Theorem aks6d1c6lem5 42210

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 2731	. . . . . . . . . . 11 ⊢ (0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)) = (0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))
22	3	fldcrngd 20652	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ CRing)
23		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾)
24	23	crngmgp 20154	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ CRing → (mulGrp‘𝐾) ∈ CMnd)
25	22, 24	syl 17	. . . . . . . . . . . 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 42208	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ (ℤring GrpHom (((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)))
28		eqid 2731	. . . . . . . . . . 11 ⊢ (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}) = (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})
29		eqid 2731	. . . . . . . . . . 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 21385	. . . . . . . . . . 11 ⊢ ℤ = (Base‘ℤring)
32		nfcv 2894	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐[𝑑](ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))
33		nfcv 2894	. . . . . . . . . . . 12 ⊢ Ⅎ𝑑[𝑐](ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))
34		eceq1 8656	. . . . . . . . . . . 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 5188	. . . . . . . . . . 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 19191	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 = (𝑋 ∘ (𝑑 ∈ ℤ ↦ [𝑑](ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))))
37		eqid 2731	. . . . . . . . . . . . . . . . 17 ⊢ (RSpan‘ℤring) = (RSpan‘ℤring)
38	25, 5, 26, 20, 16, 37	aks6d1c6isolem3 42209	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((RSpan‘ℤring)‘{𝑅}) = (◡𝐽 “ {(0g‘((mulGrp‘𝐾) ↾s 𝑈))}))
39	25, 5, 26	primrootsunit 42131	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (((mulGrp‘𝐾) PrimRoots 𝑅) = (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅) ∧ ((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel))
40	39	simprd 495	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ Abel)
41	40	ablgrpd 19693	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ Grp)
42	41	grpmndd 18854	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ Mnd)
43		0zd 12475	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 ∈ ℤ)
44		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑤 = 0) → 𝑤 = 0)
45	44	fveqeq2d 6825	. . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 = 0) → 𝑗 = 0)
48	47	oveq1d 7356	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 = 0) → (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀))
49	39	simpld 494	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ((mulGrp‘𝐾) PrimRoots 𝑅) = (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅))
50	16, 49	eleqtrd 2833	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑀 ∈ (((mulGrp‘𝐾) ↾s 𝑈) PrimRoots 𝑅))
51	40	ablcmnd 19695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ((mulGrp‘𝐾) ↾s 𝑈) ∈ CMnd)
52	5	nnnn0d 12437	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
53		eqid 2731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (.g‘((mulGrp‘𝐾) ↾s 𝑈)) = (.g‘((mulGrp‘𝐾) ↾s 𝑈))
54	51, 52, 53	isprimroot 42126	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 15	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1142	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
