aks6d1c7lem2 - Mathbox for metakunt

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Theorem aks6d1c7lem2 42929

Description: Contradiction to Claim 2 and Claim 7. We assumed in Claim 2 that there are two different prime numbers 𝑃 and 𝑄. (Contributed by metakunt, 16-May-2025.)

**Hypotheses**
Ref	Expression
aks6d1c7lem2.1	⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly₁‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘𝑓)‘𝑦)) = (((eval₁‘𝐾)‘𝑓)‘(𝑒(._g‘(mulGrp‘𝐾))𝑦)))}
aks6d1c7lem2.2	⊢ 𝑃 = (chr‘𝐾)
aks6d1c7lem2.3	⊢ (𝜑 → 𝐾 ∈ Field)
aks6d1c7lem2.4	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c7lem2.5	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks6d1c7lem2.6	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
aks6d1c7lem2.7	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks6d1c7lem2.8	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c7lem2.9	⊢ 𝐸 = (𝑘 ∈ ℕ₀, 𝑙 ∈ ℕ₀ ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
aks6d1c7lem2.10	⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
aks6d1c7lem2.11	⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))
aks6d1c7lem2.12	⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
aks6d1c7lem2.13	⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))
aks6d1c7lem2.14	⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(._g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
aks6d1c7lem2.15	⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
aks6d1c7lem2.16	⊢ 𝐻 = (ℎ ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ (((eval₁‘𝐾)‘(𝐺‘ℎ))‘𝑀))
aks6d1c7lem2.17	⊢ 𝐵 = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
aks6d1c7lem2.18	⊢ 𝐶 = (𝐸 “ ((0...𝐵) × (0...𝐵)))
aks6d1c7lem2.19	⊢ (𝜑 → (𝑄 ∈ ℙ ∧ 𝑄 ∥ 𝑁))
aks6d1c7lem2.20	⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
aks6d1c7lem2.21	⊢ 𝐺 = (𝑔 ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
aks6d1c7lem2.22	⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
aks6d1c7lem2.23	⊢ 𝑆 = {𝑠 ∈ (ℕ₀ ↑_m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}

**Assertion**
Ref	Expression
aks6d1c7lem2	⊢ (𝜑 → 𝑃 = 𝑄)

Distinct variable groups: ∼ ,𝑎 𝐴,𝑎 𝐴,𝑏,ℎ 𝐴,𝑔,𝑖,𝑥 𝐴,𝑘,𝑙,𝑠 𝑡,𝐴,𝑖,𝑥 𝐵,𝑎 𝐵,𝑔,𝑖,𝑥 𝐵,𝑘,𝑙,𝑥 𝐶,𝑎 𝐶,𝑔,𝑖,𝑥 𝐶,ℎ 𝐶,𝑘,𝑙 𝐷,𝑠 𝐸,𝑎 𝑦,𝐸,𝑒,𝑓 𝑔,𝐸,𝑖,𝑥,𝑦 𝑘,𝐸,𝑙,𝑦 𝑒,𝐺,𝑓,𝑦 𝑔,𝐺,𝑖 ℎ,𝐺 𝑡,𝐺,𝑦 𝐻,𝑎 ℎ,𝐻 𝑔,𝐻,𝑖,𝑥,𝑦 𝐻,𝑠,𝑡 𝐾,𝑎 𝐾,𝑏,ℎ 𝑒,𝐾,𝑓,𝑦 𝑔,𝐾,𝑖,𝑥 𝐾,𝑙 𝑡,𝐾 𝑀,𝑏,ℎ 𝑀,𝑙,𝑦 𝑁,𝑎 𝑁,𝑏 𝑒,𝑁,𝑓,𝑦 𝑘,𝑁,𝑙,𝑠 𝑥,𝑁 𝑃,𝑎 𝑃,𝑏,ℎ 𝑃,𝑒,𝑓,𝑦 𝑃,𝑔,𝑖,𝑥 𝑃,𝑘,𝑙,𝑠 𝑡,𝑃 𝑄,𝑎 𝑄,𝑏,ℎ 𝑄,𝑔,𝑖,𝑥,𝑦 𝑄,𝑘,𝑙,𝑠 𝑡,𝑄 𝑅,𝑎 𝑅,ℎ 𝑅,𝑒,𝑓,𝑦 𝑅,𝑔,𝑖,𝑥 𝑅,𝑘,𝑙 𝑆,𝑎 𝑆,ℎ 𝑆,𝑔,𝑖,𝑥,𝑦 𝑆,𝑠,𝑡 𝜑,𝑎 𝜑,𝑏,ℎ 𝜑,𝑔,𝑖,𝑥,𝑦 𝜑,𝑘,𝑙,𝑠 𝜑,𝑡 Allowed substitution hints: 𝜑(𝑒,𝑓) 𝐴(𝑦,𝑒,𝑓) 𝐵(𝑦,𝑡,𝑒,𝑓,ℎ,𝑠,𝑏) 𝐶(𝑦,𝑡,𝑒,𝑓,𝑠,𝑏) 𝐷(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑎,𝑏,𝑙) 𝑄(𝑒,𝑓) ∼ (𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑠,𝑏,𝑙) 𝑅(𝑡,𝑠,𝑏) 𝑆(𝑒,𝑓,𝑘,𝑏,𝑙) 𝐸(𝑡,ℎ,𝑠,𝑏) 𝐺(𝑥,𝑘,𝑠,𝑎,𝑏,𝑙) 𝐻(𝑒,𝑓,𝑘,𝑏,𝑙) 𝐾(𝑘,𝑠) 𝐿(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑠,𝑎,𝑏,𝑙) 𝑀(𝑥,𝑡,𝑒,𝑓,𝑔,𝑖,𝑘,𝑠,𝑎) 𝑁(𝑡,𝑔,ℎ,𝑖)

Proof of Theorem aks6d1c7lem2 Dummy variables 𝑐 𝑗 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 489	. 2 ⊢ ((𝜑 ∧ 𝑃 = 𝑄) → 𝑃 = 𝑄)
2		aks6d1c7lem2.1	. . . 4 ⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly₁‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘𝑓)‘𝑦)) = (((eval₁‘𝐾)‘𝑓)‘(𝑒(._g‘(mulGrp‘𝐾))𝑦)))}
3		aks6d1c7lem2.2	. . . 4 ⊢ 𝑃 = (chr‘𝐾)
4		aks6d1c7lem2.3	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Field)
5	4	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐾 ∈ Field)
6		aks6d1c7lem2.4	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℙ)
7	6	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ ℙ)
8		aks6d1c7lem2.5	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℕ)
9	8	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑅 ∈ ℕ)
10		aks6d1c7lem2.6	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
