Theorem aks6d1c4 42949

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 6900	. . 3 ⊢ (𝜑 → (Unit‘(ℤ/nℤ‘𝑅)) ∈ V)
2		aks6d1c4.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ ℕ)
3	2	nnnn0d 12580	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
4		eqid 2765	. . . . . . . . . . . 12 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
5	4	zncrng 21744	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ0 → (ℤ/nℤ‘𝑅) ∈ CRing)
6	3, 5	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ/nℤ‘𝑅) ∈ CRing)
7		crngring 20371	. . . . . . . . . 10 ⊢ ((ℤ/nℤ‘𝑅) ∈ CRing → (ℤ/nℤ‘𝑅) ∈ Ring)
8		aks6d1c4.7	. . . . . . . . . . 11 ⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
9	8	zrhrhm 21711	. . . . . . . . . 10 ⊢ ((ℤ/nℤ‘𝑅) ∈ Ring → 𝐿 ∈ (ℤring RingHom (ℤ/nℤ‘𝑅)))
10		zringbas 21653	. . . . . . . . . . 11 ⊢ ℤ = (Base‘ℤring)
11		eqid 2765	. . . . . . . . . . 11 ⊢ (Base‘(ℤ/nℤ‘𝑅)) = (Base‘(ℤ/nℤ‘𝑅))
12	10, 11	rhmf 20613	. . . . . . . . . 10 ⊢ (𝐿 ∈ (ℤring RingHom (ℤ/nℤ‘𝑅)) → 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅)))
13	6, 7, 9, 12	4syl 20	. . . . . . . . 9 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅)))
14	13	ffund 6714	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐿)
15	14	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → Fun 𝐿)
16		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
17		fvelima 6950	. . . . . . 7 ⊢ ((Fun 𝐿 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎)
18	15, 16, 17	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎)
19		simpr 490	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ (𝐿‘𝑐) = 𝑎) → (𝐿‘𝑐) = 𝑎)
20	19	eqcomd 2771	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ (𝐿‘𝑐) = 𝑎) → 𝑎 = (𝐿‘𝑐))
21		simpll 779	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝜑)
22		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0)))
23	21, 22	jca 521	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))))
24		ovexd 7454	. . . . . . . . . . . . . . . . . . 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 3461	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑘 ∈ V
27		vex 3461	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑙 ∈ V
28	26, 27	op1std 8002	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 = ⟨𝑘, 𝑙⟩ → (1st ‘𝑚) = 𝑘)
29	28	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = ⟨𝑘, 𝑙⟩ → (𝑃↑(1st ‘𝑚)) = (𝑃↑𝑘))
30	26, 27	op2ndd 8003	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 = ⟨𝑘, 𝑙⟩ → (2nd ‘𝑚) = 𝑙)
31	30	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = ⟨𝑘, 𝑙⟩ → ((𝑁 / 𝑃)↑(2nd ‘𝑚)) = ((𝑁 / 𝑃)↑𝑙))
32	29, 31	oveq12d 7437	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = ⟨𝑘, 𝑙⟩ → ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚))) = ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
33	32	mpompt 7533	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚)))) = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
34	33	eqcomi 2774	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) = (𝑚 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚))))
35	25, 34	eqtri 2788	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐸 = (𝑚 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚))))
36	24, 35	fmptd 7113	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶V)
37	36	ffund 6714	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → Fun 𝐸)
38	37	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → Fun 𝐸)
39		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0)))
40		fvelima 6950	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐸 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐)
41	38, 39, 40	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐)
42		simpr 490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (𝐸‘𝑒) = 𝑐) → (𝐸‘𝑒) = 𝑐)
43	42	eqcomd 2771	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (𝐸‘𝑒) = 𝑐) → 𝑐 = (𝐸‘𝑒))
44	43	oveq1d 7434	. . . . . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝜑)
46		simpr 490	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑒 ∈ (ℕ0 × ℕ0))
47	45, 46	jca 521	. . . . . . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → 𝑚 = 𝑒)
50	49	fveq2d 6889	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → (1st ‘𝑚) = (1st ‘𝑒))
51	50	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → (𝑃↑(1st ‘𝑚)) = (𝑃↑(1st ‘𝑒)))
