aks6d1c7lem2 - Mathbox for metakunt

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Theorem aks6d1c7lem2 42673

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 485	. 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 481	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐾 ∈ Field)
6		aks6d1c7lem2.4	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℙ)
7	6	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ ℙ)
8		aks6d1c7lem2.5	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℕ)
9	8	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑅 ∈ ℕ)
10		aks6d1c7lem2.6	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
11		eluzelz 12796	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
12	10, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
13		0red 11145	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
14		3re 12259	. . . . . . . . 9 ⊢ 3 ∈ ℝ
15	14	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 3 ∈ ℝ)
16	12	zred 12631	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17		3pos 12284	. . . . . . . . 9 ⊢ 0 < 3
18	17	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 3)
19		eluzle 12799	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
20	10, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → 3 ≤ 𝑁)
21	13, 15, 16, 18, 20	ltletrd 11304	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑁)
22	12, 21	jca 516	. . . . . 6 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
23		elnnz 12532	. . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
24	22, 23	sylibr 235	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
25	24	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑁 ∈ ℕ)
26		aks6d1c7lem2.7	. . . . 5 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
27	26	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑃 ∥ 𝑁)
28		aks6d1c7lem2.8	. . . . 5 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
29	28	adantr 481	. . . 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 16740	. . . . . . . . . . . 12 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ)
33	32	nnred 12187	. . . . . . . . . . 11 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℝ)
34		1red 11143	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
35		0le1 11671	. . . . . . . . . . . . 13 ⊢ 0 ≤ 1
36	35	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ 1)
37	32	nnge1d 12223	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ (ϕ‘𝑅))
38	13, 34, 33, 36, 37	letrd 11301	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (ϕ‘𝑅))
39	33, 38	resqrtcld 15378	. . . . . . . . . 10 ⊢ (𝜑 → (√‘(ϕ‘𝑅)) ∈ ℝ)
40		2re 12253	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
41	40	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
42		2pos 12282	. . . . . . . . . . . 12 ⊢ 0 < 2
43	42	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 2)
44		1lt2 12345	. . . . . . . . . . . . . 14 ⊢ 1 < 2
45	44	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < 2)
46	34, 45	ltned 11280	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≠ 2)
47	46	necomd 2990	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≠ 1)
48	41, 43, 16, 21, 47	relogbcld 42466	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
49	39, 48	remulcld 11173	. . . . . . . . 9 ⊢ (𝜑 → ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ∈ ℝ)
50	49	flcld 13755	. . . . . . . 8 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℤ)
51	33, 38	sqrtge0d 15381	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (√‘(ϕ‘𝑅)))
52	41	recnd 11171	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℂ)
53	13, 43	gtned 11279	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ≠ 0)
54		logb1 26758	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 log_b 1) = 0)
55	52, 53, 47, 54	syl3anc 1379	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 log_b 1) = 0)
56	55	eqcomd 2746	. . . . . . . . . . 11 ⊢ (𝜑 → 0 = (2 log_b 1))
57		2z 12557	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
58	57	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℤ)
59	41	leidd 11714	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ≤ 2)
60		0lt1 11670	. . . . . . . . . . . . 13 ⊢ 0 < 1
61	60	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 1)
62	24	nnge1d 12223	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑁)
63	58, 59, 34, 61, 16, 21, 62	logblebd 42469	. . . . . . . . . . 11 ⊢ (𝜑 → (2 log_b 1) ≤ (2 log_b 𝑁))
64	56, 63	eqbrtrd 5101	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (2 log_b 𝑁))
65	39, 48, 51, 64	mulge0d 11725	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
66		0zd 12534	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
67		flge 13762	. . . . . . . . . 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 590	. . . . . . . . 9 ⊢ (𝜑 → (0 ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ↔ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))))
69	65, 68	mpbid 233	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))
70	50, 69	jca 516	. . . . . . 7 ⊢ (𝜑 → ((⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))))
71		elnn0z 12535	. . . . . . 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 235	. . . . . 6 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℕ₀)
73	31, 72	eqeltrid 2844	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℕ₀)
74	73	adantr 481	. . . 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 481	. . . 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 481	. . . 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 481	. . . 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 495	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℙ)
88	87	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ ℙ)
89	86	simprd 496	. . . . . 6 ⊢ (𝜑 → 𝑄 ∥ 𝑁)
90	89	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑄 ∥ 𝑁)
91		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑃 ≠ 𝑄)
92	88, 90, 91	3jca 1134	. . . 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 42622	. . 3 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵))
94	24	nnzd 12548	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
95		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
96	24, 6, 26, 8, 28, 75, 76, 95	hashscontpowcl 42612	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
97	96	nn0red 12497	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ)