11		eluzelz 12873	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
12	10, 11	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
13		0red 11212	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
14		3re 12322	. . . . . . . . 9 ⊢ 3 ∈ ℝ
15	14	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 3 ∈ ℝ)
16	12	zred 12701	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17		3pos 12350	. . . . . . . . 9 ⊢ 0 < 3
18	17	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 3)
19		eluzle 12876	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
20	10, 19	syl 18	. . . . . . . 8 ⊢ (𝜑 → 3 ≤ 𝑁)
21	13, 15, 16, 18, 20	ltletrd 11371	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑁)
22	12, 21	jca 520	. . . . . 6 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
23		elnnz 12602	. . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
24	22, 23	sylibr 237	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
25	24	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑁 ∈ ℕ)
26		aks6d1c7lem2.7	. . . . 5 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
27	26	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑃 ∥ 𝑁)
28		aks6d1c7lem2.8	. . . . 5 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
29	28	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁 gcd 𝑅) = 1)
30		aks6d1c7lem2.21	. . . 4 ⊢ 𝐺 = (𝑔 ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
31		aks6d1c7lem2.12	. . . . . 6 ⊢ 𝐴 = (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
32	8	phicld 16832	. . . . . . . . . . . 12 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ)
33	32	nnred 12249	. . . . . . . . . . 11 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℝ)
34		1red 11210	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
35		0le1 11738	. . . . . . . . . . . . 13 ⊢ 0 ≤ 1
36	35	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ 1)
37	32	nnge1d 12285	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ (ϕ‘𝑅))
38	13, 34, 33, 36, 37	letrd 11368	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (ϕ‘𝑅))
39	33, 38	resqrtcld 15471	. . . . . . . . . 10 ⊢ (𝜑 → (√‘(ϕ‘𝑅)) ∈ ℝ)
40		2re 12316	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
41	40	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
42		2pos 12346	. . . . . . . . . . . 12 ⊢ 0 < 2
43	42	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 2)
44		1lt2 12414	. . . . . . . . . . . . . 14 ⊢ 1 < 2
45	44	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < 2)
46	34, 45	ltned 11347	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≠ 2)
47	46	necomd 3013	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≠ 1)
48	41, 43, 16, 21, 47	relogbcld 42722	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
49	39, 48	remulcld 11240	. . . . . . . . 9 ⊢ (𝜑 → ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ∈ ℝ)
50	49	flcld 13833	. . . . . . . 8 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℤ)
51	33, 38	sqrtge0d 15474	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (√‘(ϕ‘𝑅)))
52	41	recnd 11238	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℂ)
53	13, 43	gtned 11346	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ≠ 0)
54		logb1 26915	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 log_b 1) = 0)
55	52, 53, 47, 54	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 log_b 1) = 0)
56	55	eqcomd 2769	. . . . . . . . . . 11 ⊢ (𝜑 → 0 = (2 log_b 1))
57		2z 12627	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
58	57	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℤ)
59	41	leidd 11781	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ≤ 2)
60		0lt1 11737	. . . . . . . . . . . . 13 ⊢ 0 < 1
61	60	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 1)
