Theorem aks6d1c4 42609

Description: Claim 4 of Theorem 6.1 of the AKS inequality lemma. https://www3.nd.edu/%7eandyp/notes/AKS.pdf (Contributed by metakunt, 12-May-2025.)

Hypotheses

Ref	Expression
aks6d1c4.1	⊢ (𝜑 → 𝑁 ∈ ℕ)
aks6d1c4.2	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c4.3	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks6d1c4.4	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks6d1c4.5	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c4.6	⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
aks6d1c4.7	⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))

Assertion

Ref	Expression
aks6d1c4	⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (ϕ‘𝑅))

Distinct variable groups:   𝑘,𝑁,𝑙   𝑃,𝑘,𝑙

Allowed substitution hints:   𝜑(𝑘,𝑙)   𝑅(𝑘,𝑙)   𝐸(𝑘,𝑙)   𝐿(𝑘,𝑙)

Proof of Theorem aks6d1c4

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvexd 6842	. . 3 ⊢ (𝜑 → (Unit‘(ℤ/nℤ‘𝑅)) ∈ V)
2		aks6d1c4.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ ℕ)
3	2	nnnn0d 12489	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
4		eqid 2739	. . . . . . . . . . . 12 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
5	4	zncrng 21519	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ0 → (ℤ/nℤ‘𝑅) ∈ CRing)
6	3, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ/nℤ‘𝑅) ∈ CRing)
7		crngring 20217	. . . . . . . . . 10 ⊢ ((ℤ/nℤ‘𝑅) ∈ CRing → (ℤ/nℤ‘𝑅) ∈ Ring)
8		aks6d1c4.7	. . . . . . . . . . 11 ⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
9	8	zrhrhm 21486	. . . . . . . . . 10 ⊢ ((ℤ/nℤ‘𝑅) ∈ Ring → 𝐿 ∈ (ℤring RingHom (ℤ/nℤ‘𝑅)))
10		zringbas 21428	. . . . . . . . . . 11 ⊢ ℤ = (Base‘ℤring)
11		eqid 2739	. . . . . . . . . . 11 ⊢ (Base‘(ℤ/nℤ‘𝑅)) = (Base‘(ℤ/nℤ‘𝑅))
12	10, 11	rhmf 20455	. . . . . . . . . 10 ⊢ (𝐿 ∈ (ℤring RingHom (ℤ/nℤ‘𝑅)) → 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅)))
13	6, 7, 9, 12	4syl 19	. . . . . . . . 9 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅)))
14	13	ffund 6659	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐿)
15	14	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → Fun 𝐿)
16		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
17		fvelima 6892	. . . . . . 7 ⊢ ((Fun 𝐿 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎)
18	15, 16, 17	syl2anc 590	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎)
19		simpr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ (𝐿‘𝑐) = 𝑎) → (𝐿‘𝑐) = 𝑎)
20	19	eqcomd 2745	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ (𝐿‘𝑐) = 𝑎) → 𝑎 = (𝐿‘𝑐))
21		simpll 772	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝜑)
22		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0)))
23	21, 22	jca 516	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))))
24		ovexd 7391	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚))) ∈ V)
25		aks6d1c4.6	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
26		vex 3435	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑘 ∈ V
27		vex 3435	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑙 ∈ V
28	26, 27	op1std 7941	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 = ⟨𝑘, 𝑙⟩ → (1st ‘𝑚) = 𝑘)
29	28	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = ⟨𝑘, 𝑙⟩ → (𝑃↑(1st ‘𝑚)) = (𝑃↑𝑘))
30	26, 27	op2ndd 7942	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 = ⟨𝑘, 𝑙⟩ → (2nd ‘𝑚) = 𝑙)
31	30	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = ⟨𝑘, 𝑙⟩ → ((𝑁 / 𝑃)↑(2nd ‘𝑚)) = ((𝑁 / 𝑃)↑𝑙))
32	29, 31	oveq12d 7374	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = ⟨𝑘, 𝑙⟩ → ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚))) = ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
33	32	mpompt 7470	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚)))) = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
34	33	eqcomi 2748	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) = (𝑚 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚))))
35	25, 34	eqtri 2762	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐸 = (𝑚 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚))))
