Theorem aks6d1c4 42317

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 6847	. . 3 ⊢ (𝜑 → (Unit‘(ℤ/nℤ‘𝑅)) ∈ V)
2		aks6d1c4.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ ℕ)
3	2	nnnn0d 12460	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
4		eqid 2734	. . . . . . . . . . . 12 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
5	4	zncrng 21497	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ0 → (ℤ/nℤ‘𝑅) ∈ CRing)
6	3, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ/nℤ‘𝑅) ∈ CRing)
7		crngring 20178	. . . . . . . . . 10 ⊢ ((ℤ/nℤ‘𝑅) ∈ CRing → (ℤ/nℤ‘𝑅) ∈ Ring)
8		aks6d1c4.7	. . . . . . . . . . 11 ⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
9	8	zrhrhm 21464	. . . . . . . . . 10 ⊢ ((ℤ/nℤ‘𝑅) ∈ Ring → 𝐿 ∈ (ℤring RingHom (ℤ/nℤ‘𝑅)))
10		zringbas 21406	. . . . . . . . . . 11 ⊢ ℤ = (Base‘ℤring)
11		eqid 2734	. . . . . . . . . . 11 ⊢ (Base‘(ℤ/nℤ‘𝑅)) = (Base‘(ℤ/nℤ‘𝑅))
12	10, 11	rhmf 20418	. . . . . . . . . 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 6664	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐿)
15	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → Fun 𝐿)
16		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
17		fvelima 6897	. . . . . . 7 ⊢ ((Fun 𝐿 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎)
18	15, 16, 17	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎)
19		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ (𝐿‘𝑐) = 𝑎) → (𝐿‘𝑐) = 𝑎)
20	19	eqcomd 2740	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ (𝐿‘𝑐) = 𝑎) → 𝑎 = (𝐿‘𝑐))
21		simpll 766	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝜑)
22		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0)))
23	21, 22	jca 511	. . . . . . . . . . . 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 3442	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑘 ∈ V
27		vex 3442	. . . . . . . . . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) = (𝑚 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚))))
35	25, 34	eqtri 2757	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐸 = (𝑚 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚))))
36	24, 35	fmptd 7057	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶V)
37	36	ffund 6664	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → Fun 𝐸)
38	37	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → Fun 𝐸)
39		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0)))
40		fvelima 6897	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐸 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐)
41	38, 39, 40	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐)
42		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (𝐸‘𝑒) = 𝑐) → (𝐸‘𝑒) = 𝑐)
43	42	eqcomd 2740	. . . . . . . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝜑)
46		simpr 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑒 ∈ (ℕ0 × ℕ0))
47	45, 46	jca 511	. . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → 𝑚 = 𝑒)
50	49	fveq2d 6836	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . 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 6946	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑒) = ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))))
58		aks6d1c4.2	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑃 ∈ ℙ)
59		prmnn 16599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
60	58, 59	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑃 ∈ ℕ)
61	60	nnzd 12512	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑃 ∈ ℤ)
62	61	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑃 ∈ ℤ)
63		xp1st 7963	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑒 ∈ (ℕ0 × ℕ0) → (1st ‘𝑒) ∈ ℕ0)
64	63	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (1st ‘𝑒) ∈ ℕ0)
65	62, 64	zexpcld 14008	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑃↑(1st ‘𝑒)) ∈ ℤ)
66		aks6d1c4.3	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
67	60	nnne0d 12193	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑃 ≠ 0)
68		aks6d1c4.1	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑁 ∈ ℕ)
69	68	nnzd 12512	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑁 ∈ ℤ)
70		dvdsval2 16180	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑃 ∈ ℤ ∧ 𝑃 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
71	61, 67, 69, 70	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (𝑃 ∥ 𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