62	24	nnge1d 12285	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑁)
63	58, 59, 34, 61, 16, 21, 62	logblebd 42725	. . . . . . . . . . 11 ⊢ (𝜑 → (2 log_b 1) ≤ (2 log_b 𝑁))
64	56, 63	eqbrtrd 5134	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (2 log_b 𝑁))
65	39, 48, 51, 64	mulge0d 11792	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
66		0zd 12604	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
67		flge 13840	. . . . . . . . . 10 ⊢ ((((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ↔ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))))
68	49, 66, 67	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → (0 ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ↔ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))))
69	65, 68	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))
70	50, 69	jca 520	. . . . . . 7 ⊢ (𝜑 → ((⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))))
71		elnn0z 12605	. . . . . . 7 ⊢ ((⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℕ₀ ↔ ((⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))))
72	70, 71	sylibr 237	. . . . . 6 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℕ₀)
73	31, 72	eqeltrid 2867	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℕ₀)
74	73	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐴 ∈ ℕ₀)
75		aks6d1c7lem2.9	. . . 4 ⊢ 𝐸 = (𝑘 ∈ ℕ₀, 𝑙 ∈ ℕ₀ ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
76		aks6d1c7lem2.10	. . . 4 ⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
77		aks6d1c7lem2.22	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
78	77	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
79		aks6d1c7lem2.14	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(._g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
80	79	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(._g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
81		aks6d1c7lem2.15	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
82	81	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
83		aks6d1c7lem2.16	. . . 4 ⊢ 𝐻 = (ℎ ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ (((eval₁‘𝐾)‘(𝐺‘ℎ))‘𝑀))
84		aks6d1c7lem2.17	. . . 4 ⊢ 𝐵 = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
85		aks6d1c7lem2.18	. . . 4 ⊢ 𝐶 = (𝐸 “ ((0...𝐵) × (0...𝐵)))
86		aks6d1c7lem2.19	. . . . . . 7 ⊢ (𝜑 → (𝑄 ∈ ℙ ∧ 𝑄 ∥ 𝑁))
87	86	simpld 499	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℙ)
88	87	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ ℙ)
89	86	simprd 500	. . . . . 6 ⊢ (𝜑 → 𝑄 ∥ 𝑁)
90	89	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑄 ∥ 𝑁)
91		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑃 ≠ 𝑄)
92	88, 90, 91	3jca 1146	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑄 ∈ ℙ ∧ 𝑄 ∥ 𝑁 ∧ 𝑃 ≠ 𝑄))
93	2, 3, 5, 7, 9, 25, 27, 29, 30, 74, 75, 76, 78, 80, 82, 83, 84, 85, 92	aks6d1c2 42878	. . 3 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵))
94	24	nnzd 12618	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
95		eqid 2763	. . . . . . . . . . . . . . . 16 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
96	24, 6, 26, 8, 28, 75, 76, 95	hashscontpowcl 42868	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
97	96	nn0red 12567	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ)
98	96	nn0ge0d 12569	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
99	97, 98	resqrtcld 15471	. . . . . . . . . . . . 13 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ)
100	99	flcld 13833	. . . . . . . . . . . 12 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ)
101	97, 98	sqrtge0d 15474	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
102		flge 13840	. . . . . . . . . . . . . 14 ⊢ (((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