36	24, 35	fmptd 7055	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶V)
37	36	ffund 6659	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → Fun 𝐸)
38	37	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → Fun 𝐸)
39		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0)))
40		fvelima 6892	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐸 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐)
41	38, 39, 40	syl2anc 590	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐)
42		simpr 485	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (𝐸‘𝑒) = 𝑐) → (𝐸‘𝑒) = 𝑐)
43	42	eqcomd 2745	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (𝐸‘𝑒) = 𝑐) → 𝑐 = (𝐸‘𝑒))
44	43	oveq1d 7371	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (𝐸‘𝑒) = 𝑐) → (𝑐 gcd 𝑅) = ((𝐸‘𝑒) gcd 𝑅))
45		simplll 780	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝜑)
46		simpr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑒 ∈ (ℕ0 × ℕ0))
47	45, 46	jca 516	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)))
48	35	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝐸 = (𝑚 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚)))))
49		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → 𝑚 = 𝑒)
50	49	fveq2d 6831	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → (1st ‘𝑚) = (1st ‘𝑒))
51	50	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → (𝑃↑(1st ‘𝑚)) = (𝑃↑(1st ‘𝑒)))
52	49	fveq2d 6831	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → (2nd ‘𝑚) = (2nd ‘𝑒))
53	52	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → ((𝑁 / 𝑃)↑(2nd ‘𝑚)) = ((𝑁 / 𝑃)↑(2nd ‘𝑒)))
54	51, 53	oveq12d 7374	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚))) = ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))))
55		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑒 ∈ (ℕ0 × ℕ0))
56		ovexd 7391	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))) ∈ V)
57	48, 54, 55, 56	fvmptd 6943	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑒) = ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))))
58		aks6d1c4.2	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑃 ∈ ℙ)
59		prmnn 16634	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
60	58, 59	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑃 ∈ ℕ)
61	60	nnzd 12541	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑃 ∈ ℤ)
62	61	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑃 ∈ ℤ)
63		xp1st 7963	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑒 ∈ (ℕ0 × ℕ0) → (1st ‘𝑒) ∈ ℕ0)
64	63	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (1st ‘𝑒) ∈ ℕ0)
65	62, 64	zexpcld 14040	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑃↑(1st ‘𝑒)) ∈ ℤ)
66		aks6d1c4.3	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
67	60	nnne0d 12218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑃 ≠ 0)
68		aks6d1c4.1	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑁 ∈ ℕ)
69	68	nnzd 12541	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑁 ∈ ℤ)
70		dvdsval2 16215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑃 ∈ ℤ ∧ 𝑃 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
71	61, 67, 69, 70	syl3anc 1379	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
72	66, 71	mpbid 233	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℤ)
73	72	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑁 / 𝑃) ∈ ℤ)
74		xp2nd 7964	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑒 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑒) ∈ ℕ0)
75	74	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (2nd ‘𝑒) ∈ ℕ0)
76	73, 75	zexpcld 14040	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝑁 / 𝑃)↑(2nd ‘𝑒)) ∈ ℤ)
77	65, 76	zmulcld 12630	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))) ∈ ℤ)
78	57, 77	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑒) ∈ ℤ)
79	57	oveq1d 7371	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑒) gcd 𝑅) = (((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))) gcd 𝑅))