58		eqid 2731	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (Base‘((mulGrp‘𝐾) ↾s 𝑈)) = (Base‘((mulGrp‘𝐾) ↾s 𝑈))
59		eqid 2731	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0g‘((mulGrp‘𝐾) ↾s 𝑈)) = (0g‘((mulGrp‘𝐾) ↾s 𝑈))
60	58, 59, 53	mulg0 18982	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)) → (0(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
61	57, 60	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (0(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
62	61	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 = 0) → (0(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
63	48, 62	eqtrd 2766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑗 = 0) → (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
64		fvexd 6832	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∈ V)
65	46, 63, 43, 64	fvmptd 6931	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐽‘0) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
66	43, 45, 65	rspcedvd 3574	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ∃𝑤 ∈ ℤ (𝐽‘𝑤) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
67	41	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → ((mulGrp‘𝐾) ↾s 𝑈) ∈ Grp)
68		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℤ)
69	57	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → 𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
70	58, 53, 67, 68, 69	mulgcld 19004	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑗 ∈ ℤ) → (𝑗(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
71	70, 20	fmptd 7042	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐽:ℤ⟶(Base‘((mulGrp‘𝐾) ↾s 𝑈)))
72	71	ffnd 6647	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐽 Fn ℤ)
73		fvelrnb 6877	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐽 Fn ℤ → ((0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∈ ran 𝐽 ↔ ∃𝑤 ∈ ℤ (𝐽‘𝑤) = (0g‘((mulGrp‘𝐾) ↾s 𝑈))))
74	72, 73	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∈ ran 𝐽 ↔ ∃𝑤 ∈ ℤ (𝐽‘𝑤) = (0g‘((mulGrp‘𝐾) ↾s 𝑈))))
75	66, 74	mpbird 257	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (0g‘((mulGrp‘𝐾) ↾s 𝑈)) ∈ ran 𝐽)
76	71	frnd 6654	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ran 𝐽 ⊆ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
77		eqid 2731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽) = (((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)
78	77, 58, 59	ress0g 18665	. . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0g‘((mulGrp‘𝐾) ↾s 𝑈)) = (0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)))
80	79	sneqd 4583	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → {(0g‘((mulGrp‘𝐾) ↾s 𝑈))} = {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})
81	80	imaeq2d 6004	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (◡𝐽 “ {(0g‘((mulGrp‘𝐾) ↾s 𝑈))}) = (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))
82	38, 81	eqtr2d 2767	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}) = ((RSpan‘ℤring)‘{𝑅}))
83	82	oveq2d 7357	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})) = (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})))
84	83	eceq2d 8660	. . . . . . . . . . . . 13 ⊢ (𝜑 → [𝑑](ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})) = [𝑑](ℤring ~QG ((RSpan‘ℤring)‘{𝑅})))
85	84	mpteq2dv 5180	. . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . 15 ⊢ (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})) = (ℤring ~QG ((RSpan‘ℤring)‘{𝑅}))
