aks6d1c7lem2 - Mathbox for metakunt

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Theorem aks6d1c7lem2 42758

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 488	. 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 484	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐾 ∈ Field)
6		aks6d1c7lem2.4	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ ℙ)
7	6	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑃 ∈ ℙ)
8		aks6d1c7lem2.5	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℕ)
9	8	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑅 ∈ ℕ)
10		aks6d1c7lem2.6	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘3))
11		eluzelz 12842	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 𝑁 ∈ ℤ)
12	10, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
13		0red 11177	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
14		3re 12291	. . . . . . . . 9 ⊢ 3 ∈ ℝ
15	14	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 3 ∈ ℝ)
16	12	zred 12670	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17		3pos 12319	. . . . . . . . 9 ⊢ 0 < 3
18	17	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 3)
19		eluzle 12845	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘3) → 3 ≤ 𝑁)
20	10, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → 3 ≤ 𝑁)
21	13, 15, 16, 18, 20	ltletrd 11336	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑁)
22	12, 21	jca 519	. . . . . 6 ⊢ (𝜑 → (𝑁 ∈ ℤ ∧ 0 < 𝑁))
23		elnnz 12571	. . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
24	22, 23	sylibr 236	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
25	24	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑁 ∈ ℕ)
26		aks6d1c7lem2.7	. . . . 5 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
27	26	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑃 ∥ 𝑁)
28		aks6d1c7lem2.8	. . . . 5 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
29	28	adantr 484	. . . 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 16797	. . . . . . . . . . . 12 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℕ)
33	32	nnred 12218	. . . . . . . . . . 11 ⊢ (𝜑 → (ϕ‘𝑅) ∈ ℝ)
34		1red 11175	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
35		0le1 11703	. . . . . . . . . . . . 13 ⊢ 0 ≤ 1
36	35	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ 1)
37	32	nnge1d 12254	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ (ϕ‘𝑅))
38	13, 34, 33, 36, 37	letrd 11333	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (ϕ‘𝑅))
39	33, 38	resqrtcld 15435	. . . . . . . . . 10 ⊢ (𝜑 → (√‘(ϕ‘𝑅)) ∈ ℝ)
40		2re 12285	. . . . . . . . . . . 12 ⊢ 2 ∈ ℝ
41	40	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ∈ ℝ)
42		2pos 12315	. . . . . . . . . . . 12 ⊢ 0 < 2
43	42	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 2)
44		1lt2 12383	. . . . . . . . . . . . . 14 ⊢ 1 < 2
45	44	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 < 2)
46	34, 45	ltned 11312	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≠ 2)
47	46	necomd 3011	. . . . . . . . . . 11 ⊢ (𝜑 → 2 ≠ 1)
48	41, 43, 16, 21, 47	relogbcld 42551	. . . . . . . . . 10 ⊢ (𝜑 → (2 log_b 𝑁) ∈ ℝ)
49	39, 48	remulcld 11205	. . . . . . . . 9 ⊢ (𝜑 → ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ∈ ℝ)
50	49	flcld 13801	. . . . . . . 8 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℤ)
51	33, 38	sqrtge0d 15438	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (√‘(ϕ‘𝑅)))
52	41	recnd 11203	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℂ)
53	13, 43	gtned 11311	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ≠ 0)
54		logb1 26821	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1) → (2 log_b 1) = 0)
55	52, 53, 47, 54	syl3anc 1389	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 log_b 1) = 0)
56	55	eqcomd 2767	. . . . . . . . . . 11 ⊢ (𝜑 → 0 = (2 log_b 1))
57		2z 12596	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℤ
58	57	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ∈ ℤ)