80	2	nnzd 12541	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑅 ∈ ℤ)
81	80	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑅 ∈ ℤ)
82	77, 81	gcdcomd 16474	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))) gcd 𝑅) = (𝑅 gcd ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒)))))
83	80, 61, 69	3jca 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → (𝑅 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ))
84		aks6d1c4.5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
85	69, 80	jca 516	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ))
86		gcdcom 16473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ) → (𝑁 gcd 𝑅) = (𝑅 gcd 𝑁))
87	85, 86	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → (𝑁 gcd 𝑅) = (𝑅 gcd 𝑁))
88		eqeq1 2743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑁 gcd 𝑅) = (𝑅 gcd 𝑁) → ((𝑁 gcd 𝑅) = 1 ↔ (𝑅 gcd 𝑁) = 1))
89	87, 88	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → ((𝑁 gcd 𝑅) = 1 ↔ (𝑅 gcd 𝑁) = 1))
90	89	pm5.74i 272	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 → (𝑁 gcd 𝑅) = 1) ↔ (𝜑 → (𝑅 gcd 𝑁) = 1))
91	84, 90	mpbi 231	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (𝑅 gcd 𝑁) = 1)
92	91, 66	jca 516	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → ((𝑅 gcd 𝑁) = 1 ∧ 𝑃 ∥ 𝑁))
93		rpdvds 16620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑅 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑅 gcd 𝑁) = 1 ∧ 𝑃 ∥ 𝑁)) → (𝑅 gcd 𝑃) = 1)
94	83, 92, 93	syl2anc 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (𝑅 gcd 𝑃) = 1)
95	94	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑅 gcd 𝑃) = 1)
96	95	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd 𝑃) = 1)
97	2	ad2antrr 732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → 𝑅 ∈ ℕ)
98	60	ad2antrr 732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → 𝑃 ∈ ℕ)
99		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → (1st ‘𝑒) ∈ ℕ)
100		rprpwr 16519	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑅 ∈ ℕ ∧ 𝑃 ∈ ℕ ∧ (1st ‘𝑒) ∈ ℕ) → ((𝑅 gcd 𝑃) = 1 → (𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1))
101	97, 98, 99, 100	syl3anc 1379	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → ((𝑅 gcd 𝑃) = 1 → (𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1))
102	96, 101	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1)
103	64	anim1i 621	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ≠ 0) → ((1st ‘𝑒) ∈ ℕ0 ∧ (1st ‘𝑒) ≠ 0))
104		elnnne0 12442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((1st ‘𝑒) ∈ ℕ ↔ ((1st ‘𝑒) ∈ ℕ0 ∧ (1st ‘𝑒) ≠ 0))
105	103, 104	sylibr 235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ≠ 0) → (1st ‘𝑒) ∈ ℕ)
106	105	ex 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((1st ‘𝑒) ≠ 0 → (1st ‘𝑒) ∈ ℕ))
107	106	necon1bd 2952	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (¬ (1st ‘𝑒) ∈ ℕ → (1st ‘𝑒) = 0))
108	107	imp 407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (1st ‘𝑒) = 0)
109	108	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑃↑(1st ‘𝑒)) = (𝑃↑0))
110	109	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑃↑(1st ‘𝑒))) = (𝑅 gcd (𝑃↑0)))
111	62	zcnd 12625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑃 ∈ ℂ)
112	111	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → 𝑃 ∈ ℂ)
113	112	exp0d 14093	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑃↑0) = 1)
114	113	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑃↑0)) = (𝑅 gcd 1))
115	81	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → 𝑅 ∈ ℤ)
116		gcd1 16488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑅 ∈ ℤ → (𝑅 gcd 1) = 1)
117	115, 116	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd 1) = 1)
118	114, 117	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑃↑0)) = 1)
119	110, 118	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1)
120	102, 119	pm2.61dan 818	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1)
