aks6d1c7lem2 - Mathbox for metakunt

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Theorem aks6d1c7lem2 42548

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 484	. 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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐾 ∈ Field)
6		aks6d1c7lem2.4	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℙ)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ ℙ)
8		aks6d1c7lem2.5	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℕ)
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑅 ∈ ℕ)
10		aks6d1c7lem2.6	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
11		eluzelz 12773	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
12	10, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
13		0red 11147	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
14		3re 12237	. . . . . . . . 9 ⊢ 3 ∈ ℝ
15	14	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 3 ∈ ℝ)
16	12	zred 12608	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17		3pos 12262	. . . . . . . . 9 ⊢ 0 < 3
18	17	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 3)
19		eluzle 12776	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
20	10, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → 3 ≤ 𝑁)
21	13, 15, 16, 18, 20	ltletrd 11305	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑁)
22	12, 21	jca 511	. . . . . 6 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
23		elnnz 12510	. . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
24	22, 23	sylibr 234	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
25	24	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑁 ∈ ℕ)
26		aks6d1c7lem2.7	. . . . 5 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
27	26	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑃 ∥ 𝑁)
28		aks6d1c7lem2.8	. . . . 5 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
29	28	adantr 480	. . . 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 16711	. . . . . . . . . . . 12 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ)
33	32	nnred 12172	. . . . . . . . . . 11 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℝ)
34		1red 11145	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
35		0le1 11672	. . . . . . . . . . . . 13 ⊢ 0 ≤ 1
36	35	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ 1)
37	32	nnge1d 12205	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ (ϕ‘𝑅))
38	13, 34, 33, 36, 37	letrd 11302	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (ϕ‘𝑅))
39	33, 38	resqrtcld 15353	. . . . . . . . . 10 ⊢ (𝜑 → (√‘(ϕ‘𝑅)) ∈ ℝ)
40		2re 12231	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
41	40	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
42		2pos 12260	. . . . . . . . . . . 12 ⊢ 0 < 2
43	42	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 2)
44		1lt2 12323	. . . . . . . . . . . . . 14 ⊢ 1 < 2
45	44	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < 2)
46	34, 45	ltned 11281	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≠ 2)
47	46	necomd 2988	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≠ 1)
48	41, 43, 16, 21, 47	relogbcld 42340	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
49	39, 48	remulcld 11174	. . . . . . . . 9 ⊢ (𝜑 → ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ∈ ℝ)
50	49	flcld 13730	. . . . . . . 8 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℤ)
51	33, 38	sqrtge0d 15356	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (√‘(ϕ‘𝑅)))
52	41	recnd 11172	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℂ)
53	13, 43	gtned 11280	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ≠ 0)
54		logb1 26747	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 log_b 1) = 0)
55	52, 53, 47, 54	syl3anc 1374	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 log_b 1) = 0)
56	55	eqcomd 2743	. . . . . . . . . . 11 ⊢ (𝜑 → 0 = (2 log_b 1))
57		2z 12535	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
58	57	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℤ)
59	41	leidd 11715	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ≤ 2)
60		0lt1 11671	. . . . . . . . . . . . 13 ⊢ 0 < 1
61	60	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 1)
62	24	nnge1d 12205	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑁)
63	58, 59, 34, 61, 16, 21, 62	logblebd 42343	. . . . . . . . . . 11 ⊢ (𝜑 → (2 log_b 1) ≤ (2 log_b 𝑁))
64	56, 63	eqbrtrd 5122	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (2 log_b 𝑁))
65	39, 48, 51, 64	mulge0d 11726	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
66		0zd 12512	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
67		flge 13737	. . . . . . . . . 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 585	. . . . . . . . 9 ⊢ (𝜑 → (0 ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ↔ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))))
69	65, 68	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))
70	50, 69	jca 511	. . . . . . 7 ⊢ (𝜑 → ((⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))))
71		elnn0z 12513	. . . . . . 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 234	. . . . . 6 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℕ₀)
73	31, 72	eqeltrid 2841	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℕ₀)
74	73	adantr 480	. . . 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 480	. . . 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 480	. . . 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 480	. . . 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 494	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℙ)
88	87	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ ℙ)
89	86	simprd 495	. . . . . 6 ⊢ (𝜑 → 𝑄 ∥ 𝑁)
90	89	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑄 ∥ 𝑁)
91		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑃 ≠ 𝑄)
92	88, 90, 91	3jca 1129	. . . 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 42497	. . 3 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵))
94	24	nnzd 12526	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
95		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
96	24, 6, 26, 8, 28, 75, 76, 95	hashscontpowcl 42487	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
97	96	nn0red 12475	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ)