103	99, 66, 102	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
104	101, 103	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
105	100, 104	jca 520	. . . . . . . . . . 11 ⊢ (𝜑 → ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
106		elnn0z 12605	. . . . . . . . . . 11 ⊢ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℕ₀ ↔ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
107	105, 106	sylibr 237	. . . . . . . . . 10 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℕ₀)
108	84, 107	eqeltrid 2867	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℕ₀)
109	94, 108	zexpcld 14125	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℤ)
110	109	zred 12701	. . . . . . 7 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℝ)
111	110	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) ∈ ℝ)
112	111	rexrd 11260	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) ∈ ℝ^*)
113		aks6d1c7lem2.11	. . . . . . . . . 10 ⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))
114	96	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
115	113, 114	eqeltrid 2867	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐷 ∈ ℕ₀)
116	115, 74	nn0addcld 12570	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝐷 + 𝐴) ∈ ℕ₀)
117	115	nn0zd 12617	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐷 ∈ ℤ)
118		1zzd 12626	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 1 ∈ ℤ)
119	117, 118	zsubcld 12706	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝐷 − 1) ∈ ℤ)
120		bccl 14360	. . . . . . . 8 ⊢ (((𝐷 + 𝐴) ∈ ℕ₀ ∧ (𝐷 − 1) ∈ ℤ) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℕ₀)
121	116, 119, 120	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℕ₀)
122	121	nn0red 12567	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℝ)
123	122	rexrd 11260	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℝ^*)
124		ovexd 7447	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (ℕ₀ ↑_m (0...𝐴)) ∈ V)
125	124	mptexd 7224	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (ℎ ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ (((eval₁‘𝐾)‘(𝐺‘ℎ))‘𝑀)) ∈ V)
126	83, 125	eqeltrid 2867	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐻 ∈ V)
127	126	imaexd 7914	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ∈ V)
128		hashxrcl 14395	. . . . . 6 ⊢ ((𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ∈ V → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ∈ ℝ^*)
129	127, 128	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ∈ ℝ^*)
130		eqcom 2770	. . . . . . . . . . . 12 ⊢ (𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ↔ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = 𝐷)
131	113, 130	mpbi 233	. . . . . . . . . . 11 ⊢ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = 𝐷
132	131	fveq2i 6886	. . . . . . . . . 10 ⊢ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) = (√‘𝐷)
133	132	fveq2i 6886	. . . . . . . . 9 ⊢ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = (⌊‘(√‘𝐷))
134	84, 133	eqtri 2786	. . . . . . . 8 ⊢ 𝐵 = (⌊‘(√‘𝐷))
135	134	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐵 = (⌊‘(√‘𝐷)))
136	135	oveq2d 7428	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) = (𝑁↑(⌊‘(√‘𝐷))))
137	10	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑁 ∈ (ℤ_≥‘3))
138		aks6d1c7lem2.13	. . . . . . . 8 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))
139	138	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))
140	7, 9, 137, 27, 29, 75, 76, 113, 31, 139	aks6d1c7lem1 42928	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑(⌊‘(√‘𝐷))) < ((𝐷 + 𝐴)C(𝐷 − 1)))
141	136, 140	eqbrtrd 5134	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) < ((𝐷 + 𝐴)C(𝐷 − 1)))
142		aks6d1c7lem2.20	. . . . . . 7 ⊢ (𝜑 → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