52	49	fveq2d 6889	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → (2nd ‘𝑚) = (2nd ‘𝑒))
53	52	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → ((𝑁 / 𝑃)↑(2nd ‘𝑚)) = ((𝑁 / 𝑃)↑(2nd ‘𝑒)))
54	51, 53	oveq12d 7437	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚))) = ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))))
55		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑒 ∈ (ℕ0 × ℕ0))
56		ovexd 7454	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))) ∈ V)
57	48, 54, 55, 56	fvmptd 7001	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑒) = ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))))
58		aks6d1c4.2	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑃 ∈ ℙ)
59		prmnn 16754	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
60	58, 59	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑃 ∈ ℕ)
61	60	nnzd 12632	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑃 ∈ ℤ)
62	61	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑃 ∈ ℤ)
63		xp1st 8024	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑒 ∈ (ℕ0 × ℕ0) → (1st ‘𝑒) ∈ ℕ0)
64	63	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (1st ‘𝑒) ∈ ℕ0)
65	62, 64	zexpcld 14141	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑃↑(1st ‘𝑒)) ∈ ℤ)
66		aks6d1c4.3	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
67	60	nnne0d 12301	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑃 ≠ 0)
68		aks6d1c4.1	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑁 ∈ ℕ)
69	68	nnzd 12632	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑁 ∈ ℤ)
70		dvdsval2 16335	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑃 ∈ ℤ ∧ 𝑃 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
71	61, 67, 69, 70	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
72	66, 71	mpbid 235	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℤ)
73	72	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑁 / 𝑃) ∈ ℤ)
74		xp2nd 8025	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑒 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑒) ∈ ℕ0)
75	74	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (2nd ‘𝑒) ∈ ℕ0)
76	73, 75	zexpcld 14141	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝑁 / 𝑃)↑(2nd ‘𝑒)) ∈ ℤ)
77	65, 76	zmulcld 12722	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))) ∈ ℤ)
78	57, 77	eqeltrd 2865	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑒) ∈ ℤ)
79	57	oveq1d 7434	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑒) gcd 𝑅) = (((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))) gcd 𝑅))
80	2	nnzd 12632	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑅 ∈ ℤ)
81	80	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑅 ∈ ℤ)
82	77, 81	gcdcomd 16594	. . . . . . . . . . . . . . . . . . . . . . . . 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 1146	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → (𝑅 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ))
84		aks6d1c4.5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
85	69, 80	jca 521	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ))
86		gcdcom 16593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ) → (𝑁 gcd 𝑅) = (𝑅 gcd 𝑁))
87	85, 86	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → (𝑁 gcd 𝑅) = (𝑅 gcd 𝑁))
88		eqeq1 2769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑁 gcd 𝑅) = (𝑅 gcd 𝑁) → ((𝑁 gcd 𝑅) = 1 ↔ (𝑅 gcd 𝑁) = 1))
89	87, 88	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → ((𝑁 gcd 𝑅) = 1 ↔ (𝑅 gcd 𝑁) = 1))
90	89	pm5.74i 274	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 → (𝑁 gcd 𝑅) = 1) ↔ (𝜑 → (𝑅 gcd 𝑁) = 1))
91	84, 90	mpbi 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (𝑅 gcd 𝑁) = 1)
92	91, 66	jca 521	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → ((𝑅 gcd 𝑁) = 1 ∧ 𝑃 ∥ 𝑁))
93		rpdvds 16740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑅 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑅 gcd 𝑁) = 1 ∧ 𝑃 ∥ 𝑁)) → (𝑅 gcd 𝑃) = 1)
94	83, 92, 93	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (𝑅 gcd 𝑃) = 1)
95	94	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑅 gcd 𝑃) = 1)
96	95	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd 𝑃) = 1)
97	2	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → 𝑅 ∈ ℕ)
98	60	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → 𝑃 ∈ ℕ)