87		eqid 2731	. . . . . . . . . . . . . . 15 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
88	37, 86, 87, 13	znzrh2 21477	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℕ0 → 𝐿 = (𝑑 ∈ ℤ ↦ [𝑑](ℤring ~QG ((RSpan‘ℤring)‘{𝑅}))))
89	52, 88	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 = (𝑑 ∈ ℤ ↦ [𝑑](ℤring ~QG ((RSpan‘ℤring)‘{𝑅}))))
90	89	eqcomd 2737	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑑 ∈ ℤ ↦ [𝑑](ℤring ~QG ((RSpan‘ℤring)‘{𝑅}))) = 𝐿)
91	85, 90	eqtrd 2766	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑑 ∈ ℤ ↦ [𝑑](ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))) = 𝐿)
92	91	coeq2d 5797	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∘ (𝑑 ∈ ℤ ↦ [𝑑](ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))) = (𝑋 ∘ 𝐿))
93	36, 92	eqtrd 2766	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 = (𝑋 ∘ 𝐿))
94	93	coeq2d 5797	. . . . . . . 8 ⊢ (𝜑 → (◡𝑋 ∘ 𝐽) = (◡𝑋 ∘ (𝑋 ∘ 𝐿)))
95		coass 6208	. . . . . . . . 9 ⊢ ((◡𝑋 ∘ 𝑋) ∘ 𝐿) = (◡𝑋 ∘ (𝑋 ∘ 𝐿))
96	95	eqcomi 2740	. . . . . . . 8 ⊢ (◡𝑋 ∘ (𝑋 ∘ 𝐿)) = ((◡𝑋 ∘ 𝑋) ∘ 𝐿)
97	94, 96	eqtrdi 2782	. . . . . . 7 ⊢ (𝜑 → (◡𝑋 ∘ 𝐽) = ((◡𝑋 ∘ 𝑋) ∘ 𝐿))
98	77, 58	ressbas2 17144	. . . . . . . . . . . . 13 ⊢ (ran 𝐽 ⊆ (Base‘((mulGrp‘𝐾) ↾s 𝑈)) → ran 𝐽 = (Base‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)))
99	76, 98	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝐽 = (Base‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)))
100	21, 27, 28, 29, 30, 99	ghmqusker 19194	. . . . . . . . . . 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 2731	. . . . . . . . . . . 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 2731	. . . . . . . . . . . 12 ⊢ (Base‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽)) = (Base‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))
103	101, 102	gimf1o 19170	. . . . . . . . . . 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 17	. . . . . . . . . 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 6787	. . . . . . . . . 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 17	. . . . . . . . 9 ⊢ (𝜑 → (◡𝑋 ∘ 𝑋) = ( I ↾ (Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))))))
107	106	coeq1d 5796	. . . . . . . 8 ⊢ (𝜑 → ((◡𝑋 ∘ 𝑋) ∘ 𝐿) = (( I ↾ (Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))))) ∘ 𝐿))
108	87	zncrng 21476	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ ℕ0 → (ℤ/nℤ‘𝑅) ∈ CRing)
109	52, 108	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℤ/nℤ‘𝑅) ∈ CRing)
110		crngring 20158	. . . . . . . . . . . 12 ⊢ ((ℤ/nℤ‘𝑅) ∈ CRing → (ℤ/nℤ‘𝑅) ∈ Ring)
111	13	zrhrhm 21443	. . . . . . . . . . . 12 ⊢ ((ℤ/nℤ‘𝑅) ∈ Ring → 𝐿 ∈ (ℤring RingHom (ℤ/nℤ‘𝑅)))
112		eqid 2731	. . . . . . . . . . . . 13 ⊢ (Base‘(ℤ/nℤ‘𝑅)) = (Base‘(ℤ/nℤ‘𝑅))
113	31, 112	rhmf 20397	. . . . . . . . . . . 12 ⊢ (𝐿 ∈ (ℤring RingHom (ℤ/nℤ‘𝑅)) → 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅)))
114	109, 110, 111, 113	4syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅)))
115		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑅}))) = (ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})))
116	37, 115, 87	znbas2 21471	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ ℕ0 → (Base‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})))) = (Base‘(ℤ/nℤ‘𝑅)))
117	52, 116	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (Base‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})))) = (Base‘(ℤ/nℤ‘𝑅)))