98	96	nn0ge0d 12477	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
99	97, 98	resqrtcld 15353	. . . . . . . . . . . . 13 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ)
100	99	flcld 13730	. . . . . . . . . . . 12 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ)
101	97, 98	sqrtge0d 15356	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
102		flge 13737	. . . . . . . . . . . . . 14 ⊢ (((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
103	99, 66, 102	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
104	101, 103	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
105	100, 104	jca 511	. . . . . . . . . . 11 ⊢ (𝜑 → ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
106		elnn0z 12513	. . . . . . . . . . 11 ⊢ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℕ₀ ↔ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
107	105, 106	sylibr 234	. . . . . . . . . 10 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℕ₀)
108	84, 107	eqeltrid 2841	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℕ₀)
109	94, 108	zexpcld 14022	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℤ)
110	109	zred 12608	. . . . . . 7 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℝ)
111	110	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) ∈ ℝ)
112	111	rexrd 11194	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) ∈ ℝ^*)
113		aks6d1c7lem2.11	. . . . . . . . . 10 ⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))
114	96	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
115	113, 114	eqeltrid 2841	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐷 ∈ ℕ₀)
116	115, 74	nn0addcld 12478	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝐷 + 𝐴) ∈ ℕ₀)
117	115	nn0zd 12525	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐷 ∈ ℤ)
118		1zzd 12534	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 1 ∈ ℤ)
119	117, 118	zsubcld 12613	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝐷 − 1) ∈ ℤ)
120		bccl 14257	. . . . . . . 8 ⊢ (((𝐷 + 𝐴) ∈ ℕ₀ ∧ (𝐷 − 1) ∈ ℤ) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℕ₀)
121	116, 119, 120	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℕ₀)
122	121	nn0red 12475	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℝ)
123	122	rexrd 11194	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℝ^*)
124		ovexd 7403	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (ℕ₀ ↑_m (0...𝐴)) ∈ V)
125	124	mptexd 7180	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (ℎ ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ (((eval₁‘𝐾)‘(𝐺‘ℎ))‘𝑀)) ∈ V)
126	83, 125	eqeltrid 2841	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐻 ∈ V)
127	126	imaexd 7868	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ∈ V)
128		hashxrcl 14292	. . . . . 6 ⊢ ((𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ∈ V → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ∈ ℝ^*)
129	127, 128	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ∈ ℝ^*)
130		eqcom 2744	. . . . . . . . . . . 12 ⊢ (𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ↔ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = 𝐷)
131	113, 130	mpbi 230	. . . . . . . . . . 11 ⊢ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = 𝐷
132	131	fveq2i 6845	. . . . . . . . . 10 ⊢ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) = (√‘𝐷)
133	132	fveq2i 6845	. . . . . . . . 9 ⊢ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = (⌊‘(√‘𝐷))
134	84, 133	eqtri 2760	. . . . . . . 8 ⊢ 𝐵 = (⌊‘(√‘𝐷))
135	134	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐵 = (⌊‘(√‘𝐷)))
136	135	oveq2d 7384	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) = (𝑁↑(⌊‘(√‘𝐷))))
137	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑁 ∈ (ℤ_≥‘3))
138		aks6d1c7lem2.13	. . . . . . . 8 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))
139	138	adantr 480	. . . . . . 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 42547	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑(⌊‘(√‘𝐷))) < ((𝐷 + 𝐴)C(𝐷 − 1)))
141	136, 140	eqbrtrd 5122	. . . . 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 480	. . . . . 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 2737	. . . . . 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 2737	. . . . . 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 2899	. . . . . . 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 2899	. . . . . . 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 6023	. . . . . . . 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 4878	. . . . . . 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 5202	. . . . . 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 42544	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))))
153	112, 123, 129, 141, 152	xrltletrd 13087	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) < (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))))
154		xrltnle 11211	. . . . 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 585	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝑁↑𝐵) < (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ↔ ¬ (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵)))
156	153, 155	mpbid 232	. . 3 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ¬ (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵))
157	93, 156	pm2.21dd 195	. 2 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑃 = 𝑄)
158	1, 157	pm2.61dane 3020	1 ⊢ (𝜑 → 𝑃 = 𝑄)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 {crab 3401 Vcvv 3442 {csn 4582 ∪ cuni 4865 class class class wbr 5100 {copab 5162 ↦ cmpt 5181 × cxp 5630 ^◡ccnv 5631 ran crn 5633 “ cima 5635 ‘cfv 6500 (class class class)co 7368 ∈ cmpo 7370 ↑_m cmap 8775 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 − cmin 11376 / cdiv 11806 ℕcn 12157 2c2 12212 3c3 12213 ℕ₀cn0 12413 ℤcz 12500 ℤ_≥cuz 12763 ...cfz 13435 ⌊cfl 13722 ↑cexp 13996 Ccbc 14237 ♯chash 14265 √csqrt 15168 Σcsu 15621 ∥ cdvds 16191 gcd cgcd 16433 ℙcprime 16610 od_ℤcodz 16702 ϕcphi 16703 Basecbs 17148 ↾_s cress 17169 +_gcplusg 17189 0_gc0g 17371 Σ_g cgsu 17372 /_s cqus 17438 ._gcmg 19009 ~_QG cqg 19064 mulGrpcmgp 20087 RingIso crs 20418 Fieldcfield 20675 ℤ_ringczring 21413 ℤRHomczrh 21466 chrcchr 21468 ℤ/nℤczn 21469 algSccascl 21819 var₁cv1 22128 Poly₁cpl1 22129 eval₁ce1 22270 log_b clogb 26742 PrimRoots cprimroots 42458