59	41	leidd 11746	. . . . . . . . . . . 12 ⊢ (𝜑 → 2 ≤ 2)
60		0lt1 11702	. . . . . . . . . . . . 13 ⊢ 0 < 1
61	60	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 1)
62	24	nnge1d 12254	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ≤ 𝑁)
63	58, 59, 34, 61, 16, 21, 62	logblebd 42554	. . . . . . . . . . 11 ⊢ (𝜑 → (2 log_b 1) ≤ (2 log_b 𝑁))
64	56, 63	eqbrtrd 5119	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ (2 log_b 𝑁))
65	39, 48, 51, 64	mulge0d 11757	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))
66		0zd 12573	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
67		flge 13808	. . . . . . . . . 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 593	. . . . . . . . 9 ⊢ (𝜑 → (0 ≤ ((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)) ↔ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))))
69	65, 68	mpbid 234	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))))
70	50, 69	jca 519	. . . . . . 7 ⊢ (𝜑 → ((⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℤ ∧ 0 ≤ (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁)))))
71		elnn0z 12574	. . . . . . 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 236	. . . . . 6 ⊢ (𝜑 → (⌊‘((√‘(ϕ‘𝑅)) · (2 log_b 𝑁))) ∈ ℕ₀)
73	31, 72	eqeltrid 2865	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℕ₀)
74	73	adantr 484	. . . 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 484	. . . 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 484	. . . 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 484	. . . 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 498	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℙ)
88	87	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑄 ∈ ℙ)
89	86	simprd 499	. . . . . 6 ⊢ (𝜑 → 𝑄 ∥ 𝑁)
90	89	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑄 ∥ 𝑁)
91		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑃 ≠ 𝑄)
92	88, 90, 91	3jca 1140	. . . 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 42707	. . 3 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵))
94	24	nnzd 12587	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ)
95		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
96	24, 6, 26, 8, 28, 75, 76, 95	hashscontpowcl 42697	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
97	96	nn0red 12536	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℝ)
98	96	nn0ge0d 12538	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))
99	97, 98	resqrtcld 15435	. . . . . . . . . . . . 13 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ)
100	99	flcld 13801	. . . . . . . . . . . 12 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ)
101	97, 98	sqrtge0d 15438	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))
102		flge 13808	. . . . . . . . . . . . . 14 ⊢ (((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
103	99, 66, 102	syl2anc 593	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
104	101, 103	mpbid 234	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))))
105	100, 104	jca 519	. . . . . . . . . . 11 ⊢ (𝜑 → ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
106		elnn0z 12574	. . . . . . . . . . 11 ⊢ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℕ₀ ↔ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))))))
107	105, 106	sylibr 236	. . . . . . . . . 10 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) ∈ ℕ₀)
108	84, 107	eqeltrid 2865	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℕ₀)
109	94, 108	zexpcld 14093	. . . . . . . 8 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℤ)
110	109	zred 12670	. . . . . . 7 ⊢ (𝜑 → (𝑁↑𝐵) ∈ ℝ)
111	110	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) ∈ ℝ)
112	111	rexrd 11225	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) ∈ ℝ^*)
113		aks6d1c7lem2.11	. . . . . . . . . 10 ⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))
114	96	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ∈ ℕ₀)
