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Theorem prmind2 11594
Description: A variation on prmind 11595 assuming complete induction for primes. (Contributed by Mario Carneiro, 20-Jun-2015.)
Hypotheses
Ref Expression
prmind.1 (𝑥 = 1 → (𝜑𝜓))
prmind.2 (𝑥 = 𝑦 → (𝜑𝜒))
prmind.3 (𝑥 = 𝑧 → (𝜑𝜃))
prmind.4 (𝑥 = (𝑦 · 𝑧) → (𝜑𝜏))
prmind.5 (𝑥 = 𝐴 → (𝜑𝜂))
prmind.6 𝜓
prmind2.7 ((𝑥 ∈ ℙ ∧ ∀𝑦 ∈ (1...(𝑥 − 1))𝜒) → 𝜑)
prmind2.8 ((𝑦 ∈ (ℤ‘2) ∧ 𝑧 ∈ (ℤ‘2)) → ((𝜒𝜃) → 𝜏))
Assertion
Ref Expression
prmind2 (𝐴 ∈ ℕ → 𝜂)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝑥,𝑧,𝜒   𝜂,𝑥   𝜏,𝑥   𝜃,𝑥   𝑦,𝑧,𝜑
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑦)   𝜃(𝑦,𝑧)   𝜏(𝑦,𝑧)   𝜂(𝑦,𝑧)   𝐴(𝑦,𝑧)

Proof of Theorem prmind2
Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prmind.5 . 2 (𝑥 = 𝐴 → (𝜑𝜂))
2 oveq2 5714 . . . 4 (𝑛 = 1 → (1...𝑛) = (1...1))
32raleqdv 2590 . . 3 (𝑛 = 1 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...1)𝜑))
4 oveq2 5714 . . . 4 (𝑛 = 𝑘 → (1...𝑛) = (1...𝑘))
54raleqdv 2590 . . 3 (𝑛 = 𝑘 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝑘)𝜑))
6 oveq2 5714 . . . 4 (𝑛 = (𝑘 + 1) → (1...𝑛) = (1...(𝑘 + 1)))
76raleqdv 2590 . . 3 (𝑛 = (𝑘 + 1) → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...(𝑘 + 1))𝜑))
8 oveq2 5714 . . . 4 (𝑛 = 𝐴 → (1...𝑛) = (1...𝐴))
98raleqdv 2590 . . 3 (𝑛 = 𝐴 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝐴)𝜑))
10 prmind.6 . . . . 5 𝜓
11 elfz1eq 9656 . . . . . 6 (𝑥 ∈ (1...1) → 𝑥 = 1)
12 prmind.1 . . . . . 6 (𝑥 = 1 → (𝜑𝜓))
1311, 12syl 14 . . . . 5 (𝑥 ∈ (1...1) → (𝜑𝜓))
1410, 13mpbiri 167 . . . 4 (𝑥 ∈ (1...1) → 𝜑)
1514rgen 2444 . . 3 𝑥 ∈ (1...1)𝜑
16 peano2nn 8590 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
1716ad2antrr 475 . . . . . . . . . . . 12 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℕ)
1817nncnd 8592 . . . . . . . . . . 11 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℂ)
19 elfzuz 9643 . . . . . . . . . . . . . 14 (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ∈ (ℤ‘2))
2019ad2antrl 477 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ (ℤ‘2))
21 eluz2nn 9214 . . . . . . . . . . . . 13 (𝑦 ∈ (ℤ‘2) → 𝑦 ∈ ℕ)
2220, 21syl 14 . . . . . . . . . . . 12 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℕ)
2322nncnd 8592 . . . . . . . . . . 11 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℂ)
2422nnap0d 8624 . . . . . . . . . . 11 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 # 0)
2518, 23, 24divcanap2d 8413 . . . . . . . . . 10 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 · ((𝑘 + 1) / 𝑦)) = (𝑘 + 1))
26 simprr 502 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∥ (𝑘 + 1))
2722nnzd 9024 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℤ)
2822nnne0d 8623 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≠ 0)
2917nnzd 9024 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℤ)
30 dvdsval2 11291 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℤ ∧ 𝑦 ≠ 0 ∧ (𝑘 + 1) ∈ ℤ) → (𝑦 ∥ (𝑘 + 1) ↔ ((𝑘 + 1) / 𝑦) ∈ ℤ))
3127, 28, 29, 30syl3anc 1184 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ∥ (𝑘 + 1) ↔ ((𝑘 + 1) / 𝑦) ∈ ℤ))
3226, 31mpbid 146 . . . . . . . . . . . 12 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℤ)
3323mulid2d 7656 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) = 𝑦)
34 elfzle2 9649 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ≤ ((𝑘 + 1) − 1))
3534ad2antrl 477 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≤ ((𝑘 + 1) − 1))
36 nncn 8586 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
3736ad2antrr 475 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℂ)
38 ax-1cn 7588 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
39 pncan 7839 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