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-ofr 7633 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-ec 8647 df-qs 8651 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-dju 9825 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-xnn0 12487 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-fac 14209 df-bc 14238 df-hash 14266 df-shft 15002 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-limsup 15406 df-clim 15423 df-rlim 15424 df-sum 15622 df-prod 15839 df-fallfac 15942 df-ef 16002 df-sin 16004 df-cos 16005 df-pi 16007 df-dvds 16192 df-gcd 16434 df-prm 16611 df-odz 16704 df-phi 16705 df-pc 16777 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-pws 17381 df-xrs 17435 df-qtop 17440 df-imas 17441 df-qus 17442 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19010 df-subg 19065 df-nsg 19066 df-eqg 19067 df-ghm 19154 df-gim 19200 df-cntz 19258 df-od 19469 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-srg 20134 df-ring 20182 df-cring 20183 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-dvr 20349 df-rhm 20420 df-rim 20421 df-nzr 20458 df-subrng 20491 df-subrg 20515 df-rlreg 20639 df-domn 20640 df-idom 20641 df-drng 20676 df-field 20677 df-lmod 20825 df-lss 20895 df-lsp 20935 df-sra 21137 df-rgmod 21138 df-lidl 21175 df-rsp 21176 df-2idl 21217 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-zring 21414 df-zrh 21470 df-chr 21472 df-zn 21473 df-assa 21820 df-asp 21821 df-ascl 21822 df-psr 21877 df-mvr 21878 df-mpl 21879 df-opsr 21881 df-evls 22041 df-evl 22042 df-psr1 22132 df-vr1 22133 df-ply1 22134 df-coe1 22135 df-evl1 22272 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-limc 25835 df-dv 25836 df-mdeg 26028 df-deg1 26029 df-mon1 26104 df-uc1p 26105 df-q1p 26106 df-r1p 26107 df-log 26533 df-cxp 26534 df-logb 26743 df-primroots 42459

This theorem is referenced by: aks6d1c7lem3 42549

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