143	142	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ∀𝑏 ∈ (1...𝐴)(𝑏 gcd 𝑁) = 1)
144		aks6d1c7lem2.23	. . . . . 6 ⊢ 𝑆 = {𝑠 ∈ (ℕ₀ ↑_m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}
145		eqid 2763	. . . . . 6 ⊢ (𝑐 ∈ ℤ ↦ (𝑐(._g‘((mulGrp‘𝐾) ↾_s {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))}))𝑀)) = (𝑐 ∈ ℤ ↦ (𝑐(._g‘((mulGrp‘𝐾) ↾_s {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))}))𝑀))
146		eqid 2763	. . . . . 6 ⊢ {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))} = {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))}
147		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑏∪ ((𝑐 ∈ ℤ ↦ (𝑐(._g‘((mulGrp‘𝐾) ↾_s {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))}))𝑀)) “ ℎ)
148		nfcv 2925	. . . . . . 7 ⊢ Ⅎℎ∪ ((𝑐 ∈ ℤ ↦ (𝑐(._g‘((mulGrp‘𝐾) ↾_s {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))}))𝑀)) “ 𝑏)
149		imaeq2 6060	. . . . . . . 8 ⊢ (ℎ = 𝑏 → ((𝑐 ∈ ℤ ↦ (𝑐(._g‘((mulGrp‘𝐾) ↾_s {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))}))𝑀)) “ ℎ) = ((𝑐 ∈ ℤ ↦ (𝑐(._g‘((mulGrp‘𝐾) ↾_s {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))}))𝑀)) “ 𝑏))
150	149	unieqd 4886	. . . . . . 7 ⊢ (ℎ = 𝑏 → ∪ ((𝑐 ∈ ℤ ↦ (𝑐(._g‘((mulGrp‘𝐾) ↾_s {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))}))𝑀)) “ ℎ) = ∪ ((𝑐 ∈ ℤ ↦ (𝑐(._g‘((mulGrp‘𝐾) ↾_s {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))}))𝑀)) “ 𝑏))
151	147, 148, 150	cbvmpt 5214	. . . . . 6 ⊢ (ℎ ∈ (Base‘(ℤ_ring /_s (ℤ_ring ~_QG (^◡(𝑐 ∈ ℤ ↦ (𝑐(._g‘((mulGrp‘𝐾) ↾_s {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))}))𝑀)) “ {(0_g‘(((mulGrp‘𝐾) ↾_s {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))}) ↾_s ran (𝑐 ∈ ℤ ↦ (𝑐(._g‘((mulGrp‘𝐾) ↾_s {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))}))𝑀))))})))) ↦ ∪ ((𝑐 ∈ ℤ ↦ (𝑐(._g‘((mulGrp‘𝐾) ↾_s {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))}))𝑀)) “ ℎ)) = (𝑏 ∈ (Base‘(ℤ_ring /_s (ℤ_ring ~_QG (^◡(𝑐 ∈ ℤ ↦ (𝑐(._g‘((mulGrp‘𝐾) ↾_s {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))}))𝑀)) “ {(0_g‘(((mulGrp‘𝐾) ↾_s {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))}) ↾_s ran (𝑐 ∈ ℤ ↦ (𝑐(._g‘((mulGrp‘𝐾) ↾_s {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))}))𝑀))))})))) ↦ ∪ ((𝑐 ∈ ℤ ↦ (𝑐(._g‘((mulGrp‘𝐾) ↾_s {𝑗 ∈ (Base‘(mulGrp‘𝐾)) ∣ ∃𝑚 ∈ (Base‘(mulGrp‘𝐾))(𝑚(+_g‘(mulGrp‘𝐾))𝑗) = (0_g‘(mulGrp‘𝐾))}))𝑀)) “ 𝑏))
152	2, 3, 5, 7, 9, 25, 27, 29, 143, 30, 31, 75, 76, 78, 80, 82, 83, 113, 144, 145, 146, 151	aks6d1c6lem5 42925	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))))
153	112, 123, 129, 141, 152	xrltletrd 13187	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) < (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))))
154		xrltnle 11277	. . . . 5 ⊢ (((𝑁↑𝐵) ∈ ℝ^* ∧ (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ∈ ℝ^*) → ((𝑁↑𝐵) < (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ↔ ¬ (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵)))
155	112, 129, 154	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝑁↑𝐵) < (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ↔ ¬ (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵)))
156	153, 155	mpbid 235	. . 3 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ¬ (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵))
157	93, 156	pm2.21dd 198	. 2 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑃 = 𝑄)
