Theorem aks6d1c4 42891

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 6896	. . 3 ⊢ (𝜑 → (Unit‘(ℤ/nℤ‘𝑅)) ∈ V)
2		aks6d1c4.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ ℕ)
3	2	nnnn0d 12560	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
4		eqid 2763	. . . . . . . . . . . 12 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
5	4	zncrng 21694	. . . . . . . . . . 11 ⊢ (𝑅 ∈ ℕ0 → (ℤ/nℤ‘𝑅) ∈ CRing)
6	3, 5	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ/nℤ‘𝑅) ∈ CRing)
7		crngring 20322	. . . . . . . . . 10 ⊢ ((ℤ/nℤ‘𝑅) ∈ CRing → (ℤ/nℤ‘𝑅) ∈ Ring)
8		aks6d1c4.7	. . . . . . . . . . 11 ⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
9	8	zrhrhm 21661	. . . . . . . . . 10 ⊢ ((ℤ/nℤ‘𝑅) ∈ Ring → 𝐿 ∈ (ℤring RingHom (ℤ/nℤ‘𝑅)))
10		zringbas 21603	. . . . . . . . . . 11 ⊢ ℤ = (Base‘ℤring)
11		eqid 2763	. . . . . . . . . . 11 ⊢ (Base‘(ℤ/nℤ‘𝑅)) = (Base‘(ℤ/nℤ‘𝑅))
12	10, 11	rhmf 20563	. . . . . . . . . 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 6710	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐿)
15	14	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → Fun 𝐿)
16		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
17		fvelima 6946	. . . . . . 7 ⊢ ((Fun 𝐿 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎)
18	15, 16, 17	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) → ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎)
19		simpr 489	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ (𝐿‘𝑐) = 𝑎) → (𝐿‘𝑐) = 𝑎)
20	19	eqcomd 2769	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ (𝐿‘𝑐) = 𝑎) → 𝑎 = (𝐿‘𝑐))
21		simpll 778	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝜑)
22		simpr 489	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0)))
23	21, 22	jca 520	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))))
24		ovexd 7445	. . . . . . . . . . . . . . . . . . 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 3459	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑘 ∈ V
27		vex 3459	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑙 ∈ V
28	26, 27	op1std 7992	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 = ⟨𝑘, 𝑙⟩ → (1st ‘𝑚) = 𝑘)
29	28	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = ⟨𝑘, 𝑙⟩ → (𝑃↑(1st ‘𝑚)) = (𝑃↑𝑘))
30	26, 27	op2ndd 7993	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 = ⟨𝑘, 𝑙⟩ → (2nd ‘𝑚) = 𝑙)
31	30	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = ⟨𝑘, 𝑙⟩ → ((𝑁 / 𝑃)↑(2nd ‘𝑚)) = ((𝑁 / 𝑃)↑𝑙))
32	29, 31	oveq12d 7428	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = ⟨𝑘, 𝑙⟩ → ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚))) = ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
33	32	mpompt 7524	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚)))) = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
34	33	eqcomi 2772	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙))) = (𝑚 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚))))
35	25, 34	eqtri 2786	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐸 = (𝑚 ∈ (ℕ0 × ℕ0) ↦ ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚))))
36	24, 35	fmptd 7109	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶V)
37	36	ffund 6710	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → Fun 𝐸)
38	37	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → Fun 𝐸)
39		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0)))
40		fvelima 6946	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝐸 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐)
41	38, 39, 40	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐)
42		simpr 489	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (𝐸‘𝑒) = 𝑐) → (𝐸‘𝑒) = 𝑐)
43	42	eqcomd 2769	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (𝐸‘𝑒) = 𝑐) → 𝑐 = (𝐸‘𝑒))
44	43	oveq1d 7425	. . . . . . . . . . . . . . . . . . 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 786	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝜑)
46		simpr 489	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑒 ∈ (ℕ0 × ℕ0))
47	45, 46	jca 520	. . . . . . . . . . . . . . . . . . . . . 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 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → 𝑚 = 𝑒)
50	49	fveq2d 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → (1st ‘𝑚) = (1st ‘𝑒))
51	50	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → (𝑃↑(1st ‘𝑚)) = (𝑃↑(1st ‘𝑒)))
52	49	fveq2d 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → (2nd ‘𝑚) = (2nd ‘𝑒))
53	52	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → ((𝑁 / 𝑃)↑(2nd ‘𝑚)) = ((𝑁 / 𝑃)↑(2nd ‘𝑒)))