4037, 38, 39sylancl 407 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) − 1) = 𝑘)
4135, 40breqtrd 3899 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦𝑘)
42 nnz 8925 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
4342ad2antrr 475 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℤ)
44 zleltp1 8961 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑦𝑘𝑦 < (𝑘 + 1)))
4527, 43, 44syl2anc 406 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦𝑘𝑦 < (𝑘 + 1)))
4641, 45mpbid 146 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 < (𝑘 + 1))
4733, 46eqbrtrd 3895 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) < (𝑘 + 1))
48 1red 7653 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 ∈ ℝ)
4917nnred 8591 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℝ)
5022nnred 8591 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ)
5122nngt0d 8622 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < 𝑦)
52 ltmuldiv 8490 . . . . . . . . . . . . . 14 ((1 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) → ((1 · 𝑦) < (𝑘 + 1) ↔ 1 < ((𝑘 + 1) / 𝑦)))
5348, 49, 50, 51, 52syl112anc 1188 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((1 · 𝑦) < (𝑘 + 1) ↔ 1 < ((𝑘 + 1) / 𝑦)))
5447, 53mpbid 146 . . . . . . . . . . . 12 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 < ((𝑘 + 1) / 𝑦))
55 eluz2b1 9245 . . . . . . . . . . . 12 (((𝑘 + 1) / 𝑦) ∈ (ℤ‘2) ↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 1 < ((𝑘 + 1) / 𝑦)))
5632, 54, 55sylanbrc 411 . . . . . . . . . . 11 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ (ℤ‘2))
57 prmind.2 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝜑𝜒))
58 simplr 500 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ∀𝑥 ∈ (1...𝑘)𝜑)
59 fznn 9710 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℤ → (𝑦 ∈ (1...𝑘) ↔ (𝑦 ∈ ℕ ∧ 𝑦𝑘)))
6043, 59syl 14 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ∈ (1...𝑘) ↔ (𝑦 ∈ ℕ ∧ 𝑦𝑘)))
6122, 41, 60mpbir2and 896 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ (1...𝑘))
6257, 58, 61rspcdva 2749 . . . . . . . . . . . 12 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝜒)
63 vex 2644 . . . . . . . . . . . . . . 15 𝑧 ∈ V
64 prmind.3 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝜑𝜃))
6563, 64sbcie 2895 . . . . . . . . . . . . . 14 ([𝑧 / 𝑥]𝜑𝜃)
66 dfsbcq 2864 . . . . . . . . . . . . . 14 (𝑧 = ((𝑘 + 1) / 𝑦) → ([𝑧 / 𝑥]𝜑[((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
6765, 66syl5bbr 193 . . . . . . . . . . . . 13 (𝑧 = ((𝑘 + 1) / 𝑦) → (𝜃[((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
6864cbvralv 2612 . . . . . . . . . . . . . 14 (∀𝑥 ∈ (1...𝑘)𝜑 ↔ ∀𝑧 ∈ (1...𝑘)𝜃)
6958, 68sylib 121 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ∀𝑧 ∈ (1...𝑘)𝜃)
7017nnrpd 9329 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℝ+)
7122nnrpd 9329 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ+)
7270, 71rpdivcld 9348 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℝ+)
7372rpgt0d 9333 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < ((𝑘 + 1) / 𝑦))
74 elnnz 8916 . . . . . . . . . . . . . . 15 (((𝑘 + 1) / 𝑦) ∈ ℕ ↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 0 < ((𝑘 + 1) / 𝑦)))
7532, 73, 74sylanbrc 411 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℕ)
7617nnap0d 8624 . . . . . . . . . . . . . . . . . 18 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) # 0)
7718, 76dividapd 8407 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) = 1)
78 eluz2gt1 9246 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (ℤ‘2) → 1 < 𝑦)
7920, 78syl 14 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 < 𝑦)
