16585  df-prm 16762  df-phi 16857  df-struct 17239  df-sets 17256  df-slot 17274  df-ndx 17286  df-base 17302  df-ress 17323  df-plusg 17355  df-mulr 17356  df-starv 17357  df-sca 17358  df-vsca 17359  df-ip 17360  df-tset 17361  df-ple 17362  df-ds 17364  df-unif 17365  df-hom 17366  df-cco 17367  df-rest 17507  df-topn 17508  df-0g 17526  df-gsum 17527  df-topgen 17528  df-pt 17529  df-prds 17532  df-pws 17534  df-xrs 17588  df-qtop 17593  df-imas 17594  df-qus 17595  df-xps 17596  df-mre 17670  df-mrc 17671  df-acs 17673  df-mgm 18730  df-sgrp 18821  df-mnd 18837  df-mhm 18891  df-submnd 18892  df-grp 19060  df-minusg 19061  df-sbg 19062  df-mulg 19191  df-subg 19246  df-nsg 19247  df-eqg 19248  df-ghm 19341  df-gim 19386  df-cntz 19444  df-od 19655  df-cmn 19909  df-abl 19910  df-mgp 20274  df-rng 20288  df-ur 20321  df-srg 20326  df-ring 20374  df-cring 20375  df-oppr 20478  df-dvdsr 20498  df-unit 20499  df-invr 20529  df-dvr 20542  df-rhm 20613  df-rim 20614  df-nzr 20673  df-subrng 20708  df-subrg 20732  df-rlreg 20856  df-domn 20857  df-idom 20858  df-drng 20892  df-field 20893  df-lmod 21046  df-lss 21116  df-lsp 21156  df-sra 21357  df-rgmod 21358  df-lidl 21395  df-rsp 21396  df-2idl 21452  df-psmet 21577  df-xmet 21578  df-met 21579  df-bl 21580  df-mopn 21581  df-fbas 21582  df-fg 21583  df-cnfld 21586  df-zring 21660  df-zrh 21716  df-chr 21718  df-zn 21719  df-assa 22068  df-asp 22069  df-ascl 22070  df-psr 22124  df-mvr 22125  df-mpl 22126  df-opsr 22128  df-evls 22290  df-evl 22291  df-psr1 22405  df-vr1 22406  df-ply1 22407  df-coe1 22408  df-evl1 22541  df-top 23119  df-topon 23136  df-topsp 23158  df-bases 23171  df-cld 23244  df-ntr 23245  df-cls 23246  df-nei 23323  df-lp 23361  df-perf 23362  df-cn 23452  df-cnp 23453  df-haus 23540  df-tx 23788  df-hmeo 23981  df-fil 24072  df-fm 24164  df-flim 24165  df-flf 24166  df-xms 24546  df-ms 24547  df-tms 24548  df-cncf 25106  df-limc 26093  df-dv 26094  df-mdeg 26280  df-deg1 26281  df-mon1 26356  df-uc1p 26357  df-q1p 26358  df-r1p 26359  df-log 26793  df-logb 27002  df-primroots 42958

This theorem is used by:  aks6d1c7lem2  43047