118	117	feq3d 6631	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿:ℤ⟶(Base‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})))) ↔ 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅))))
119	114, 118	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘(ℤring /s (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})))))
120	82	eqcomd 2737	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((RSpan‘ℤring)‘{𝑅}) = (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))
121	120	oveq2d 7357	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℤring ~QG ((RSpan‘ℤring)‘{𝑅})) = (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))
122	121	oveq2d 7357	. . . . . . . . . . . 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 6821	. . . . . . . . . . 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 6631	. . . . . . . . . 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 232	. . . . . . . . 9 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))})))))
126		fcoi2 6693	. . . . . . . . 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 17	. . . . . . . 8 ⊢ (𝜑 → (( I ↾ (Base‘(ℤring /s (ℤring ~QG (◡𝐽 “ {(0g‘(((mulGrp‘𝐾) ↾s 𝑈) ↾s ran 𝐽))}))))) ∘ 𝐿) = 𝐿)
128	107, 127	eqtrd 2766	. . . . . . 7 ⊢ (𝜑 → ((◡𝑋 ∘ 𝑋) ∘ 𝐿) = 𝐿)
129	97, 128	eqtr2d 2767	. . . . . 6 ⊢ (𝜑 → 𝐿 = (◡𝑋 ∘ 𝐽))
130	129	imaeq1d 6003	. . . . 5 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) = ((◡𝑋 ∘ 𝐽) “ (𝐸 “ (ℕ0 × ℕ0))))
131		imaco 6193	. . . . . 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 2766	. . . 4 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) = (◡𝑋 “ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))))
134	133	fveq2d 6821	. . 3 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) = (♯‘(◡𝑋 “ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))))))
135		simplll 774	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) ∧ 𝑢 ∈ ℤ) ∧ (𝐽‘𝑢) = 𝑤) → 𝜑)
136		simplr 768	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) ∧ 𝑢 ∈ ℤ) ∧ (𝐽‘𝑢) = 𝑤) → 𝑢 ∈ ℤ)
137	135, 136	jca 511	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) ∧ 𝑢 ∈ ℤ) ∧ (𝐽‘𝑢) = 𝑤) → (𝜑 ∧ 𝑢 ∈ ℤ))
138		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → 𝑧 ∈ (0...(𝑅 − 1)))
139		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) ∧ 𝑣 = 𝑧) → 𝑣 = 𝑧)
140	139	fveqeq2d 6825	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) ∧ 𝑗 = 𝑧) → 𝑗 = 𝑧)
143	142	oveq1d 7356	. . . . . . . . . . . . . . . . . 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 13421	. . . . . . . . . . . . . . . . . . 19 ⊢ (0...(𝑅 − 1)) ⊆ ℤ
145	144, 138	sselid 3927	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → 𝑧 ∈ ℤ)
146		ovexd 7376	. . . . . . . . . . . . . . . . . 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 6931	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) ∧ 𝑗 = 𝑢) → 𝑗 = 𝑢)
149	148	oveq1d 7356	. . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑢 ∈ ℤ) → 𝑢 ∈ ℤ)
151	150	ad3antrrr 730	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → 𝑢 ∈ ℤ)
152		ovexd 7376	. . . . . . . . . . . . . . . . . . 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 6931	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → 𝑢 = ((𝑦 · 𝑅) + 𝑧))
155	154	oveq1d 7356	. . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → ((mulGrp‘𝐾) ↾s 𝑈) ∈ Grp)
157		simplr 768	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → 𝑦 ∈ ℤ)
158	5	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑢 ∈ ℤ) → 𝑅 ∈ ℕ)