99		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → (1st ‘𝑒) ∈ ℕ)
100		rprpwr 16639	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑅 ∈ ℕ ∧ 𝑃 ∈ ℕ ∧ (1st ‘𝑒) ∈ ℕ) → ((𝑅 gcd 𝑃) = 1 → (𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1))
101	97, 98, 99, 100	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → ((𝑅 gcd 𝑃) = 1 → (𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1))
102	96, 101	mpd 16	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1)
103	64	anim1i 627	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ≠ 0) → ((1st ‘𝑒) ∈ ℕ0 ∧ (1st ‘𝑒) ≠ 0))
104		elnnne0 12533	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((1st ‘𝑒) ∈ ℕ ↔ ((1st ‘𝑒) ∈ ℕ0 ∧ (1st ‘𝑒) ≠ 0))
105	103, 104	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ≠ 0) → (1st ‘𝑒) ∈ ℕ)
106	105	ex 418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((1st ‘𝑒) ≠ 0 → (1st ‘𝑒) ∈ ℕ))
107	106	necon1bd 2978	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (¬ (1st ‘𝑒) ∈ ℕ → (1st ‘𝑒) = 0))
108	107	imp 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (1st ‘𝑒) = 0)
109	108	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑃↑(1st ‘𝑒)) = (𝑃↑0))
110	109	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑃↑(1st ‘𝑒))) = (𝑅 gcd (𝑃↑0)))
111	62	zcnd 12717	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑃 ∈ ℂ)
112	111	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → 𝑃 ∈ ℂ)
113	112	exp0d 14194	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑃↑0) = 1)
114	113	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑃↑0)) = (𝑅 gcd 1))
115	81	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → 𝑅 ∈ ℤ)
116		gcd1 16608	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑅 ∈ ℤ → (𝑅 gcd 1) = 1)
117	115, 116	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd 1) = 1)
118	114, 117	eqtrd 2800	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑃↑0)) = 1)
119	110, 118	eqtrd 2800	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1)
120	102, 119	pm2.61dan 825	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1)
121	80, 72, 69	3jca 1146	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → (𝑅 ∈ ℤ ∧ (𝑁 / 𝑃) ∈ ℤ ∧ 𝑁 ∈ ℤ))
122	68	nnred 12263	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → 𝑁 ∈ ℝ)
123	122	recnd 11252	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 𝑁 ∈ ℂ)
124	60	nnred 12263	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → 𝑃 ∈ ℝ)
125	124	recnd 11252	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 𝑃 ∈ ℂ)
126	68	nngt0d 12300	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → 0 < 𝑁)
127	126	gt0ne0d 11793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 𝑁 ≠ 0)
128	123, 125, 127, 67	ddcand 12026	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (𝑁 / (𝑁 / 𝑃)) = 𝑃)
129	128, 61	eqeltrd 2865	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝑁 / (𝑁 / 𝑃)) ∈ ℤ)
130	60	nngt0d 12300	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → 0 < 𝑃)
131	122, 124, 126, 130	divgt0d 12165	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 0 < (𝑁 / 𝑃))
132	131	gt0ne0d 11793	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (𝑁 / 𝑃) ≠ 0)
133		dvdsval2 16335	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑁 / 𝑃) ∈ ℤ ∧ (𝑁 / 𝑃) ≠ 0 ∧ 𝑁 ∈ ℤ) → ((𝑁 / 𝑃) ∥ 𝑁 ↔ (𝑁 / (𝑁 / 𝑃)) ∈ ℤ))
134	72, 132, 69, 133	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → ((𝑁 / 𝑃) ∥ 𝑁 ↔ (𝑁 / (𝑁 / 𝑃)) ∈ ℤ))
135	129, 134	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (𝑁 / 𝑃) ∥ 𝑁)
136	91, 135	jca 521	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → ((𝑅 gcd 𝑁) = 1 ∧ (𝑁 / 𝑃) ∥ 𝑁))
137		rpdvds 16740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (𝑅 gcd (𝑁 / 𝑃)) = 1)
139	138	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑅 gcd (𝑁 / 𝑃)) = 1)
140	139	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑁 / 𝑃)) = 1)
141	2	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → 𝑅 ∈ ℕ)
142	72, 131	jca 521	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → ((𝑁 / 𝑃) ∈ ℤ ∧ 0 < (𝑁 / 𝑃)))
143		elnnz 12616	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑁 / 𝑃) ∈ ℕ ↔ ((𝑁 / 𝑃) ∈ ℤ ∧ 0 < (𝑁 / 𝑃)))
144	142, 143	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℕ)
145	144	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑁 / 𝑃) ∈ ℕ)
146	145	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → (𝑁 / 𝑃) ∈ ℕ)