121	80, 72, 69	3jca 1134	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → (𝑅 ∈ ℤ ∧ (𝑁 / 𝑃) ∈ ℤ ∧ 𝑁 ∈ ℤ))
122	68	nnred 12180	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → 𝑁 ∈ ℝ)
123	122	recnd 11164	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 𝑁 ∈ ℂ)
124	60	nnred 12180	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → 𝑃 ∈ ℝ)
125	124	recnd 11164	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 𝑃 ∈ ℂ)
126	68	nngt0d 12217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → 0 < 𝑁)
127	126	gt0ne0d 11705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 𝑁 ≠ 0)
128	123, 125, 127, 67	ddcand 11942	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (𝑁 / (𝑁 / 𝑃)) = 𝑃)
129	128, 61	eqeltrd 2839	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝑁 / (𝑁 / 𝑃)) ∈ ℤ)
130	60	nngt0d 12217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → 0 < 𝑃)
131	122, 124, 126, 130	divgt0d 12082	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 0 < (𝑁 / 𝑃))
132	131	gt0ne0d 11705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (𝑁 / 𝑃) ≠ 0)
133		dvdsval2 16215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑁 / 𝑃) ∈ ℤ ∧ (𝑁 / 𝑃) ≠ 0 ∧ 𝑁 ∈ ℤ) → ((𝑁 / 𝑃) ∥ 𝑁 ↔ (𝑁 / (𝑁 / 𝑃)) ∈ ℤ))
134	72, 132, 69, 133	syl3anc 1379	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → ((𝑁 / 𝑃) ∥ 𝑁 ↔ (𝑁 / (𝑁 / 𝑃)) ∈ ℤ))
135	129, 134	mpbird 258	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (𝑁 / 𝑃) ∥ 𝑁)
136	91, 135	jca 516	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → ((𝑅 gcd 𝑁) = 1 ∧ (𝑁 / 𝑃) ∥ 𝑁))
137		rpdvds 16620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑅 ∈ ℤ ∧ (𝑁 / 𝑃) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑅 gcd 𝑁) = 1 ∧ (𝑁 / 𝑃) ∥ 𝑁)) → (𝑅 gcd (𝑁 / 𝑃)) = 1)
138	121, 136, 137	syl2anc 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (𝑅 gcd (𝑁 / 𝑃)) = 1)
139	138	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑅 gcd (𝑁 / 𝑃)) = 1)
140	139	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑁 / 𝑃)) = 1)
141	2	ad2antrr 732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → 𝑅 ∈ ℕ)
142	72, 131	jca 516	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → ((𝑁 / 𝑃) ∈ ℤ ∧ 0 < (𝑁 / 𝑃)))
143		elnnz 12525	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑁 / 𝑃) ∈ ℕ ↔ ((𝑁 / 𝑃) ∈ ℤ ∧ 0 < (𝑁 / 𝑃)))
144	142, 143	sylibr 235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℕ)
145	144	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑁 / 𝑃) ∈ ℕ)
146	145	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → (𝑁 / 𝑃) ∈ ℕ)
147		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → (2nd ‘𝑒) ∈ ℕ)
148		rprpwr 16519	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑅 ∈ ℕ ∧ (𝑁 / 𝑃) ∈ ℕ ∧ (2nd ‘𝑒) ∈ ℕ) → ((𝑅 gcd (𝑁 / 𝑃)) = 1 → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1))
149	141, 146, 147, 148	syl3anc 1379	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → ((𝑅 gcd (𝑁 / 𝑃)) = 1 → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1))
150	140, 149	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1)
151	75	anim1i 621	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ≠ 0) → ((2nd ‘𝑒) ∈ ℕ0 ∧ (2nd ‘𝑒) ≠ 0))
152		elnnne0 12442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((2nd ‘𝑒) ∈ ℕ ↔ ((2nd ‘𝑒) ∈ ℕ0 ∧ (2nd ‘𝑒) ≠ 0))
153	151, 152	sylibr 235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ≠ 0) → (2nd ‘𝑒) ∈ ℕ)
154	153	ex 413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((2nd ‘𝑒) ≠ 0 → (2nd ‘𝑒) ∈ ℕ))
155	154	necon1bd 2952	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (¬ (2nd ‘𝑒) ∈ ℕ → (2nd ‘𝑒) = 0))
156	155	imp 407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (2nd ‘𝑒) = 0)
157	156	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → ((𝑁 / 𝑃)↑(2nd ‘𝑒)) = ((𝑁 / 𝑃)↑0))
158	157	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = (𝑅 gcd ((𝑁 / 𝑃)↑0)))
159	123	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑁 ∈ ℂ)
160	159	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → 𝑁 ∈ ℂ)
161	111	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → 𝑃 ∈ ℂ)