159	158	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → 𝑅 ∈ ℕ)
160	159	nnzd 12490	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → 𝑅 ∈ ℤ)
161	157, 160	zmulcld 12578	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → (𝑦 · 𝑅) ∈ ℤ)
162	144	sseli 3925	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ (0...(𝑅 − 1)) → 𝑧 ∈ ℤ)
163	162	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → 𝑧 ∈ ℤ)
164	57	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → 𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈)))
165	161, 163, 164	3jca 1128	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → ((𝑦 · 𝑅) ∈ ℤ ∧ 𝑧 ∈ ℤ ∧ 𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈))))
166		eqid 2731	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (+g‘((mulGrp‘𝐾) ↾s 𝑈)) = (+g‘((mulGrp‘𝐾) ↾s 𝑈))
167	58, 53, 166	mulgdir 19014	. . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) → (𝑦 ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝑀 ∈ (Base‘((mulGrp‘𝐾) ↾s 𝑈))))
170	58, 53	mulgass 19019	. . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . 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 1143	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
173	172	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑢 ∈ ℤ) → (𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
174	173	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) → (𝑅(.g‘((mulGrp‘𝐾) ↾s 𝑈))𝑀) = (0g‘((mulGrp‘𝐾) ↾s 𝑈)))
175	174	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7357	. . . . . . . . . . . . . . . . . . . . . . . . 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 19010	. . . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . . . . . . 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 7356	. . . . . . . . . . . . . . . . . . . . . 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 19004	. . . . . . . . . . . . . . . . . . . . . . 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 18877	. . . . . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑢 ∈ ℤ) ∧ 𝑦 ∈ ℤ) ∧ 𝑧 ∈ (0...(𝑅 − 1))) ∧ 𝑢 = ((𝑦 · 𝑅) + 𝑧)) → (𝐽‘𝑧) = (𝐽‘𝑢))
190	138, 140, 189	rspcedvd 3574	. . . . . . . . . . . . . . 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 42137	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ ℤ) → ∃𝑦 ∈ ℤ ∃𝑧 ∈ (0...(𝑅 − 1))𝑢 = ((𝑦 · 𝑅) + 𝑧))
192	190, 191	r19.29vva 3192	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ ℤ) → ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = (𝐽‘𝑢))
193	137, 192	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) ∧ 𝑢 ∈ ℤ) ∧ (𝐽‘𝑢) = 𝑤) → ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = (𝐽‘𝑢))
194		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) ∧ 𝑢 ∈ ℤ) ∧ (𝐽‘𝑢) = 𝑤) → (𝐽‘𝑢) = 𝑤)
195	194	eqcomd 2737	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) ∧ 𝑢 ∈ ℤ) ∧ (𝐽‘𝑢) = 𝑤) → 𝑤 = (𝐽‘𝑢))
196	195	eqeq2d 2742	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) ∧ 𝑢 ∈ ℤ) ∧ (𝐽‘𝑢) = 𝑤) → ((𝐽‘𝑣) = 𝑤 ↔ (𝐽‘𝑣) = (𝐽‘𝑢)))
197	196	rexbidv 3156	. . . . . . . . . . . . 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 257	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) ∧ 𝑢 ∈ ℤ) ∧ (𝐽‘𝑢) = 𝑤) → ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = 𝑤)
199		ssidd 3953	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ℤ ⊆ ℤ)
200		fvelimab 6889	. . . . . . . . . . . . . . 15 ⊢ ((𝐽 Fn ℤ ∧ ℤ ⊆ ℤ) → (𝑤 ∈ (𝐽 “ ℤ) ↔ ∃𝑢 ∈ ℤ (𝐽‘𝑢) = 𝑤))
201	72, 199, 200	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑤 ∈ (𝐽 “ ℤ) ↔ ∃𝑢 ∈ ℤ (𝐽‘𝑢) = 𝑤))