147		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → (2nd ‘𝑒) ∈ ℕ)
148		rprpwr 16639	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → ((𝑅 gcd (𝑁 / 𝑃)) = 1 → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1))
150	140, 149	mpd 16	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1)
151	75	anim1i 627	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ≠ 0) → ((2nd ‘𝑒) ∈ ℕ0 ∧ (2nd ‘𝑒) ≠ 0))
152		elnnne0 12533	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((2nd ‘𝑒) ∈ ℕ ↔ ((2nd ‘𝑒) ∈ ℕ0 ∧ (2nd ‘𝑒) ≠ 0))
153	151, 152	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ≠ 0) → (2nd ‘𝑒) ∈ ℕ)
154	153	ex 418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((2nd ‘𝑒) ≠ 0 → (2nd ‘𝑒) ∈ ℕ))
155	154	necon1bd 2978	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (¬ (2nd ‘𝑒) ∈ ℕ → (2nd ‘𝑒) = 0))
156	155	imp 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (2nd ‘𝑒) = 0)
157	156	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → ((𝑁 / 𝑃)↑(2nd ‘𝑒)) = ((𝑁 / 𝑃)↑0))
158	157	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = (𝑅 gcd ((𝑁 / 𝑃)↑0)))
159	123	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑁 ∈ ℂ)
160	159	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → 𝑁 ∈ ℂ)
161	111	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → 𝑃 ∈ ℂ)
162	67	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → 𝑃 ≠ 0)
163	160, 161, 162	divcld 12006	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑁 / 𝑃) ∈ ℂ)
164	163	exp0d 14194	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → ((𝑁 / 𝑃)↑0) = 1)
165	164	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd ((𝑁 / 𝑃)↑0)) = (𝑅 gcd 1))
166	158, 165	eqtrd 2800	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = (𝑅 gcd 1))
167	81	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → 𝑅 ∈ ℤ)
168	167, 116	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd 1) = 1)
169	166, 168	eqtrd 2800	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1)
170	150, 169	pm2.61dan 825	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1)
171	120, 170	jca 521	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1 ∧ (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1))
172		rpmul 16739	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑅 gcd ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒)))) = 1)
175	82, 174	eqtrd 2800	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))) gcd 𝑅) = 1)
176	79, 175	eqtrd 2800	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑒) gcd 𝑅) = 1)
177	78, 176	jca 521	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑒) ∈ ℤ ∧ ((𝐸‘𝑒) gcd 𝑅) = 1))
178	47, 177	syl 18	. . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . 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 501	. . . . . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (𝐸‘𝑒) = 𝑐) → 𝑐 ∈ ℤ)
184	181, 183	jca 521	. . . . . . . . . . . . . . . . 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 1947	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑒(𝐸‘𝑑) = 𝑐
186		nfv 1947	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑑(𝐸‘𝑒) = 𝑐
187		fveqeq2 6894	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = 𝑒 → ((𝐸‘𝑑) = 𝑐 ↔ (𝐸‘𝑒) = 𝑐))
188	185, 186, 187	cbvrexw 3310	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐 ↔ ∃𝑒 ∈ (ℕ0 × ℕ0)(𝐸‘𝑒) = 𝑐)
189	188	bilani 510	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) → ∃𝑒 ∈ (ℕ0 × ℕ0)(𝐸‘𝑒) = 𝑐)
190	184, 189	r19.29a 3175	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) → ((𝑐 gcd 𝑅) = 1 ∧ 𝑐 ∈ ℤ))
191	190	ex 418	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐 → ((𝑐 gcd 𝑅) = 1 ∧ 𝑐 ∈ ℤ)))
192	41, 191	mpd 16	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → ((𝑐 gcd 𝑅) = 1 ∧ 𝑐 ∈ ℤ))
193	192	simpld 500	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (𝑐 gcd 𝑅) = 1)
194	3	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝑅 ∈ ℕ0)
195	192	simprd 501	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝑐 ∈ ℤ)
196		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ (Unit‘(ℤ/nℤ‘𝑅)) = (Unit‘(ℤ/nℤ‘𝑅))
197	4, 196, 8	znunit 21763	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℕ0 ∧ 𝑐 ∈ ℤ) → ((𝐿‘𝑐) ∈ (Unit‘(ℤ/nℤ‘𝑅)) ↔ (𝑐 gcd 𝑅) = 1))
198	194, 195, 197	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → ((𝐿‘𝑐) ∈ (Unit‘(ℤ/nℤ‘𝑅)) ↔ (𝑐 gcd 𝑅) = 1))