72	66, 71	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℤ)
73	72	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑁 / 𝑃) ∈ ℤ)
74		xp2nd 7964	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑒 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑒) ∈ ℕ0)
75	74	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (2nd ‘𝑒) ∈ ℕ0)
76	73, 75	zexpcld 14008	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝑁 / 𝑃)↑(2nd ‘𝑒)) ∈ ℤ)
77	65, 76	zmulcld 12600	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))) ∈ ℤ)
78	57, 77	eqeltrd 2834	. . . . . . . . . . . . . . . . . . . . . . 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 12512	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑅 ∈ ℤ)
81	80	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑅 ∈ ℤ)
82	77, 81	gcdcomd 16439	. . . . . . . . . . . . . . . . . . . . . . . . 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 1128	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → (𝑅 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ))
84		aks6d1c4.5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
85	69, 80	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ))
86		gcdcom 16438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 271	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 → (𝑁 gcd 𝑅) = 1) ↔ (𝜑 → (𝑅 gcd 𝑁) = 1))
91	84, 90	mpbi 230	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (𝑅 gcd 𝑁) = 1)
92	91, 66	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → ((𝑅 gcd 𝑁) = 1 ∧ 𝑃 ∥ 𝑁))
93		rpdvds 16585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑅 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑅 gcd 𝑁) = 1 ∧ 𝑃 ∥ 𝑁)) → (𝑅 gcd 𝑃) = 1)
94	83, 92, 93	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (𝑅 gcd 𝑃) = 1)
95	94	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑅 gcd 𝑃) = 1)
96	95	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd 𝑃) = 1)
97	2	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → 𝑅 ∈ ℕ)
98	60	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → 𝑃 ∈ ℕ)
99		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → (1st ‘𝑒) ∈ ℕ)
100		rprpwr 16484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑅 ∈ ℕ ∧ 𝑃 ∈ ℕ ∧ (1st ‘𝑒) ∈ ℕ) → ((𝑅 gcd 𝑃) = 1 → (𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1))
101	97, 98, 99, 100	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 615	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ≠ 0) → ((1st ‘𝑒) ∈ ℕ0 ∧ (1st ‘𝑒) ≠ 0))
104		elnnne0 12413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((1st ‘𝑒) ∈ ℕ ↔ ((1st ‘𝑒) ∈ ℕ0 ∧ (1st ‘𝑒) ≠ 0))
105	103, 104	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ≠ 0) → (1st ‘𝑒) ∈ ℕ)
106	105	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((1st ‘𝑒) ≠ 0 → (1st ‘𝑒) ∈ ℕ))
107	106	necon1bd 2948	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (¬ (1st ‘𝑒) ∈ ℕ → (1st ‘𝑒) = 0))
108	107	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑃 ∈ ℂ)
112	111	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → 𝑃 ∈ ℂ)
113	112	exp0d 14061	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → 𝑅 ∈ ℤ)
116		gcd1 16453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑃↑0)) = 1)
119	110, 118	eqtrd 2769	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1)
120	102, 119	pm2.61dan 812	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1)
121	80, 72, 69	3jca 1128	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → (𝑅 ∈ ℤ ∧ (𝑁 / 𝑃) ∈ ℤ ∧ 𝑁 ∈ ℤ))
122	68	nnred 12158	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → 𝑁 ∈ ℝ)
123	122	recnd 11158	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 𝑁 ∈ ℂ)
124	60	nnred 12158	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → 𝑃 ∈ ℝ)
125	124	recnd 11158	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 𝑃 ∈ ℂ)
126	68	nngt0d 12192	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → 0 < 𝑁)
127	126	gt0ne0d 11699	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 𝑁 ≠ 0)
128	123, 125, 127, 67	ddcand 11935	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (𝑁 / (𝑁 / 𝑃)) = 𝑃)
129	128, 61	eqeltrd 2834	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝑁 / (𝑁 / 𝑃)) ∈ ℤ)
130	60	nngt0d 12192	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → 0 < 𝑃)
131	122, 124, 126, 130	divgt0d 12075	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 0 < (𝑁 / 𝑃))
132	131	gt0ne0d 11699	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (𝑁 / 𝑃) ≠ 0)