162	67	ad2antrr 732	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → 𝑃 ≠ 0)
163	160, 161, 162	divcld 11922	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑁 / 𝑃) ∈ ℂ)
164	163	exp0d 14093	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → ((𝑁 / 𝑃)↑0) = 1)
165	164	oveq2d 7372	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd ((𝑁 / 𝑃)↑0)) = (𝑅 gcd 1))
166	158, 165	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = (𝑅 gcd 1))
167	81	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → 𝑅 ∈ ℤ)
168	167, 116	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd 1) = 1)
169	166, 168	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1)
170	150, 169	pm2.61dan 818	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1)
171	120, 170	jca 516	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1 ∧ (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1))
172		rpmul 16619	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑅 ∈ ℤ ∧ (𝑃↑(1st ‘𝑒)) ∈ ℤ ∧ ((𝑁 / 𝑃)↑(2nd ‘𝑒)) ∈ ℤ) → (((𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1 ∧ (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1) → (𝑅 gcd ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒)))) = 1))
173	81, 65, 76, 172	syl3anc 1379	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (((𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1 ∧ (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1) → (𝑅 gcd ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒)))) = 1))
174	171, 173	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑅 gcd ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒)))) = 1)
175	82, 174	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))) gcd 𝑅) = 1)
176	79, 175	eqtrd 2774	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑒) gcd 𝑅) = 1)
177	78, 176	jca 516	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑒) ∈ ℤ ∧ ((𝐸‘𝑒) gcd 𝑅) = 1))
178	47, 177	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑒) ∈ ℤ ∧ ((𝐸‘𝑒) gcd 𝑅) = 1))
179	178	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (𝐸‘𝑒) = 𝑐) → ((𝐸‘𝑒) ∈ ℤ ∧ ((𝐸‘𝑒) gcd 𝑅) = 1))
180	179	simprd 496	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (𝐸‘𝑒) = 𝑐) → ((𝐸‘𝑒) gcd 𝑅) = 1)
181	44, 180	eqtrd 2774	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (𝐸‘𝑒) = 𝑐) → (𝑐 gcd 𝑅) = 1)
182	179	simpld 495	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (𝐸‘𝑒) = 𝑐) → (𝐸‘𝑒) ∈ ℤ)
183	43, 182	eqeltrd 2839	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (𝐸‘𝑒) = 𝑐) → 𝑐 ∈ ℤ)
184	181, 183	jca 516	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (𝐸‘𝑒) = 𝑐) → ((𝑐 gcd 𝑅) = 1 ∧ 𝑐 ∈ ℤ))
185		nfv 1921	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑒(𝐸‘𝑑) = 𝑐
186		nfv 1921	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑑(𝐸‘𝑒) = 𝑐
187		fveqeq2 6836	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = 𝑒 → ((𝐸‘𝑑) = 𝑐 ↔ (𝐸‘𝑒) = 𝑐))
188	185, 186, 187	cbvrexw 3282	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐 ↔ ∃𝑒 ∈ (ℕ0 × ℕ0)(𝐸‘𝑒) = 𝑐)
189	188	bilani 505	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) → ∃𝑒 ∈ (ℕ0 × ℕ0)(𝐸‘𝑒) = 𝑐)
190	184, 189	r19.29a 3147	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) → ((𝑐 gcd 𝑅) = 1 ∧ 𝑐 ∈ ℤ))
191	190	ex 413	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐 → ((𝑐 gcd 𝑅) = 1 ∧ 𝑐 ∈ ℤ)))
192	41, 191	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → ((𝑐 gcd 𝑅) = 1 ∧ 𝑐 ∈ ℤ))
193	192	simpld 495	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (𝑐 gcd 𝑅) = 1)
194	3	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝑅 ∈ ℕ0)
195	192	simprd 496	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝑐 ∈ ℤ)
196		eqid 2739	. . . . . . . . . . . . . . 15 ⊢ (Unit‘(ℤ/nℤ‘𝑅)) = (Unit‘(ℤ/nℤ‘𝑅))
197	4, 196, 8	znunit 21538	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℕ0 ∧ 𝑐 ∈ ℤ) → ((𝐿‘𝑐) ∈ (Unit‘(ℤ/nℤ‘𝑅)) ↔ (𝑐 gcd 𝑅) = 1))
198	194, 195, 197	syl2anc 590	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → ((𝐿‘𝑐) ∈ (Unit‘(ℤ/nℤ‘𝑅)) ↔ (𝑐 gcd 𝑅) = 1))