158	1, 157	pm2.61dane 3045	1 ⊢ (𝜑 → 𝑃 = 𝑄)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 {crab 3416 Vcvv 3455 {csn 4590 ∪ cuni 4873 class class class wbr 5110 {copab 5174 ↦ cmpt 5193 × cxp 5661 ^◡ccnv 5662 ran crn 5664 “ cima 5666 ‘cfv 6538 (class class class)co 7412 ∈ cmpo 7414 ↑_m cmap 8825 ℂcc 11099 ℝcr 11100 0cc0 11101 1c1 11102 + caddc 11104 · cmul 11106 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 − cmin 11442 / cdiv 11872 ℕcn 12234 2c2 12296 3c3 12297 ℕ₀cn0 12505 ℤcz 12592 ℤ_≥cuz 12863 ...cfz 13536 ⌊cfl 13825 ↑cexp 14099 Ccbc 14340 ♯chash 14368 √csqrt 15286 Σcsu 15739 ∥ cdvds 16311 gcd cgcd 16553 ℙcprime 16730 od_ℤcodz 16823 ϕcphi 16824 Basecbs 17270 ↾_s cress 17291 +_gcplusg 17311 0_gc0g 17493 Σ_g cgsu 17494 /_s cqus 17560 ._gcmg 19134 ~_QG cqg 19189 mulGrpcmgp 20217 RingIso crs 20553 Fieldcfield 20815 ℤ_ringczring 21577 ℤRHomczrh 21630 chrcchr 21632 ℤ/nℤczn 21633 algSccascl 21983 var₁cv1 22317 Poly₁cpl1 22318 eval₁ce1 22455 log_b clogb 26910 PrimRoots cprimroots 42839

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 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179 ax-addf 11180 ax-mulf 11181

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 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 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 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-ofr 7677 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8158 df-tpos 8223 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-oadd 8458 df-er 8695 df-ec 8697 df-qs 8701 df-map 8827 df-pm 8828 df-ixp 8897 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-fsupp 9323 df-fi 9372 df-sup 9403 df-inf 9404 df-oi 9473 df-dju 9888 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-xnn0 12579 df-z 12593 df-dec 12713 df-uz 12864 df-q 12974 df-rp 13018 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13377 df-ioc 13378 df-ico 13379 df-icc 13380 df-fz 13537 df-fzo 13685 df-fl 13827 df-mod 13905 df-seq 14040 df-exp 14100 df-fac 14312 df-bc 14341 df-hash 14369 df-shft 15106 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-limsup 15524 df-clim 15541 df-rlim 15542 df-sum 15740 df-prod 15960 df-fallfac 16063 df-ef 16122 df-sin 16124 df-cos 16125 df-pi 16127 df-dvds 16312 df-gcd 16554 df-prm 16731 df-odz 16825 df-phi 16826 df-pc 16898 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17476 df-topn 17477 df-0g 17495 df-gsum 17496 df-topgen 17497 df-pt 17498 df-prds 17501 df-pws 17503 df-xrs 17557 df-qtop 17562 df-imas 17563 df-qus 17564 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-mhm 18842 df-submnd 18843 df-grp 19004 df-minusg 19005 df-sbg 19006 df-mulg 19135 df-subg 19190 df-nsg 19191 df-eqg 19192 df-ghm 19285 df-gim 19330 df-cntz 19388 df-od 19599 df-cmn 19853 df-abl 19854 df-mgp 20218 df-rng 20232 df-ur 20265 df-srg 20270 df-ring 20318 df-cring 20319 df-oppr 20420 df-dvdsr 20440 df-unit 20441 df-invr 20471 df-dvr 20484 df-rhm 20555 df-rim 20556 df-nzr 20597 df-subrng 20632 df-subrg 20656 df-rlreg 20780 df-domn 20781 df-idom 20782 df-drng 20816 df-field 20817 df-lmod 20964 df-lss 21034 df-lsp 21074 df-sra 21275 df-rgmod 21276 df-lidl 21313 df-rsp 21314 df-2idl 21370 df-psmet 21495 df-xmet 21496 df-met 21497 df-bl 21498 df-mopn 21499 df-fbas 21500 df-fg 21501 df-cnfld 21504 df-zring 21578 df-zrh 21634 df-chr 21636 df-zn 21637 df-assa 21984 df-asp 21985 df-ascl 21986 df-psr 22040 df-mvr 22041 df-mpl 22042 df-opsr 22044 df-evls 22206 df-evl 22207 df-psr1 22321 df-vr1 22322 df-ply1 22323 df-coe1 22324 df-evl1 22457 df-top 23032 df-topon 23049 df-topsp 23071 df-bases 23084 df-cld 23157 df-ntr 23158 df-cls 23159 df-nei 23236 df-lp 23274 df-perf 23275 df-cn 23365 df-cnp 23366 df-haus 23453 df-tx 23700 df-hmeo 23893 df-fil 23984 df-fm 24076 df-flim 24077 df-flf 24078 df-xms 24458 df-ms 24459 df-tms 24460 df-cncf 25018 df-limc 26006 df-dv 26007 df-mdeg 26193 df-deg1 26194 df-mon1 26269 df-uc1p 26270 df-q1p 26271 df-r1p 26272 df-log 26702 df-cxp 26703 df-logb 26911 df-primroots 42840

This theorem is referenced by: aks6d1c7lem3 42930

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