54	51, 53	oveq12d 7428	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ 𝑚 = 𝑒) → ((𝑃↑(1st ‘𝑚)) · ((𝑁 / 𝑃)↑(2nd ‘𝑚))) = ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))))
55		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑒 ∈ (ℕ0 × ℕ0))
56		ovexd 7445	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))) ∈ V)
57	48, 54, 55, 56	fvmptd 6997	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑒) = ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))))
58		aks6d1c4.2	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑃 ∈ ℙ)
59		prmnn 16727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
60	58, 59	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑃 ∈ ℕ)
61	60	nnzd 12612	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑃 ∈ ℤ)
62	61	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑃 ∈ ℤ)
63		xp1st 8014	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑒 ∈ (ℕ0 × ℕ0) → (1st ‘𝑒) ∈ ℕ0)
64	63	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (1st ‘𝑒) ∈ ℕ0)
65	62, 64	zexpcld 14119	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑃↑(1st ‘𝑒)) ∈ ℤ)
66		aks6d1c4.3	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
67	60	nnne0d 12281	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑃 ≠ 0)
68		aks6d1c4.1	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝑁 ∈ ℕ)
69	68	nnzd 12612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑁 ∈ ℤ)
70		dvdsval2 16308	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑁 / 𝑃) ∈ ℤ)
74		xp2nd 8015	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑒 ∈ (ℕ0 × ℕ0) → (2nd ‘𝑒) ∈ ℕ0)
75	74	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (2nd ‘𝑒) ∈ ℕ0)
76	73, 75	zexpcld 14119	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝑁 / 𝑃)↑(2nd ‘𝑒)) ∈ ℤ)
77	65, 76	zmulcld 12701	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))) ∈ ℤ)
78	57, 77	eqeltrd 2863	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑒) ∈ ℤ)
79	57	oveq1d 7425	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑒) gcd 𝑅) = (((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))) gcd 𝑅))
80	2	nnzd 12612	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑅 ∈ ℤ)
81	80	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑅 ∈ ℤ)
82	77, 81	gcdcomd 16567	. . . . . . . . . . . . . . . . . . . . . . . . 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 520	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ))
86		gcdcom 16566	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 520	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → ((𝑅 gcd 𝑁) = 1 ∧ 𝑃 ∥ 𝑁))
93		rpdvds 16713	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑅 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑅 gcd 𝑁) = 1 ∧ 𝑃 ∥ 𝑁)) → (𝑅 gcd 𝑃) = 1)
94	83, 92, 93	syl2anc 595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (𝑅 gcd 𝑃) = 1)
95	94	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑅 gcd 𝑃) = 1)
96	95	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd 𝑃) = 1)
97	2	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → 𝑅 ∈ ℕ)
98	60	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → 𝑃 ∈ ℕ)
99		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ∈ ℕ) → (1st ‘𝑒) ∈ ℕ)
100		rprpwr 16612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 626	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (1st ‘𝑒) ≠ 0) → ((1st ‘𝑒) ∈ ℕ0 ∧ (1st ‘𝑒) ≠ 0))
104		elnnne0 12513	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((1st ‘𝑒) ≠ 0 → (1st ‘𝑒) ∈ ℕ))
107	106	necon1bd 2976	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (¬ (1st ‘𝑒) ∈ ℕ → (1st ‘𝑒) = 0))
108	107	imp 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (1st ‘𝑒) = 0)
109	108	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑃↑(1st ‘𝑒)) = (𝑃↑0))
110	109	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑃↑(1st ‘𝑒))) = (𝑅 gcd (𝑃↑0)))
111	62	zcnd 12696	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑃 ∈ ℂ)
112	111	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → 𝑃 ∈ ℂ)
113	112	exp0d 14172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑃↑0) = 1)
114	113	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑃↑0)) = (𝑅 gcd 1))
115	81	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → 𝑅 ∈ ℤ)
116		gcd1 16581	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑃↑0)) = 1)
119	110, 118	eqtrd 2798	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (1st ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1)
120	102, 119	pm2.61dan 824	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 12243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → 𝑁 ∈ ℝ)