199	193, 198	mpbird 260	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (𝐿‘𝑐) ∈ (Unit‘(ℤ/nℤ‘𝑅)))
200	23, 199	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (𝐿‘𝑐) ∈ (Unit‘(ℤ/nℤ‘𝑅)))
201	200	adantr 486	. . . . . . . . . 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 2865	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ (𝐿‘𝑐) = 𝑎) → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅)))
203		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑐(𝐿‘𝑏) = 𝑎
204		nfv 1947	. . . . . . . . . . 11 ⊢ Ⅎ𝑏(𝐿‘𝑐) = 𝑎
205		fveqeq2 6894	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑐 → ((𝐿‘𝑏) = 𝑎 ↔ (𝐿‘𝑐) = 𝑎))
206	203, 204, 205	cbvrexw 3310	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎 ↔ ∃𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑐) = 𝑎)
207	206	bilani 510	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) → ∃𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑐) = 𝑎)
208	202, 207	r19.29a 3175	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅)))
209	208	ex 418	. . . . . . 7 ⊢ (𝜑 → (∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎 → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅))))
210	209	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → (∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎 → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅))))
211	18, 210	mpd 16	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅)))
212	211	ex 418	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅))))
213	212	ssrdv 3944	. . 3 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ⊆ (Unit‘(ℤ/nℤ‘𝑅)))
214		hashss 14463	. . 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 596	. 2 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(Unit‘(ℤ/nℤ‘𝑅))))
216	4, 196	znunithash 21764	. . 3 ⊢ (𝑅 ∈ ℕ → (♯‘(Unit‘(ℤ/nℤ‘𝑅))) = (ϕ‘𝑅))
217	2, 216	syl 18	. 2 ⊢ (𝜑 → (♯‘(Unit‘(ℤ/nℤ‘𝑅))) = (ϕ‘𝑅))
218	215, 217	breqtrd 5139	1 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (ϕ‘𝑅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∃wrex 3091  Vcvv 3457   ⊆ wss 3906  ⟨cop 4597   class class class wbr 5111   ↦ cmpt 5194   × cxp 5661   “ cima 5666  Fun wfun 6534  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  1^st c1st 7990  2^nd c2nd 7991  ℂcc 11113  0cc0 11115  1c1 11116   · cmul 11120   < clt 11258   ≤ cle 11259   / cdiv 11886  ℕcn 12248  ℕ₀cn0 12519  ℤcz 12606  ↑cexp 14115  ♯chash 14384   ∥ cdvds 16332   gcd cgcd 16574  ℙcprime 16751  ϕcphi 16845  Basecbs 17291  Ringcrg 20359  CRingccrg 20360  Unitcui 20483   RingHom crh 20597  ℤ_ringczring 21646  ℤRHomczrh 21699  ℤ/nℤczn 21702

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192  ax-pre-sup 11193  ax-addf 11194  ax-mulf 11195

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-tpos 8228  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-oadd 8463  df-er 8700  df-ec 8702  df-qs 8706  df-map 8832  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-sup 9409  df-inf 9410  df-card 9941  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-div 11887  df-nn 12249  df-2 12318  df-3 12319  df-4 12320  df-5 12321  df-6 12322  df-7 12323  df-8 12324  df-9 12325  df-n0 12520  df-xnn0 12593  df-z 12607  df-dec 12728  df-uz 12879  df-rp 13033  df-fz 13552  df-fzo 13700  df-fl 13843  df-mod 13921  df-seq 14056  df-exp 14116  df-hash 14385  df-cj 15174  df-re 15175  df-im 15176  df-sqrt 15310  df-abs 15311  df-dvds 16333  df-gcd 16575  df-prm 16752  df-phi 16847  df-struct 17229  df-sets 17246  df-slot 17264  df-ndx 17276  df-base 17292  df-ress 17313  df-plusg 17345  df-mulr 17346  df-starv 17347  df-sca 17348  df-vsca 17349  df-ip 17350  df-tset 17351  df-ple 17352  df-ds 17354  df-unif 17355  df-0g 17516  df-imas 17584  df-qus 17585  df-mgm 18720  df-sgrp 18809  df-mnd 18825  df-mhm 18878  df-grp 19047  df-minusg 19048  df-sbg 19049  df-mulg 19178  df-subg 19233  df-nsg 19234  df-eqg 19235  df-ghm 19328  df-cmn 19896  df-abl 19897  df-mgp 20261  df-rng 20275  df-ur 20308  df-ring 20361  df-cring 20362  df-oppr 20465  df-dvdsr 20485  df-unit 20486  df-rhm 20600  df-subrng 20695  df-subrg 20719  df-lmod 21033  df-lss 21103  df-lsp 21143  df-sra 21344  df-rgmod 21345  df-lidl 21382  df-rsp 21383  df-2idl 21439  df-cnfld 21573  df-zring 21647  df-zrh 21703  df-zn 21706

This theorem is used by:  aks6d1c7lem1  43005