8077, 79eqbrtrd 3895 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) < 𝑦)
8117nngt0d 8622 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < (𝑘 + 1))
82 ltdiv23 8508 . . . . . . . . . . . . . . . . 17 (((𝑘 + 1) ∈ ℝ ∧ ((𝑘 + 1) ∈ ℝ ∧ 0 < (𝑘 + 1)) ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) → (((𝑘 + 1) / (𝑘 + 1)) < 𝑦 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1)))
8349, 49, 81, 50, 51, 82syl122anc 1193 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (((𝑘 + 1) / (𝑘 + 1)) < 𝑦 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1)))
8480, 83mpbid 146 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) < (𝑘 + 1))
85 zleltp1 8961 . . . . . . . . . . . . . . . 16 ((((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (((𝑘 + 1) / 𝑦) ≤ 𝑘 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1)))
8632, 43, 85syl2anc 406 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (((𝑘 + 1) / 𝑦) ≤ 𝑘 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1)))
8784, 86mpbird 166 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ≤ 𝑘)
88 fznn 9710 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℤ → (((𝑘 + 1) / 𝑦) ∈ (1...𝑘) ↔ (((𝑘 + 1) / 𝑦) ∈ ℕ ∧ ((𝑘 + 1) / 𝑦) ≤ 𝑘)))
8943, 88syl 14 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (((𝑘 + 1) / 𝑦) ∈ (1...𝑘) ↔ (((𝑘 + 1) / 𝑦) ∈ ℕ ∧ ((𝑘 + 1) / 𝑦) ≤ 𝑘)))
9075, 87, 89mpbir2and 896 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ (1...𝑘))
9167, 69, 90rspcdva 2749 . . . . . . . . . . . 12 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)
9262, 91jca 302 . . . . . . . . . . 11 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝜒[((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
9367anbi2d 455 . . . . . . . . . . . . . 14 (𝑧 = ((𝑘 + 1) / 𝑦) → ((𝜒𝜃) ↔ (𝜒[((𝑘 + 1) / 𝑦) / 𝑥]𝜑)))
94 oveq2 5714 . . . . . . . . . . . . . . 15 (𝑧 = ((𝑘 + 1) / 𝑦) → (𝑦 · 𝑧) = (𝑦 · ((𝑘 + 1) / 𝑦)))
9594sbceq1d 2867 . . . . . . . . . . . . . 14 (𝑧 = ((𝑘 + 1) / 𝑦) → ([(𝑦 · 𝑧) / 𝑥]𝜑[(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑))
9693, 95imbi12d 233 . . . . . . . . . . . . 13 (𝑧 = ((𝑘 + 1) / 𝑦) → (((𝜒𝜃) → [(𝑦 · 𝑧) / 𝑥]𝜑) ↔ ((𝜒[((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)))
9796imbi2d 229 . . . . . . . . . . . 12 (𝑧 = ((𝑘 + 1) / 𝑦) → ((𝑦 ∈ (ℤ‘2) → ((𝜒𝜃) → [(𝑦 · 𝑧) / 𝑥]𝜑)) ↔ (𝑦 ∈ (ℤ‘2) → ((𝜒[((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑))))
98 prmind2.8 . . . . . . . . . . . . . . 15 ((𝑦 ∈ (ℤ‘2) ∧ 𝑧 ∈ (ℤ‘2)) → ((𝜒𝜃) → 𝜏))
9998ancoms 266 . . . . . . . . . . . . . 14 ((𝑧 ∈ (ℤ‘2) ∧ 𝑦 ∈ (ℤ‘2)) → ((𝜒𝜃) → 𝜏))
100 eluzelz 9185 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ℤ‘2) → 𝑦 ∈ ℤ)
101100adantl 273 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ (ℤ‘2) ∧ 𝑦 ∈ (ℤ‘2)) → 𝑦 ∈ ℤ)
102 eluzelz 9185 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ (ℤ‘2) → 𝑧 ∈ ℤ)
103102adantr 272 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ (ℤ‘2) ∧ 𝑦 ∈ (ℤ‘2)) → 𝑧 ∈ ℤ)
104101, 103zmulcld 9031 . . . . . . . . . . . . . . 15 ((𝑧 ∈ (ℤ‘2) ∧ 𝑦 ∈ (ℤ‘2)) → (𝑦 · 𝑧) ∈ ℤ)
105 prmind.4 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 · 𝑧) → (𝜑𝜏))
106105sbcieg 2893 . . . . . . . . . . . . . . 15 ((𝑦 · 𝑧) ∈ ℤ → ([(𝑦 · 𝑧) / 𝑥]𝜑𝜏))
107104, 106syl 14 . . . . . . . . . . . . . 14 ((𝑧 ∈ (ℤ‘2) ∧ 𝑦 ∈ (ℤ‘2)) → ([(𝑦 · 𝑧) / 𝑥]𝜑𝜏))
10899, 107sylibrd 168 . . . . . . . . . . . . 13 ((𝑧 ∈ (ℤ‘2) ∧ 𝑦 ∈ (ℤ‘2)) → ((𝜒𝜃) → [(𝑦 · 𝑧) / 𝑥]𝜑))
109108ex 114 . . . . . . . . . . . 12 (𝑧 ∈ (ℤ‘2) → (𝑦 ∈ (ℤ‘2) → ((𝜒𝜃) → [(𝑦 · 𝑧) / 𝑥]𝜑)))
11097, 109vtoclga 2707 . . . . . . . . . . 11 (((𝑘 + 1) / 𝑦) ∈ (ℤ‘2) → (𝑦 ∈ (ℤ‘2) → ((𝜒[((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)))
11156, 20, 92, 110syl3c 63 . . . . . . . . . 10 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)