123	122	recnd 11232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 𝑁 ∈ ℂ)
124	60	nnred 12243	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → 𝑃 ∈ ℝ)
125	124	recnd 11232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 𝑃 ∈ ℂ)
126	68	nngt0d 12280	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → 0 < 𝑁)
127	126	gt0ne0d 11773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 𝑁 ≠ 0)
128	123, 125, 127, 67	ddcand 12006	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (𝑁 / (𝑁 / 𝑃)) = 𝑃)
129	128, 61	eqeltrd 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝜑 → (𝑁 / (𝑁 / 𝑃)) ∈ ℤ)
130	60	nngt0d 12280	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝜑 → 0 < 𝑃)
131	122, 124, 126, 130	divgt0d 12145	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝜑 → 0 < (𝑁 / 𝑃))
132	131	gt0ne0d 11773	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝜑 → (𝑁 / 𝑃) ≠ 0)
133		dvdsval2 16308	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 520	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → ((𝑅 gcd 𝑁) = 1 ∧ (𝑁 / 𝑃) ∥ 𝑁))
137		rpdvds 16713	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → (𝑅 gcd (𝑁 / 𝑃)) = 1)
139	138	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑅 gcd (𝑁 / 𝑃)) = 1)
140	139	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd (𝑁 / 𝑃)) = 1)
141	2	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → 𝑅 ∈ ℕ)
142	72, 131	jca 520	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → ((𝑁 / 𝑃) ∈ ℤ ∧ 0 < (𝑁 / 𝑃)))
143		elnnz 12596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑁 / 𝑃) ∈ ℕ ↔ ((𝑁 / 𝑃) ∈ ℤ ∧ 0 < (𝑁 / 𝑃)))
144	142, 143	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → (𝑁 / 𝑃) ∈ ℕ)
145	144	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑁 / 𝑃) ∈ ℕ)
146	145	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → (𝑁 / 𝑃) ∈ ℕ)
147		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ∈ ℕ) → (2nd ‘𝑒) ∈ ℕ)
148		rprpwr 16612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 626	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (2nd ‘𝑒) ≠ 0) → ((2nd ‘𝑒) ∈ ℕ0 ∧ (2nd ‘𝑒) ≠ 0))
152		elnnne0 12513	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((2nd ‘𝑒) ≠ 0 → (2nd ‘𝑒) ∈ ℕ))
155	154	necon1bd 2976	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (¬ (2nd ‘𝑒) ∈ ℕ → (2nd ‘𝑒) = 0))
156	155	imp 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (2nd ‘𝑒) = 0)
157	156	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → ((𝑁 / 𝑃)↑(2nd ‘𝑒)) = ((𝑁 / 𝑃)↑0))
158	157	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = (𝑅 gcd ((𝑁 / 𝑃)↑0)))
159	123	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → 𝑁 ∈ ℂ)
160	159	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → 𝑁 ∈ ℂ)
161	111	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → 𝑃 ∈ ℂ)
162	67	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → 𝑃 ≠ 0)
163	160, 161, 162	divcld 11986	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑁 / 𝑃) ∈ ℂ)
164	163	exp0d 14172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → ((𝑁 / 𝑃)↑0) = 1)
165	164	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd ((𝑁 / 𝑃)↑0)) = (𝑅 gcd 1))
166	158, 165	eqtrd 2798	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = (𝑅 gcd 1))
167	81	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ ¬ (2nd ‘𝑒) ∈ ℕ) → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1)
170	150, 169	pm2.61dan 824	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1)
171	120, 170	jca 520	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝑅 gcd (𝑃↑(1st ‘𝑒))) = 1 ∧ (𝑅 gcd ((𝑁 / 𝑃)↑(2nd ‘𝑒))) = 1))
172		rpmul 16712	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → (((𝑃↑(1st ‘𝑒)) · ((𝑁 / 𝑃)↑(2nd ‘𝑒))) gcd 𝑅) = 1)
176	79, 175	eqtrd 2798	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑒 ∈ (ℕ0 × ℕ0)) → ((𝐸‘𝑒) gcd 𝑅) = 1)
177	78, 176	jca 520	. . . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . . 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 499	. . . . . . . . . . . . . . . . . . 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 2863	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) ∧ 𝑒 ∈ (ℕ0 × ℕ0)) ∧ (𝐸‘𝑒) = 𝑐) → 𝑐 ∈ ℤ)
184	181, 183	jca 520	. . . . . . . . . . . . . . . . 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 1944	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑒(𝐸‘𝑑) = 𝑐
186		nfv 1944	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑑(𝐸‘𝑒) = 𝑐
187		fveqeq2 6890	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = 𝑒 → ((𝐸‘𝑑) = 𝑐 ↔ (𝐸‘𝑒) = 𝑐))