133		dvdsval2 16180	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝑁 / 𝑃) ∈ ℤ ∧ (𝑁 / 𝑃) ≠ 0 ∧ 𝑁 ∈ ℤ) → ((𝑁 / 𝑃) ∥ 𝑁 ↔ (𝑁 / (𝑁 / 𝑃)) ∈ ℤ))
134	72, 132, 69, 133	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → ((𝑁 / 𝑃) ∥ 𝑁 ↔ (𝑁 / (𝑁 / 𝑃)) ∈ ℤ))
135	129, 134	mpbird 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (𝑁 / 𝑃) ∥ 𝑁)
136	91, 135	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → ((𝑅 gcd 𝑁) = 1 ∧ (𝑁 / 𝑃) ∥ 𝑁))
137		rpdvds 16585	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (𝑅 gcd (𝑁 / 𝑃)) = 1)
139	138	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑅 gcd (𝑁 / 𝑃)) = 1)
140	139	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑁 / 𝑃)) = 1)
141	2	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → 𝑅 ∈ ℕ)
142	72, 131	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → ((𝑁 / 𝑃) ∈ ℤ ∧ 0 < (𝑁 / 𝑃)))
143		elnnz 12496	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑁 / 𝑃) ∈ ℕ ↔ ((𝑁 / 𝑃) ∈ ℤ ∧ 0 < (𝑁 / 𝑃)))
144	142, 143	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℕ)
145	144	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑁 / 𝑃) ∈ ℕ)
146	145	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → (𝑁 / 𝑃) ∈ ℕ)
147		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → (2nd ‘𝑒) ∈ ℕ)
148		rprpwr 16484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 615	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ≠ 0) → ((2nd ‘𝑒) ∈ ℕ0 ∧ (2nd ‘𝑒) ≠ 0))
152		elnnne0 12413	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((2nd ‘𝑒) ∈ ℕ ↔ ((2nd ‘𝑒) ∈ ℕ0 ∧ (2nd ‘𝑒) ≠ 0))
153	151, 152	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ≠ 0) → (2nd ‘𝑒) ∈ ℕ)
154	153	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((2nd ‘𝑒) ≠ 0 → (2nd ‘𝑒) ∈ ℕ))
155	154	necon1bd 2948	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (¬ (2nd ‘𝑒) ∈ ℕ → (2nd ‘𝑒) = 0))
156	155	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑁 ∈ ℂ)
160	159	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → 𝑁 ∈ ℂ)
161	111	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → 𝑃 ∈ ℂ)
162	67	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → 𝑃 ≠ 0)
163	160, 161, 162	divcld 11915	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑁 / 𝑃) ∈ ℂ)
164	163	exp0d 14061	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = (𝑅 gcd 1))
167	81	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1)
170	150, 169	pm2.61dan 812	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1)
171	120, 170	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1 ∧ (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1))
172		rpmul 16584	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))) gcd 𝑅) = 1)
176	79, 175	eqtrd 2769	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑒) gcd 𝑅) = 1)
177	78, 176	jca 511	. . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (𝐸‘𝑒) = 𝑐) → 𝑐 ∈ ℤ)
184	181, 183	jca 511	. . . . . . . . . . . . . . . . 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 1915	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑒(𝐸‘𝑑) = 𝑐
186		nfv 1915	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑑(𝐸‘𝑒) = 𝑐
187		fveqeq2 6841	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 = 𝑒 → ((𝐸‘𝑑) = 𝑐 ↔ (𝐸‘𝑒) = 𝑐))
188	185, 186, 187	cbvrexw 3277	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐 ↔ ∃𝑒 ∈ (ℕ0 × ℕ0)(𝐸‘𝑒) = 𝑐)
189	188	biimpi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐 → ∃𝑒 ∈ (ℕ0 × ℕ0)(𝐸‘𝑒) = 𝑐)
190	189	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) → ∃𝑒 ∈ (ℕ0 × ℕ0)(𝐸‘𝑒) = 𝑐)
191	184, 190	r19.29a 3142	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) → ((𝑐 gcd 𝑅) = 1 ∧ 𝑐 ∈ ℤ))
192	191	ex 412	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐 → ((𝑐 gcd 𝑅) = 1 ∧ 𝑐 ∈ ℤ)))
193	41, 192	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → ((𝑐 gcd 𝑅) = 1 ∧ 𝑐 ∈ ℤ))
194	193	simpld 494	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (𝑐 gcd 𝑅) = 1)
195	3	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝑅 ∈ ℕ0)
196	193	simprd 495	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝑐 ∈ ℤ)
197		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (Unit‘(ℤ/nℤ‘𝑅)) = (Unit‘(ℤ/nℤ‘𝑅))
198	4, 197, 8	znunit 21516	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℕ0 ∧ 𝑐 ∈ ℤ) → ((𝐿‘𝑐) ∈ (Unit‘(ℤ/nℤ‘𝑅)) ↔ (𝑐 gcd 𝑅) = 1))
199	195, 196, 198	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → ((𝐿‘𝑐) ∈ (Unit‘(ℤ/nℤ‘𝑅)) ↔ (𝑐 gcd 𝑅) = 1))