202	201	biimpd 229	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑤 ∈ (𝐽 “ ℤ) → ∃𝑢 ∈ ℤ (𝐽‘𝑢) = 𝑤))
203	202	imp 406	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) → ∃𝑢 ∈ ℤ (𝐽‘𝑢) = 𝑤)
204	198, 203	r19.29a 3140	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) → ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = 𝑤)
205	144	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...(𝑅 − 1)) ⊆ ℤ)
206		fvelimab 6889	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑤 ∈ (𝐽 “ (0...(𝑅 − 1))) ↔ ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = 𝑤))
208	207	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) → (𝑤 ∈ (𝐽 “ (0...(𝑅 − 1))) ↔ ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = 𝑤))
209	204, 208	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ ℤ)) → 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1))))
210	209	ex 412	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ (𝐽 “ ℤ) → 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))))
211	210	ssrdv 3935	. . . . . . . 8 ⊢ (𝜑 → (𝐽 “ ℤ) ⊆ (𝐽 “ (0...(𝑅 − 1))))
212	207	biimpd 229	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑤 ∈ (𝐽 “ (0...(𝑅 − 1))) → ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = 𝑤))
213	212	imp 406	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))) → ∃𝑣 ∈ (0...(𝑅 − 1))(𝐽‘𝑣) = 𝑤)
214	144	sseli 3925	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (0...(𝑅 − 1)) → 𝑣 ∈ ℤ)
215	214	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ (0...(𝑅 − 1)) ∧ (𝐽‘𝑣) = 𝑤) → 𝑣 ∈ ℤ)
216	215	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))) ∧ (𝑣 ∈ (0...(𝑅 − 1)) ∧ (𝐽‘𝑣) = 𝑤)) → 𝑣 ∈ ℤ)
217		simprr 772	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))) ∧ (𝑣 ∈ (0...(𝑅 − 1)) ∧ (𝐽‘𝑣) = 𝑤)) → (𝐽‘𝑣) = 𝑤)
218	213, 216, 217	reximssdv 3150	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))) → ∃𝑣 ∈ ℤ (𝐽‘𝑣) = 𝑤)
219	72	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))) → 𝐽 Fn ℤ)
220		ssidd 3953	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))) → ℤ ⊆ ℤ)
221		fvelimab 6889	. . . . . . . . . . . 12 ⊢ ((𝐽 Fn ℤ ∧ ℤ ⊆ ℤ) → (𝑤 ∈ (𝐽 “ ℤ) ↔ ∃𝑣 ∈ ℤ (𝐽‘𝑣) = 𝑤))
222	219, 220, 221	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))) → (𝑤 ∈ (𝐽 “ ℤ) ↔ ∃𝑣 ∈ ℤ (𝐽‘𝑣) = 𝑤))
223	218, 222	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 “ (0...(𝑅 − 1)))) → 𝑤 ∈ (𝐽 “ ℤ))
224	223	ex 412	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ (𝐽 “ (0...(𝑅 − 1))) → 𝑤 ∈ (𝐽 “ ℤ)))
225	224	ssrdv 3935	. . . . . . . 8 ⊢ (𝜑 → (𝐽 “ (0...(𝑅 − 1))) ⊆ (𝐽 “ ℤ))
226	211, 225	eqssd 3947	. . . . . . 7 ⊢ (𝜑 → (𝐽 “ ℤ) = (𝐽 “ (0...(𝑅 − 1))))
227	72	fnfund 6577	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐽)
228		fzfid 13875	. . . . . . . 8 ⊢ (𝜑 → (0...(𝑅 − 1)) ∈ Fin)
229		imafi 9194	. . . . . . . 8 ⊢ ((Fun 𝐽 ∧ (0...(𝑅 − 1)) ∈ Fin) → (𝐽 “ (0...(𝑅 − 1))) ∈ Fin)
230	227, 228, 229	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝐽 “ (0...(𝑅 − 1))) ∈ Fin)
231	226, 230	eqeltrd 2831	. . . . . 6 ⊢ (𝜑 → (𝐽 “ ℤ) ∈ Fin)
232	6, 4, 7, 12	aks6d1c2p1 42151	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶ℕ)
233		nnssz 12485	. . . . . . . . . . . 12 ⊢ ℕ ⊆ ℤ
234	233	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ℕ ⊆ ℤ)
235	232, 234	jca 511	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸:(ℕ0 × ℕ0)⟶ℕ ∧ ℕ ⊆ ℤ))
236		fss 6662	. . . . . . . . . 10 ⊢ ((𝐸:(ℕ0 × ℕ0)⟶ℕ ∧ ℕ ⊆ ℤ) → 𝐸:(ℕ0 × ℕ0)⟶ℤ)