188	185, 186, 187	cbvrexw 3308	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐 ↔ ∃𝑒 ∈ (ℕ0 × ℕ0)(𝐸‘𝑒) = 𝑐)
189	188	bilani 509	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) → ∃𝑒 ∈ (ℕ0 × ℕ0)(𝐸‘𝑒) = 𝑐)
190	184, 189	r19.29a 3173	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ ∃𝑑 ∈ (ℕ0 × ℕ0)(𝐸‘𝑑) = 𝑐) → ((𝑐 gcd 𝑅) = 1 ∧ 𝑐 ∈ ℤ))
191	190	ex 417	. . . . . . . . . . . . . . 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 499	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (𝑐 gcd 𝑅) = 1)
194	3	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝑅 ∈ ℕ0)
195	192	simprd 500	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) → 𝑐 ∈ ℤ)
196		eqid 2763	. . . . . . . . . . . . . . 15 ⊢ (Unit‘(ℤ/nℤ‘𝑅)) = (Unit‘(ℤ/nℤ‘𝑅))
197	4, 196, 8	znunit 21713	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ ℕ0 ∧ 𝑐 ∈ ℤ) → ((𝐿‘𝑐) ∈ (Unit‘(ℤ/nℤ‘𝑅)) ↔ (𝑐 gcd 𝑅) = 1))
198	194, 195, 197	syl2anc 595	. . . . . . . . . . . . 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 485	. . . . . . . . . 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 2863	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) ∧ 𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))) ∧ (𝐿‘𝑐) = 𝑎) → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅)))
203		nfv 1944	. . . . . . . . . . 11 ⊢ Ⅎ𝑐(𝐿‘𝑏) = 𝑎
204		nfv 1944	. . . . . . . . . . 11 ⊢ Ⅎ𝑏(𝐿‘𝑐) = 𝑎
205		fveqeq2 6890	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑐 → ((𝐿‘𝑏) = 𝑎 ↔ (𝐿‘𝑐) = 𝑎))
206	203, 204, 205	cbvrexw 3308	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎 ↔ ∃𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑐) = 𝑎)
207	206	bilani 509	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) → ∃𝑐 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑐) = 𝑎)
208	202, 207	r19.29a 3173	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎) → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅)))
209	208	ex 417	. . . . . . 7 ⊢ (𝜑 → (∃𝑏 ∈ (𝐸 “ (ℕ0 × ℕ0))(𝐿‘𝑏) = 𝑎 → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅))))
210	209	adantr 485	. . . . . 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 417	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) → 𝑎 ∈ (Unit‘(ℤ/nℤ‘𝑅))))
213	212	ssrdv 3943	. . 3 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ⊆ (Unit‘(ℤ/nℤ‘𝑅)))
214		hashss 14441	. . 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 595	. 2 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (♯‘(Unit‘(ℤ/nℤ‘𝑅))))
216	4, 196	znunithash 21714	. . 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 5137	1 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ≤ (ϕ‘𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089  Vcvv 3455   ⊆ wss 3905  ⟨cop 4595   class class class wbr 5109   ↦ cmpt 5192   × cxp 5659   “ cima 5664  Fun wfun 6530  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7980  2^nd c2nd 7981  ℂcc 11093  0cc0 11095  1c1 11096   · cmul 11100   < clt 11238   ≤ cle 11239   / cdiv 11866  ℕcn 12228  ℕ₀cn0 12499  ℤcz 12586  ↑cexp 14093  ♯chash 14362   ∥ cdvds 16305   gcd cgcd 16547  ℙcprime 16724  ϕcphi 16818  Basecbs 17264  Ringcrg 20310  CRingccrg 20311  Unitcui 20433   RingHom crh 20547  ℤ_ringczring 21596  ℤRHomczrh 21649  ℤ/nℤczn 21652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173  ax-addf 11174  ax-mulf 11175

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-er 8690  df-ec 8692  df-qs 8696  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-xnn0 12573  df-z 12587  df-dec 12707  df-uz 12858  df-rp 13012  df-fz 13531  df-fzo 13679  df-fl 13821  df-mod 13899  df-seq 14034  df-exp 14094  df-hash 14363  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-dvds 16306  df-gcd 16548  df-prm 16725  df-phi 16820  df-struct 17202  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-mulr 17319  df-starv 17320  df-sca 17321  df-vsca 17322  df-ip 17323  df-tset 17324  df-ple 17325  df-ds 17327  df-unif 17328  df-0g 17489  df-imas 17557  df-qus 17558  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-mhm 18836  df-grp 18998  df-minusg 18999  df-sbg 19000  df-mulg 19129  df-subg 19184  df-nsg 19185  df-eqg 19186  df-ghm 19279  df-cmn 19847  df-abl 19848  df-mgp 20212  df-rng 20226  df-ur 20259  df-ring 20312  df-cring 20313  df-oppr 20415  df-dvdsr 20435  df-unit 20436  df-rhm 20550  df-subrng 20645  df-subrg 20669  df-lmod 20983  df-lss 21053  df-lsp 21093  df-sra 21294  df-rgmod 21295  df-lidl 21332  df-rsp 21333  df-2idl 21389  df-cnfld 21523  df-zring 21597  df-zrh 21653  df-zn 21656

This theorem is referenced by:  aks6d1c7lem1  42947