98	96	nn0ge0d 12499	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
99	97, 98	resqrtcld 15378	. . . . . . . . . . . . 13 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ)
100	99	flcld 13755	. . . . . . . . . . . 12 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ)
101	97, 98	sqrtge0d 15381	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
102		flge 13762	. . . . . . . . . . . . . 14 ⊢ (((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
103	99, 66, 102	syl2anc 590	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
104	101, 103	mpbid 233	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
105	100, 104	jca 516	. . . . . . . . . . 11 ⊢ (𝜑 → ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
106		elnn0z 12535	. . . . . . . . . . 11 ⊢ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℕ₀ ↔ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
107	105, 106	sylibr 235	. . . . . . . . . 10 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℕ₀)
108	84, 107	eqeltrid 2844	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℕ₀)
109	94, 108	zexpcld 14047	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℤ)
110	109	zred 12631	. . . . . . 7 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℝ)
111	110	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) ∈ ℝ)
112	111	rexrd 11193	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) ∈ ℝ^*)
113		aks6d1c7lem2.11	. . . . . . . . . 10 ⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))
114	96	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
115	113, 114	eqeltrid 2844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐷 ∈ ℕ₀)
116	115, 74	nn0addcld 12500	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝐷 + 𝐴) ∈ ℕ₀)
117	115	nn0zd 12547	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐷 ∈ ℤ)
118		1zzd 12556	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 1 ∈ ℤ)
119	117, 118	zsubcld 12636	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝐷 − 1) ∈ ℤ)
120		bccl 14282	. . . . . . . 8 ⊢ (((𝐷 + 𝐴) ∈ ℕ₀ ∧ (𝐷 − 1) ∈ ℤ) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℕ₀)
121	116, 119, 120	syl2anc 590	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℕ₀)
122	121	nn0red 12497	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℝ)
123	122	rexrd 11193	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℝ^*)
124		ovexd 7398	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (ℕ₀ ↑_m (0...𝐴)) ∈ V)
125	124	mptexd 7175	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (ℎ ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ (((eval₁‘𝐾)‘(𝐺‘ℎ))‘𝑀)) ∈ V)
126	83, 125	eqeltrid 2844	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐻 ∈ V)
127	126	imaexd 7863	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ∈ V)
128		hashxrcl 14317	. . . . . 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 2747	. . . . . . . . . . . 12 ⊢ (𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ↔ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = 𝐷)
131	113, 130	mpbi 231	. . . . . . . . . . 11 ⊢ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = 𝐷
132	131	fveq2i 6837	. . . . . . . . . 10 ⊢ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) = (√‘𝐷)
133	132	fveq2i 6837	. . . . . . . . 9 ⊢ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = (⌊‘(√‘𝐷))
134	84, 133	eqtri 2763	. . . . . . . 8 ⊢ 𝐵 = (⌊‘(√‘𝐷))
135	134	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐵 = (⌊‘(√‘𝐷)))
136	135	oveq2d 7379	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) = (𝑁↑(⌊‘(√‘𝐷))))
137	10	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑁 ∈ (ℤ_≥‘3))
138		aks6d1c7lem2.13	. . . . . . . 8 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))
139	138	adantr 481	. . . . . . 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 42672	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑(⌊‘(√‘𝐷))) < ((𝐷 + 𝐴)C(𝐷 − 1)))
141	136, 140	eqbrtrd 5101	. . . . 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 481	. . . . . 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 2740	. . . . . 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 2740	. . . . . 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 2902	. . . . . . 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 2902	. . . . . . 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 6015	. . . . . . . 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 4858	. . . . . . 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 5181	. . . . . 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 42669	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))))
153	112, 123, 129, 141, 152	xrltletrd 13110	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) < (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))))
154		xrltnle 11210	. . . . 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 590	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝑁↑𝐵) < (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ↔ ¬ (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵)))
156	153, 155	mpbid 233	. . 3 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ¬ (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵))
157	93, 156	pm2.21dd 196	. 2 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑃 = 𝑄)
158	1, 157	pm2.61dane 3022	1 ⊢ (𝜑 → 𝑃 = 𝑄)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 ∀wral 3054 ∃wrex 3064 {crab 3392 Vcvv 3432 {csn 4562 ∪ cuni 4845 class class class wbr 5079 {copab 5141 ↦ cmpt 5160 × cxp 5623 ^◡ccnv 5624 ran crn 5626 “ cima 5628 ‘cfv 6492 (class class class)co 7363 ∈ cmpo 7365 ↑_m cmap 8770 ℂcc 11034 ℝcr 11035 0cc0 11036 1c1 11037 + caddc 11039 · cmul 11041 ℝ^*cxr 11176 < clt 11177 ≤ cle 11178 − cmin 11375 / cdiv 11805 ℕcn 12172 2c2 12234 3c3 12235 ℕ₀cn0 12435 ℤcz 12522 ℤ_≥cuz 12786 ...cfz 13459 ⌊cfl 13747 ↑cexp 14021 Ccbc 14262 ♯chash 14290 √csqrt 15193 Σcsu 15646 ∥ cdvds 16219 gcd cgcd 16461 ℙcprime 16638 od_ℤcodz 16731 ϕcphi 16732 Basecbs 17177 ↾_s cress 17198 +_gcplusg 17218 0_gc0g 17400 Σ_g cgsu 17401 /_s cqus 17467 ._gcmg 19041 ~_QG cqg 19096 mulGrpcmgp 20119 RingIso crs 20448 Fieldcfield 20709 ℤ_ringczring 21428 ℤRHomczrh 21481 chrcchr 21483 ℤ/nℤczn 21484 algSccascl 21834 var₁cv1 22168 Poly₁cpl1 22169 eval₁ce1 22307 log_b clogb 26753 PrimRoots cprimroots 42583