237	235, 236	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶ℤ)
238	237	frnd 6654	. . . . . . . 8 ⊢ (𝜑 → ran 𝐸 ⊆ ℤ)
239	232	ffnd 6647	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 Fn (ℕ0 × ℕ0))
240		fnima 6606	. . . . . . . . . 10 ⊢ (𝐸 Fn (ℕ0 × ℕ0) → (𝐸 “ (ℕ0 × ℕ0)) = ran 𝐸)
241	239, 240	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 “ (ℕ0 × ℕ0)) = ran 𝐸)
242	241	sseq1d 3961	. . . . . . . 8 ⊢ (𝜑 → ((𝐸 “ (ℕ0 × ℕ0)) ⊆ ℤ ↔ ran 𝐸 ⊆ ℤ))
243	238, 242	mpbird 257	. . . . . . 7 ⊢ (𝜑 → (𝐸 “ (ℕ0 × ℕ0)) ⊆ ℤ)
244		imass2 6046	. . . . . . 7 ⊢ ((𝐸 “ (ℕ0 × ℕ0)) ⊆ ℤ → (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) ⊆ (𝐽 “ ℤ))
245	243, 244	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) ⊆ (𝐽 “ ℤ))
246	231, 245	ssfid 9148	. . . . 5 ⊢ (𝜑 → (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ Fin)
247		dff1o2 6763	. . . . . . . 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 216	. . . . . . 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 1143	. . . . . 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 17	. . . . 5 ⊢ (𝜑 → Fun ◡𝑋)
251		imadomfi 42035	. . . . 5 ⊢ (((𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ Fin ∧ Fun ◡𝑋) → (◡𝑋 “ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))) ≼ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))))
252	246, 250, 251	syl2anc 584	. . . 4 ⊢ (𝜑 → (◡𝑋 “ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))) ≼ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))))
253		hashdomi 14282	. . . 4 ⊢ ((◡𝑋 “ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))) ≼ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))) → (♯‘(◡𝑋 “ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))))) ≤ (♯‘(𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))))
254	252, 253	syl 17	. . 3 ⊢ (𝜑 → (♯‘(◡𝑋 “ (𝐽 “ (𝐸 “ (ℕ0 × ℕ0))))) ≤ (♯‘(𝐽 “ (𝐸 “ (ℕ0 × ℕ0)))))
255	134, 254	eqbrtrd 5108	. 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 42206	1 ⊢ (𝜑 → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  {crab 3395  Vcvv 3436   ⊆ wss 3897  {csn 4571  ∪ cuni 4854   class class class wbr 5086  {copab 5148   ↦ cmpt 5167   I cid 5505   × cxp 5609  ^◡ccnv 5610  ran crn 5612   ↾ cres 5613   “ cima 5614   ∘ ccom 5615  Fun wfun 6470   Fn wfn 6471  ⟶wf 6472  –1-1-onto→wf1o 6475  ‘cfv 6476  (class class class)co 7341   ∈ cmpo 7343  [cec 8615   ↑_m cmap 8745   ≼ cdom 8862  Fincfn 8864  0cc0 11001  1c1 11002   + caddc 11004   · cmul 11006   ≤ cle 11142   − cmin 11339   / cdiv 11769  ℕcn 12120  2c2 12175  ℕ₀cn0 12376  ℤcz 12463  ...cfz 13402  ⌊cfl 13689  ↑cexp 13963  Ccbc 14204  ♯chash 14232  √csqrt 15135  Σcsu 15588   ∥ cdvds 16158   gcd cgcd 16400  ℙcprime 16577  ϕcphi 16670  Basecbs 17115   ↾_s cress 17136  +_gcplusg 17156  0_gc0g 17338   Σ_g cgsu 17339   /_s cqus 17404  Mndcmnd 18637  Grpcgrp 18841  ._gcmg 18975   ~_QG cqg 19030   GrpIso cgim 19164  CMndccmn 19687  Abelcabl 19688  mulGrpcmgp 20053  Ringcrg 20146  CRingccrg 20147   RingHom crh 20382   RingIso crs 20383  Fieldcfield 20640  RSpancrsp 21139  ℤ_ringczring 21378  ℤRHomczrh 21431  chrcchr 21433  ℤ/nℤczn 21434  algSccascl 21784  var₁cv1 22083  Poly₁cpl1 22084  eval₁ce1 22224   log_b clogb 26696   PrimRoots cprimroots 42124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-inf2 9526  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078  ax-pre-sup 11079  ax-addf 11080  ax-mulf 11081