200	194, 199	mpbird 257	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (𝐿‘𝑐) ∈ (Unit‘(ℤ/nℤ‘𝑅)))
201	23, 200	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (𝐿‘𝑐) ∈ (Unit‘(ℤ/nℤ‘𝑅)))
202	201	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ (𝐿‘𝑐) = 𝑎) → (𝐿‘𝑐) ∈ (Unit‘(ℤ/nℤ‘𝑅)))
203	20, 202	eqeltrd 2834	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ (𝐿‘𝑐) = 𝑎) → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅)))
204		nfv 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑐(𝐿‘𝑏) = 𝑎
205		nfv 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑏(𝐿‘𝑐) = 𝑎
206		fveqeq2 6841	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑐 → ((𝐿‘𝑏) = 𝑎 ↔ (𝐿‘𝑐) = 𝑎))
207	204, 205, 206	cbvrexw 3277	. . . . . . . . . . 11 ⊢ (∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎 ↔ ∃𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑐) = 𝑎)
208	207	biimpi 216	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎 → ∃𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑐) = 𝑎)
209	208	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) → ∃𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑐) = 𝑎)
210	203, 209	r19.29a 3142	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅)))
211	210	ex 412	. . . . . . 7 ⊢ (𝜑 → (∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎 → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅))))
212	211	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → (∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎 → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅))))
213	18, 212	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅)))
214	213	ex 412	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅))))
215	214	ssrdv 3937	. . 3 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ⊆ (Unit‘(ℤ/nℤ‘𝑅)))
216		hashss 14330	. . 3 ⊢ (((Unit‘(ℤ/nℤ‘𝑅)) ∈ V ∧ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ⊆ (Unit‘(ℤ/nℤ‘𝑅))) → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(Unit‘(ℤ/nℤ‘𝑅))))
217	1, 215, 216	syl2anc 584	. 2 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(Unit‘(ℤ/nℤ‘𝑅))))
218	4, 197	znunithash 21517	. . 3 ⊢ (𝑅 ∈ ℕ → (♯‘(Unit‘(ℤ/nℤ‘𝑅))) = (ϕ‘𝑅))
219	2, 218	syl 17	. 2 ⊢ (𝜑 → (♯‘(Unit‘(ℤ/nℤ‘𝑅))) = (ϕ‘𝑅))
220	217, 219	breqtrd 5122	1 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (ϕ‘𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∃wrex 3058  Vcvv 3438   ⊆ wss 3899  ⟨cop 4584   class class class wbr 5096   ↦ cmpt 5177   × cxp 5620   “ cima 5625  Fun wfun 6484  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7929  2^nd c2nd 7930  ℂcc 11022  0cc0 11024  1c1 11025   · cmul 11029   < clt 11164   ≤ cle 11165   / cdiv 11792  ℕcn 12143  ℕ₀cn0 12399  ℤcz 12486  ↑cexp 13982  ♯chash 14251   ∥ cdvds 16177   gcd cgcd 16419  ℙcprime 16596  ϕcphi 16689  Basecbs 17134  Ringcrg 20166  CRingccrg 20167  Unitcui 20289   RingHom crh 20403  ℤ_ringczring 21399  ℤRHomczrh 21452  ℤ/nℤczn 21455

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102  ax-addf 11103  ax-mulf 11104

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  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 8763  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-sup 9343  df-inf 9344  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-9 12213  df-n0 12400  df-xnn0 12473  df-z 12487  df-dec 12606  df-uz 12750  df-rp 12904  df-fz 13422  df-fzo 13569  df-fl 13710  df-mod 13788  df-seq 13923  df-exp 13983  df-hash 14252  df-cj 15020  df-re 15021  df-im 15022  df-sqrt 15156  df-abs 15157  df-dvds 16178  df-gcd 16420  df-prm 16597  df-phi 16691  df-struct 17072  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-plusg 17188  df-mulr 17189  df-starv 17190  df-sca 17191  df-vsca 17192  df-ip 17193  df-tset 17194  df-ple 17195  df-ds 17197  df-unif 17198  df-0g 17359  df-imas 17427  df-qus 17428  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-mhm 18706  df-grp 18864  df-minusg 18865  df-sbg 18866  df-mulg 18996  df-subg 19051  df-nsg 19052  df-eqg 19053  df-ghm 19140  df-cmn 19709  df-abl 19710  df-mgp 20074  df-rng 20086  df-ur 20115  df-ring 20168  df-cring 20169  df-oppr 20271  df-dvdsr 20291  df-unit 20292  df-rhm 20406  df-subrng 20477  df-subrg 20501  df-lmod 20811  df-lss 20881  df-lsp 20921  df-sra 21123  df-rgmod 21124  df-lidl 21161  df-rsp 21162  df-2idl 21203  df-cnfld 21308  df-zring 21400  df-zrh 21456  df-zn 21459

This theorem is referenced by:  aks6d1c7lem1  42373