aks6d1c7lem2 - Mathbox for metakunt

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Theorem aks6d1c7lem2 42138

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 12913	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
12	10, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
13		0red 11293	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
14		3re 12373	. . . . . . . . 9 ⊢ 3 ∈ ℝ
15	14	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 3 ∈ ℝ)
16	12	zred 12747	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17		3pos 12398	. . . . . . . . 9 ⊢ 0 < 3
18	17	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 3)
19		eluzle 12916	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
20	10, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → 3 ≤ 𝑁)
21	13, 15, 16, 18, 20	ltletrd 11450	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑁)
22	12, 21	jca 511	. . . . . 6 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
23		elnnz 12649	. . . . . 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 16819	. . . . . . . . . . . 12 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ)
33	32	nnred 12308	. . . . . . . . . . 11 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℝ)
34		1red 11291	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
35		0le1 11813	. . . . . . . . . . . . 13 ⊢ 0 ≤ 1
36	35	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ 1)
37	32	nnge1d 12341	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ (ϕ‘𝑅))
38	13, 34, 33, 36, 37	letrd 11447	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (ϕ‘𝑅))
39	33, 38	resqrtcld 15466	. . . . . . . . . 10 ⊢ (𝜑 → (√‘(ϕ‘𝑅)) ∈ ℝ)
40		2re 12367	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
41	40	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
42		2pos 12396	. . . . . . . . . . . 12 ⊢ 0 < 2
43	42	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 2)
44		1lt2 12464	. . . . . . . . . . . . . 14 ⊢ 1 < 2
45	44	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < 2)
46	34, 45	ltned 11426	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≠ 2)
47	46	necomd 3002	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≠ 1)
48	41, 43, 16, 21, 47	relogbcld 41929	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
49	39, 48	remulcld 11320	. . . . . . . . 9 ⊢ (𝜑 → ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ∈ ℝ)
50	49	flcld 13849	. . . . . . . 8 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℤ)
51	33, 38	sqrtge0d 15469	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (√‘(ϕ‘𝑅)))
52	41	recnd 11318	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℂ)
53	13, 43	gtned 11425	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ≠ 0)
54		logb1 26830	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 log_b 1) = 0)
55	52, 53, 47, 54	syl3anc 1371	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 log_b 1) = 0)
56	55	eqcomd 2746	. . . . . . . . . . 11 ⊢ (𝜑 → 0 = (2 log_b 1))
57		2z 12675	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
58	57	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℤ)
59	41	leidd 11856	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ≤ 2)
60		0lt1 11812	. . . . . . . . . . . . 13 ⊢ 0 < 1
61	60	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 1)
62	24	nnge1d 12341	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑁)
63	58, 59, 34, 61, 16, 21, 62	logblebd 41932	. . . . . . . . . . 11 ⊢ (𝜑 → (2 log_b 1) ≤ (2 log_b 𝑁))
64	56, 63	eqbrtrd 5188	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (2 log_b 𝑁))
65	39, 48, 51, 64	mulge0d 11867	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
66		0zd 12651	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
67		flge 13856	. . . . . . . . . 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 583	. . . . . . . . 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 12652	. . . . . . 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 2848	. . . . 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 1128	. . . 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 42087	. . 3 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵))
94	24	nnzd 12666	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
95		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
96	24, 6, 26, 8, 28, 75, 76, 95	hashscontpowcl 42077	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
97	96	nn0red 12614	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ)
98	96	nn0ge0d 12616	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
99	97, 98	resqrtcld 15466	. . . . . . . . . . . . 13 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ)
100	99	flcld 13849	. . . . . . . . . . . 12 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ)
101	97, 98	sqrtge0d 15469	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
102		flge 13856	. . . . . . . . . . . . . 14 ⊢ (((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
103	99, 66, 102	syl2anc 583	. . . . . . . . . . . . 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 12652	. . . . . . . . . . 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 2848	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℕ₀)
109	94, 108	zexpcld 14138	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℤ)
110	109	zred 12747	. . . . . . 7 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℝ)
111	110	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) ∈ ℝ)
112	111	rexrd 11340	. . . . 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 2848	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐷 ∈ ℕ₀)
116	115, 74	nn0addcld 12617	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝐷 + 𝐴) ∈ ℕ₀)
117	115	nn0zd 12665	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐷 ∈ ℤ)
118		1zzd 12674	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 1 ∈ ℤ)
119	117, 118	zsubcld 12752	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝐷 − 1) ∈ ℤ)
120		bccl 14371	. . . . . . . 8 ⊢ (((𝐷 + 𝐴) ∈ ℕ₀ ∧ (𝐷 − 1) ∈ ℤ) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℕ₀)
121	116, 119, 120	syl2anc 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℕ₀)
122	121	nn0red 12614	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℝ)
123	122	rexrd 11340	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℝ^*)
124		ovexd 7483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (ℕ₀ ↑_m (0...𝐴)) ∈ V)
125	124	mptexd 7261	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (ℎ ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ (((eval₁‘𝐾)‘(𝐺‘ℎ))‘𝑀)) ∈ V)
126	83, 125	eqeltrid 2848	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐻 ∈ V)
127	126	imaexd 7956	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ∈ V)
128		hashxrcl 14406	. . . . . 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 230	. . . . . . . . . . 11 ⊢ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = 𝐷
132	131	fveq2i 6923	. . . . . . . . . 10 ⊢ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) = (√‘𝐷)
133	132	fveq2i 6923	. . . . . . . . 9 ⊢ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = (⌊‘(√‘𝐷))
134	84, 133	eqtri 2768	. . . . . . . 8 ⊢ 𝐵 = (⌊‘(√‘𝐷))
135	134	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐵 = (⌊‘(√‘𝐷)))
136	135	oveq2d 7464	. . . . . 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 42137	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑(⌊‘(√‘𝐷))) < ((𝐷 + 𝐴)C(𝐷 − 1)))
141	136, 140	eqbrtrd 5188	. . . . 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 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 2908	. . . . . . 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 2908	. . . . . . 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 6085	. . . . . . . 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 4944	. . . . . . 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 5277	. . . . . 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 42134	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))))
153	112, 123, 129, 141, 152	xrltletrd 13223	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) < (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))))
154		xrltnle 11357	. . . . 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 583	. . . 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 3035	1 ⊢ (𝜑 → 𝑃 = 𝑄)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 {crab 3443 Vcvv 3488 {csn 4648 ∪ cuni 4931 class class class wbr 5166 {copab 5228 ↦ cmpt 5249 × cxp 5698 ^◡ccnv 5699 ran crn 5701 “ cima 5703 ‘cfv 6573 (class class class)co 7448 ∈ cmpo 7450 ↑_m cmap 8884 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℕcn 12293 2c2 12348 3c3 12349 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 ...cfz 13567 ⌊cfl 13841 ↑cexp 14112 Ccbc 14351 ♯chash 14379 √csqrt 15282 Σcsu 15734 ∥ cdvds 16302 gcd cgcd 16540 ℙcprime 16718 od_ℤcodz 16810 ϕcphi 16811 Basecbs 17258 ↾_s cress 17287 +_gcplusg 17311 0_gc0g 17499 Σ_g cgsu 17500 /_s cqus 17565 ._gcmg 19107 ~_QG cqg 19162 mulGrpcmgp 20161 RingIso crs 20496 Fieldcfield 20752 ℤ_ringczring 21480 ℤRHomczrh 21533 chrcchr 21535 ℤ/nℤczn 21536 algSccascl 21895 var₁cv1 22198 Poly₁cpl1 22199 eval₁ce1 22339 log_b clogb 26825 PrimRoots cprimroots 42048