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-iin 4939  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-se 5565  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-isom 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-ofr 7606  df-om 7792  df-1st 7916  df-2nd 7917  df-supp 8086  df-tpos 8151  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-oadd 8384  df-er 8617  df-ec 8619  df-qs 8623  df-map 8747  df-pm 8748  df-ixp 8817  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-fsupp 9241  df-fi 9290  df-sup 9321  df-inf 9322  df-oi 9391  df-dju 9789  df-card 9827  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-div 11770  df-nn 12121  df-2 12183  df-3 12184  df-4 12185  df-5 12186  df-6 12187  df-7 12188  df-8 12189  df-9 12190  df-n0 12377  df-xnn0 12450  df-z 12464  df-dec 12584  df-uz 12728  df-q 12842  df-rp 12886  df-xneg 13006  df-xadd 13007  df-xmul 13008  df-ioo 13244  df-ioc 13245  df-ico 13246  df-icc 13247  df-fz 13403  df-fzo 13550  df-fl 13691  df-mod 13769  df-seq 13904  df-exp 13964  df-fac 14176  df-bc 14205  df-hash 14233  df-shft 14969  df-cj 15001  df-re 15002  df-im 15003  df-sqrt 15137  df-abs 15138  df-limsup 15373  df-clim 15390  df-rlim 15391  df-sum 15589  df-ef 15969  df-sin 15971  df-cos 15972  df-pi 15974  df-dvds 16159  df-gcd 16401  df-prm 16578  df-phi 16672  df-struct 17053  df-sets 17070  df-slot 17088  df-ndx 17100  df-base 17116  df-ress 17137  df-plusg 17169  df-mulr 17170  df-starv 17171  df-sca 17172  df-vsca 17173  df-ip 17174  df-tset 17175  df-ple 17176  df-ds 17178  df-unif 17179  df-hom 17180  df-cco 17181  df-rest 17321  df-topn 17322  df-0g 17340  df-gsum 17341  df-topgen 17342  df-pt 17343  df-prds 17346  df-pws 17348  df-xrs 17401  df-qtop 17406  df-imas 17407  df-qus 17408  df-xps 17409  df-mre 17483  df-mrc 17484  df-acs 17486  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-mhm 18686  df-submnd 18687  df-grp 18844  df-minusg 18845  df-sbg 18846  df-mulg 18976  df-subg 19031  df-nsg 19032  df-eqg 19033  df-ghm 19120  df-gim 19166  df-cntz 19224  df-od 19435  df-cmn 19689  df-abl 19690  df-mgp 20054  df-rng 20066  df-ur 20095  df-srg 20100  df-ring 20148  df-cring 20149  df-oppr 20250  df-dvdsr 20270  df-unit 20271  df-invr 20301  df-dvr 20314  df-rhm 20385  df-rim 20386  df-nzr 20423  df-subrng 20456  df-subrg 20480  df-rlreg 20604  df-domn 20605  df-idom 20606  df-drng 20641  df-field 20642  df-lmod 20790  df-lss 20860  df-lsp 20900  df-sra 21102  df-rgmod 21103  df-lidl 21140  df-rsp 21141  df-2idl 21182  df-psmet 21278  df-xmet 21279  df-met 21280  df-bl 21281  df-mopn 21282  df-fbas 21283  df-fg 21284  df-cnfld 21287  df-zring 21379  df-zrh 21435  df-chr 21437  df-zn 21438  df-assa 21785  df-asp 21786  df-ascl 21787  df-psr 21841  df-mvr 21842  df-mpl 21843  df-opsr 21845  df-evls 22004  df-evl 22005  df-psr1 22087  df-vr1 22088  df-ply1 22089  df-coe1 22090  df-evl1 22226  df-top 22804  df-topon 22821  df-topsp 22843  df-bases 22856  df-cld 22929  df-ntr 22930  df-cls 22931  df-nei 23008  df-lp 23046  df-perf 23047  df-cn 23137  df-cnp 23138  df-haus 23225  df-tx 23472  df-hmeo 23665  df-fil 23756  df-fm 23848  df-flim 23849  df-flf 23850  df-xms 24230  df-ms 24231  df-tms 24232  df-cncf 24793  df-limc 25789  df-dv 25790  df-mdeg 25982  df-deg1 25983  df-mon1 26058  df-uc1p 26059  df-q1p 26060  df-r1p 26061  df-log 26487  df-logb 26697  df-primroots 42125

This theorem is referenced by:  aks6d1c7lem2  42214