199	193, 198	mpbird 258	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (𝐿‘𝑐) ∈ (Unit‘(ℤ/nℤ‘𝑅)))
200	23, 199	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (𝐿‘𝑐) ∈ (Unit‘(ℤ/nℤ‘𝑅)))
201	200	adantr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ (𝐿‘𝑐) = 𝑎) → (𝐿‘𝑐) ∈ (Unit‘(ℤ/nℤ‘𝑅)))
202	20, 201	eqeltrd 2839	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ (𝐿‘𝑐) = 𝑎) → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅)))
203		nfv 1921	. . . . . . . . . . 11 ⊢ Ⅎ𝑐(𝐿‘𝑏) = 𝑎
204		nfv 1921	. . . . . . . . . . 11 ⊢ Ⅎ𝑏(𝐿‘𝑐) = 𝑎
205		fveqeq2 6836	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑐 → ((𝐿‘𝑏) = 𝑎 ↔ (𝐿‘𝑐) = 𝑎))
206	203, 204, 205	cbvrexw 3282	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎 ↔ ∃𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑐) = 𝑎)
207	206	bilani 505	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) → ∃𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑐) = 𝑎)
208	202, 207	r19.29a 3147	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅)))
209	208	ex 413	. . . . . . 7 ⊢ (𝜑 → (∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎 → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅))))
210	209	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → (∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎 → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅))))
211	18, 210	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅)))
212	211	ex 413	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅))))
213	212	ssrdv 3921	. . 3 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ⊆ (Unit‘(ℤ/nℤ‘𝑅)))
214		hashss 14362	. . 3 ⊢ (((Unit‘(ℤ/nℤ‘𝑅)) ∈ V ∧ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ⊆ (Unit‘(ℤ/nℤ‘𝑅))) → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(Unit‘(ℤ/nℤ‘𝑅))))
215	1, 213, 214	syl2anc 590	. 2 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(Unit‘(ℤ/nℤ‘𝑅))))
216	4, 196	znunithash 21539	. . 3 ⊢ (𝑅 ∈ ℕ → (♯‘(Unit‘(ℤ/nℤ‘𝑅))) = (ϕ‘𝑅))
217	2, 216	syl 17	. 2 ⊢ (𝜑 → (♯‘(Unit‘(ℤ/nℤ‘𝑅))) = (ϕ‘𝑅))
218	215, 217	breqtrd 5098	1 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (ϕ‘𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∃wrex 3063  Vcvv 3431   ⊆ wss 3883  ⟨cop 4561   class class class wbr 5072   ↦ cmpt 5153   × cxp 5616   “ cima 5621  Fun wfun 6479  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929  2^nd c2nd 7930  ℂcc 11027  0cc0 11029  1c1 11030   · cmul 11034   < clt 11170   ≤ cle 11171   / cdiv 11798  ℕcn 12165  ℕ₀cn0 12428  ℤcz 12515  ↑cexp 14014  ♯chash 14283   ∥ cdvds 16212   gcd cgcd 16454  ℙcprime 16631  ϕcphi 16725  Basecbs 17170  Ringcrg 20205  CRingccrg 20206  Unitcui 20326   RingHom crh 20440  ℤ_ringczring 21421  ℤRHomczrh 21474  ℤ/nℤczn 21477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107  ax-addf 11108  ax-mulf 11109

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-er 8633  df-ec 8635  df-qs 8639  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-xnn0 12502  df-z 12516  df-dec 12636  df-uz 12780  df-rp 12934  df-fz 13453  df-fzo 13600  df-fl 13742  df-mod 13820  df-seq 13955  df-exp 14015  df-hash 14284  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-dvds 16213  df-gcd 16455  df-prm 16632  df-phi 16727  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-starv 17226  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ds 17233  df-unif 17234  df-0g 17395  df-imas 17463  df-qus 17464  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-grp 18903  df-minusg 18904  df-sbg 18905  df-mulg 19035  df-subg 19090  df-nsg 19091  df-eqg 19092  df-ghm 19179  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-cring 20208  df-oppr 20308  df-dvdsr 20328  df-unit 20329  df-rhm 20443  df-subrng 20518  df-subrg 20542  df-lmod 20852  df-lss 20922  df-lsp 20962  df-sra 21163  df-rgmod 21164  df-lidl 21201  df-rsp 21202  df-2idl 21243  df-cnfld 21348  df-zring 21422  df-zrh 21478  df-zn 21481

This theorem is referenced by:  aks6d1c7lem1  42665