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114 ax-addf 11115 ax-mulf 11116

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-ofr 7628 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-tpos 8173 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-oadd 8406 df-er 8640 df-ec 8642 df-qs 8646 df-map 8772 df-pm 8773 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-fi 9321 df-sup 9352 df-inf 9353 df-oi 9422 df-dju 9823 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-xnn0 12509 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-ioo 13300 df-ioc 13301 df-ico 13302 df-icc 13303 df-fz 13460 df-fzo 13607 df-fl 13749 df-mod 13827 df-seq 13962 df-exp 14022 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15027 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-limsup 15431 df-clim 15448 df-rlim 15449 df-sum 15647 df-prod 15867 df-fallfac 15970 df-ef 16030 df-sin 16032 df-cos 16033 df-pi 16035 df-dvds 16220 df-gcd 16462 df-prm 16639 df-odz 16733 df-phi 16734 df-pc 16806 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17383 df-topn 17384 df-0g 17402 df-gsum 17403 df-topgen 17404 df-pt 17405 df-prds 17408 df-pws 17410 df-xrs 17464 df-qtop 17469 df-imas 17470 df-qus 17471 df-xps 17472 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-mhm 18749 df-submnd 18750 df-grp 18910 df-minusg 18911 df-sbg 18912 df-mulg 19042 df-subg 19097 df-nsg 19098 df-eqg 19099 df-ghm 19186 df-gim 19232 df-cntz 19290 df-od 19501 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-srg 20166 df-ring 20214 df-cring 20215 df-oppr 20315 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-rhm 20450 df-rim 20451 df-nzr 20492 df-subrng 20525 df-subrg 20549 df-rlreg 20673 df-domn 20674 df-idom 20675 df-drng 20710 df-field 20711 df-lmod 20859 df-lss 20929 df-lsp 20969 df-sra 21170 df-rgmod 21171 df-lidl 21208 df-rsp 21209 df-2idl 21250 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-fbas 21351 df-fg 21352 df-cnfld 21355 df-zring 21429 df-zrh 21485 df-chr 21487 df-zn 21488 df-assa 21835 df-asp 21836 df-ascl 21837 df-psr 21891 df-mvr 21892 df-mpl 21893 df-opsr 21895 df-evls 22057 df-evl 22058 df-psr1 22172 df-vr1 22173 df-ply1 22174 df-coe1 22175 df-evl1 22309 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cld 23009 df-ntr 23010 df-cls 23011 df-nei 23088 df-lp 23126 df-perf 23127 df-cn 23217 df-cnp 23218 df-haus 23305 df-tx 23552 df-hmeo 23745 df-fil 23836 df-fm 23928 df-flim 23929 df-flf 23930 df-xms 24310 df-ms 24311 df-tms 24312 df-cncf 24870 df-limc 25858 df-dv 25859 df-mdeg 26045 df-deg1 26046 df-mon1 26121 df-uc1p 26122 df-q1p 26123 df-r1p 26124 df-log 26545 df-cxp 26546 df-logb 26754 df-primroots 42584

This theorem is referenced by: aks6d1c7lem3 42674

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