115	113, 114	eqeltrid 2865	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐷 ∈ ℕ₀)
116	115, 74	nn0addcld 12539	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝐷 + 𝐴) ∈ ℕ₀)
117	115	nn0zd 12586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐷 ∈ ℤ)
118		1zzd 12595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 1 ∈ ℤ)
119	117, 118	zsubcld 12675	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝐷 − 1) ∈ ℤ)
120		bccl 14328	. . . . . . . 8 ⊢ (((𝐷 + 𝐴) ∈ ℕ₀ ∧ (𝐷 − 1) ∈ ℤ) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℕ₀)
121	116, 119, 120	syl2anc 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℕ₀)
122	121	nn0red 12536	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℝ)
123	122	rexrd 11225	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ∈ ℝ^*)
124		ovexd 7425	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (ℕ₀ ↑_m (0...𝐴)) ∈ V)
125	124	mptexd 7202	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (ℎ ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ (((eval₁‘𝐾)‘(𝐺‘ℎ))‘𝑀)) ∈ V)
126	83, 125	eqeltrid 2865	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐻 ∈ V)
127	126	imaexd 7891	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝐻 “ (ℕ₀ ↑_m (0...𝐴))) ∈ V)
128		hashxrcl 14363	. . . . . 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 2768	. . . . . . . . . . . 12 ⊢ (𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) ↔ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = 𝐷)
131	113, 130	mpbi 232	. . . . . . . . . . 11 ⊢ (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))) = 𝐷
132	131	fveq2i 6864	. . . . . . . . . 10 ⊢ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))) = (√‘𝐷)
133	132	fveq2i 6864	. . . . . . . . 9 ⊢ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀)))))) = (⌊‘(√‘𝐷))
134	84, 133	eqtri 2784	. . . . . . . 8 ⊢ 𝐵 = (⌊‘(√‘𝐷))
135	134	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝐵 = (⌊‘(√‘𝐷)))
136	135	oveq2d 7406	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) = (𝑁↑(⌊‘(√‘𝐷))))
137	10	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑁 ∈ (ℤ_≥‘3))
138		aks6d1c7lem2.13	. . . . . . . 8 ⊢ (𝜑 → ((2 log_b 𝑁)↑2) < ((od_ℤ‘𝑅)‘𝑁))
139	138	adantr 484	. . . . . . 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 42757	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑(⌊‘(√‘𝐷))) < ((𝐷 + 𝐴)C(𝐷 − 1)))
141	136, 140	eqbrtrd 5119	. . . . 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 484	. . . . . 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 2761	. . . . . 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 2761	. . . . . 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 2923	. . . . . . 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 2923	. . . . . . 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 6040	. . . . . . . 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 4875	. . . . . . 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 5199	. . . . . 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 42754	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝐷 + 𝐴)C(𝐷 − 1)) ≤ (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))))
153	112, 123, 129, 141, 152	xrltletrd 13156	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → (𝑁↑𝐵) < (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))))
154		xrltnle 11242	. . . . 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 593	. . . 4 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ((𝑁↑𝐵) < (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ↔ ¬ (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵)))
156	153, 155	mpbid 234	. . 3 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → ¬ (♯‘(𝐻 “ (ℕ₀ ↑_m (0...𝐴)))) ≤ (𝑁↑𝐵))
157	93, 156	pm2.21dd 197	. 2 ⊢ ((𝜑 ∧ 𝑃 ≠ 𝑄) → 𝑃 = 𝑄)