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-ec 8765 df-qs 8769 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-prod 15952 df-fallfac 16055 df-ef 16115 df-sin 16117 df-cos 16118 df-pi 16120 df-dvds 16303 df-gcd 16541 df-prm 16719 df-odz 16812 df-phi 16813 df-pc 16884 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 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 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-pws 17509 df-xrs 17562 df-qtop 17567 df-imas 17568 df-qus 17569 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-nsg 19164 df-eqg 19165 df-ghm 19253 df-gim 19299 df-cntz 19357 df-od 19570 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-srg 20214 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-rhm 20498 df-rim 20499 df-nzr 20539 df-subrng 20572 df-subrg 20597 df-rlreg 20716 df-domn 20717 df-idom 20718 df-drng 20753 df-field 20754 df-lmod 20882 df-lss 20953 df-lsp 20993 df-sra 21195 df-rgmod 21196 df-lidl 21241 df-rsp 21242 df-2idl 21283 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-zring 21481 df-zrh 21537 df-chr 21539 df-zn 21540 df-assa 21896 df-asp 21897 df-ascl 21898 df-psr 21952 df-mvr 21953 df-mpl 21954 df-opsr 21956 df-evls 22121 df-evl 22122 df-psr1 22202 df-vr1 22203 df-ply1 22204 df-coe1 22205 df-evl1 22341 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922 df-mdeg 26114 df-deg1 26115 df-mon1 26190 df-uc1p 26191 df-q1p 26192 df-r1p 26193 df-log 26616 df-cxp 26617 df-logb 26826 df-primroots 42049

This theorem is referenced by: aks6d1c7lem3 42139

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