11225, 111sbceq1dd 2868 . . . . . . . . 9 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [(𝑘 + 1) / 𝑥]𝜑)
113112rexlimdvaa 2509 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑))
114 ralnex 2385 . . . . . . . . 9 (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ ¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1))
115 simpl 108 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℕ)
116 elnnuz 9212 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ‘1))
117115, 116sylib 121 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ (ℤ‘1))
118 eluzp1p1 9201 . . . . . . . . . . . . 13 (𝑘 ∈ (ℤ‘1) → (𝑘 + 1) ∈ (ℤ‘(1 + 1)))
119117, 118syl 14 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ‘(1 + 1)))
120 df-2 8637 . . . . . . . . . . . . 13 2 = (1 + 1)
121120fveq2i 5356 . . . . . . . . . . . 12 (ℤ‘2) = (ℤ‘(1 + 1))
122119, 121syl6eleqr 2193 . . . . . . . . . . 11 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ‘2))
123 isprm3 11592 . . . . . . . . . . . 12 ((𝑘 + 1) ∈ ℙ ↔ ((𝑘 + 1) ∈ (ℤ‘2) ∧ ∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1)))
124123baibr 873 . . . . . . . . . . 11 ((𝑘 + 1) ∈ (ℤ‘2) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ (𝑘 + 1) ∈ ℙ))
125122, 124syl 14 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ (𝑘 + 1) ∈ ℙ))
126 simpr 109 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑥 ∈ (1...𝑘)𝜑)
12757cbvralv 2612 . . . . . . . . . . . . 13 (∀𝑥 ∈ (1...𝑘)𝜑 ↔ ∀𝑦 ∈ (1...𝑘)𝜒)
128126, 127sylib 121 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑦 ∈ (1...𝑘)𝜒)
129115nncnd 8592 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℂ)
130129, 38, 39sylancl 407 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘)
131130oveq2d 5722 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (1...((𝑘 + 1) − 1)) = (1...𝑘))
132131raleqdv 2590 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 ↔ ∀𝑦 ∈ (1...𝑘)𝜒))
133128, 132mpbird 166 . . . . . . . . . . 11 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒)
134 nfcv 2240 . . . . . . . . . . . 12 𝑥(𝑘 + 1)
135 nfv 1476 . . . . . . . . . . . . 13 𝑥𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒
136 nfsbc1v 2880 . . . . . . . . . . . . 13 𝑥[(𝑘 + 1) / 𝑥]𝜑
137135, 136nfim 1519 . . . . . . . . . . . 12 𝑥(∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒[(𝑘 + 1) / 𝑥]𝜑)
138 oveq1 5713 . . . . . . . . . . . . . . 15 (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1))
139138oveq2d 5722 . . . . . . . . . . . . . 14 (𝑥 = (𝑘 + 1) → (1...(𝑥 − 1)) = (1...((𝑘 + 1) − 1)))
140139raleqdv 2590 . . . . . . . . . . . . 13 (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (1...(𝑥 − 1))𝜒 ↔ ∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒))
141 sbceq1a 2871 . . . . . . . . . . . . 13 (𝑥 = (𝑘 + 1) → (𝜑[(𝑘 + 1) / 𝑥]𝜑))
142140, 141imbi12d 233 . . . . . . . . . . . 12 (𝑥 = (𝑘 + 1) → ((∀𝑦 ∈ (1...(𝑥 − 1))𝜒𝜑) ↔ (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒[(𝑘 + 1) / 𝑥]𝜑)))
143 prmind2.7 . . . . . . . . . . . . 13 ((𝑥 ∈ ℙ ∧ ∀𝑦 ∈ (1...(𝑥 − 1))𝜒) → 𝜑)
144143ex 114 . . . . . . . . . . . 12 (𝑥 ∈ ℙ → (∀𝑦 ∈ (1...(𝑥 − 1))𝜒𝜑))
145134, 137, 142, 144vtoclgaf 2706 . . . . . . . . . . 11 ((𝑘 + 1) ∈ ℙ → (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒[(𝑘 + 1) / 𝑥]𝜑))
146133, 145syl5com 29 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) ∈ ℙ → [(𝑘 + 1) / 𝑥]𝜑))
147125, 146sylbid 149 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑))
148114, 147syl5bir 152 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑))
149 2z 8934 . . . . . . . . . . 11 2 ∈ ℤ
150149a1i 9 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 2 ∈ ℤ)
151115nnzd 9024 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℤ)