158	1, 157	pm2.61dane 3043	1 ⊢ (𝜑 → 𝑃 = 𝑄)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 ∃wrex 3085 {crab 3413 Vcvv 3453 {csn 4579 ∪ cuni 4862 class class class wbr 5097 {copab 5159 ↦ cmpt 5178 × cxp 5641 ^◡ccnv 5642 ran crn 5644 “ cima 5646 ‘cfv 6515 (class class class)co 7390 ∈ cmpo 7392 ↑_m cmap 8801 ℂcc 11064 ℝcr 11065 0cc0 11066 1c1 11067 + caddc 11069 · cmul 11071 ℝ^*cxr 11208 < clt 11209 ≤ cle 11210 − cmin 11407 / cdiv 11837 ℕcn 12203 2c2 12265 3c3 12266 ℕ₀cn0 12474 ℤcz 12561 ℤ_≥cuz 12832 ...cfz 13505 ⌊cfl 13793 ↑cexp 14067 Ccbc 14308 ♯chash 14336 √csqrt 15250 Σcsu 15703 ∥ cdvds 16276 gcd cgcd 16518 ℙcprime 16695 od_ℤcodz 16788 ϕcphi 16789 Basecbs 17235 ↾_s cress 17256 +_gcplusg 17276 0_gc0g 17458 Σ_g cgsu 17459 /_s cqus 17525 ._gcmg 19099 ~_QG cqg 19154 mulGrpcmgp 20176 RingIso crs 20505 Fieldcfield 20766 ℤ_ringczring 21485 ℤRHomczrh 21538 chrcchr 21540 ℤ/nℤczn 21541 algSccascl 21891 var₁cv1 22225 Poly₁cpl1 22226 eval₁ce1 22364 log_b clogb 26816 PrimRoots cprimroots 42668

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-inf2 9589 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143 ax-pre-sup 11144 ax-addf 11145 ax-mulf 11146

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7654 df-ofr 7655 df-om 7841 df-1st 7964 df-2nd 7965 df-supp 8134 df-tpos 8199 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-oadd 8434 df-er 8671 df-ec 8673 df-qs 8677 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9301 df-fi 9350 df-sup 9381 df-inf 9382 df-oi 9451 df-dju 9852 df-card 9890 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-nn 12204 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12475 df-xnn0 12548 df-z 12562 df-dec 12682 df-uz 12833 df-q 12943 df-rp 12987 df-xneg 13107 df-xadd 13108 df-xmul 13109 df-ioo 13346 df-ioc 13347 df-ico 13348 df-icc 13349 df-fz 13506 df-fzo 13653 df-fl 13795 df-mod 13873 df-seq 14008 df-exp 14068 df-fac 14280 df-bc 14309 df-hash 14337 df-shft 15073 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-limsup 15488 df-clim 15505 df-rlim 15506 df-sum 15704 df-prod 15924 df-fallfac 16027 df-ef 16087 df-sin 16089 df-cos 16090 df-pi 16092 df-dvds 16277 df-gcd 16519 df-prm 16696 df-odz 16790 df-phi 16791 df-pc 16863 df-struct 17173 df-sets 17190 df-slot 17208 df-ndx 17220 df-base 17236 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-pws 17468 df-xrs 17522 df-qtop 17527 df-imas 17528 df-qus 17529 df-xps 17530 df-mre 17604 df-mrc 17605 df-acs 17607 df-mgm 18664 df-sgrp 18743 df-mnd 18759 df-mhm 18807 df-submnd 18808 df-grp 18968 df-minusg 18969 df-sbg 18970 df-mulg 19100 df-subg 19155 df-nsg 19156 df-eqg 19157 df-ghm 19244 df-gim 19289 df-cntz 19347 df-od 19558 df-cmn 19812 df-abl 19813 df-mgp 20177 df-rng 20189 df-ur 20218 df-srg 20223 df-ring 20271 df-cring 20272 df-oppr 20372 df-dvdsr 20392 df-unit 20393 df-invr 20423 df-dvr 20436 df-rhm 20507 df-rim 20508 df-nzr 20549 df-subrng 20582 df-subrg 20606 df-rlreg 20730 df-domn 20731 df-idom 20732 df-drng 20767 df-field 20768 df-lmod 20916 df-lss 20986 df-lsp 21026 df-sra 21227 df-rgmod 21228 df-lidl 21265 df-rsp 21266 df-2idl 21307 df-psmet 21403 df-xmet 21404 df-met 21405 df-bl 21406 df-mopn 21407 df-fbas 21408 df-fg 21409 df-cnfld 21412 df-zring 21486 df-zrh 21542 df-chr 21544 df-zn 21545 df-assa 21892 df-asp 21893 df-ascl 21894 df-psr 21948 df-mvr 21949 df-mpl 21950 df-opsr 21952 df-evls 22114 df-evl 22115 df-psr1 22229 df-vr1 22230 df-ply1 22231 df-coe1 22232 df-evl1 22366 df-top 22941 df-topon 22958 df-topsp 22980 df-bases 22993 df-cld 23066 df-ntr 23067 df-cls 23068 df-nei 23145 df-lp 23183 df-perf 23184 df-cn 23274 df-cnp 23275 df-haus 23362 df-tx 23609 df-hmeo 23802 df-fil 23893 df-fm 23985 df-flim 23986 df-flf 23987 df-xms 24367 df-ms 24368 df-tms 24369 df-cncf 24927 df-limc 25915 df-dv 25916 df-mdeg 26102 df-deg1 26103 df-mon1 26178 df-uc1p 26179 df-q1p 26180 df-r1p 26181 df-log 26608 df-cxp 26609 df-logb 26817 df-primroots 42669

This theorem is referenced by: aks6d1c7lem3 42759

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