152151peano2zd 9028 . . . . . . . . . . 11 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈ ℤ)
153 1zzd 8933 . . . . . . . . . . 11 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 1 ∈ ℤ)
154152, 153zsubcld 9030 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) − 1) ∈ ℤ)
15519, 21syl 14 . . . . . . . . . . 11 (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ∈ ℕ)
156 dvdsdc 11296 . . . . . . . . . . 11 ((𝑦 ∈ ℕ ∧ (𝑘 + 1) ∈ ℤ) → DECID 𝑦 ∥ (𝑘 + 1))
157155, 152, 156syl2anr 286 . . . . . . . . . 10 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ 𝑦 ∈ (2...((𝑘 + 1) − 1))) → DECID 𝑦 ∥ (𝑘 + 1))
158150, 154, 157exfzdc 9858 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → DECID𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1))
159 exmiddc 788 . . . . . . . . 9 (DECID𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → (∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) ∨ ¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1)))
160158, 159syl 14 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) ∨ ¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1)))
161113, 148, 160mpjaod 679 . . . . . . 7 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → [(𝑘 + 1) / 𝑥]𝜑)
162161ex 114 . . . . . 6 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑[(𝑘 + 1) / 𝑥]𝜑))
163 ralsnsg 3508 . . . . . . 7 ((𝑘 + 1) ∈ ℕ → (∀𝑥 ∈ {(𝑘 + 1)}𝜑[(𝑘 + 1) / 𝑥]𝜑))
16416, 163syl 14 . . . . . 6 (𝑘 ∈ ℕ → (∀𝑥 ∈ {(𝑘 + 1)}𝜑[(𝑘 + 1) / 𝑥]𝜑))
165162, 164sylibrd 168 . . . . 5 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → ∀𝑥 ∈ {(𝑘 + 1)}𝜑))
166165ancld 321 . . . 4 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑)))
167 fzsuc 9690 . . . . . . 7 (𝑘 ∈ (ℤ‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
168116, 167sylbi 120 . . . . . 6 (𝑘 ∈ ℕ → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
169168raleqdv 2590 . . . . 5 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})𝜑))
170 ralunb 3204 . . . . 5 (∀𝑥 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})𝜑 ↔ (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑))
171169, 170syl6bb 195 . . . 4 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...(𝑘 + 1))𝜑 ↔ (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑)))
172166, 171sylibrd 168 . . 3 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → ∀𝑥 ∈ (1...(𝑘 + 1))𝜑))
1733, 5, 7, 9, 15, 172nnind 8594 . 2 (𝐴 ∈ ℕ → ∀𝑥 ∈ (1...𝐴)𝜑)
174 elfz1end 9676 . . 3 (𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴))
175174biimpi 119 . 2 (𝐴 ∈ ℕ → 𝐴 ∈ (1...𝐴))
1761, 173, 175rspcdva 2749 1 (𝐴 ∈ ℕ → 𝜂)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 670  DECID wdc 786   = wceq 1299  wcel 1448  wne 2267  wral 2375  wrex 2376  [wsbc 2862  cun 3019  {csn 3474   class class class wbr 3875  cfv 5059  (class class class)co 5706  cc 7498  cr 7499  0cc0 7500  1c1 7501   + caddc 7503   · cmul 7505   < clt 7672  cle 7673  cmin 7804   / cdiv 8293  cn 8578  2c2 8629  cz 8906  cuz 9176  ...cfz 9631  cdvds 11288  cprime 11581
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614  ax-caucvg 7615
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-ilim 4229  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-frec 6218  df-1o 6243  df-2o 6244  df-er 6359  df-en 6565  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-2 8637  df-3 8638  df-4 8639  df-n0 8830  df-z 8907  df-uz 9177  df-q 9262  df-rp 9292  df-fz 9632  df-fzo 9761  df-fl 9884  df-mod 9937  df-seqfrec 10060  df-exp 10134  df-cj 10455  df-re 10456  df-im 10457  df-rsqrt 10610  df-abs 10611  df-dvds 11289  df-prm 11582
This theorem is referenced by:  prmind  11595
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