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Theorem prmind2 12151
Description: A variation on prmind 12152 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 5903 . . . 4 (𝑛 = 1 → (1...𝑛) = (1...1))
32raleqdv 2692 . . 3 (𝑛 = 1 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...1)𝜑))
4 oveq2 5903 . . . 4 (𝑛 = 𝑘 → (1...𝑛) = (1...𝑘))
54raleqdv 2692 . . 3 (𝑛 = 𝑘 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝑘)𝜑))
6 oveq2 5903 . . . 4 (𝑛 = (𝑘 + 1) → (1...𝑛) = (1...(𝑘 + 1)))
76raleqdv 2692 . . 3 (𝑛 = (𝑘 + 1) → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...(𝑘 + 1))𝜑))
8 oveq2 5903 . . . 4 (𝑛 = 𝐴 → (1...𝑛) = (1...𝐴))
98raleqdv 2692 . . 3 (𝑛 = 𝐴 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝐴)𝜑))
10 prmind.6 . . . . 5 𝜓
11 elfz1eq 10064 . . . . . 6 (𝑥 ∈ (1...1) → 𝑥 = 1)
12 prmind.1 . . . . . 6 (𝑥 = 1 → (𝜑𝜓))
1311, 12syl 14 . . . . 5 (𝑥 ∈ (1...1) → (𝜑𝜓))
1410, 13mpbiri 168 . . . 4 (𝑥 ∈ (1...1) → 𝜑)
1514rgen 2543 . . 3 𝑥 ∈ (1...1)𝜑
16 peano2nn 8960 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
1716ad2antrr 488 . . . . . . . . . . . 12 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℕ)
1817nncnd 8962 . . . . . . . . . . 11 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℂ)
19 elfzuz 10050 . . . . . . . . . . . . . 14 (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ∈ (ℤ‘2))
2019ad2antrl 490 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ (ℤ‘2))
21 eluz2nn 9595 . . . . . . . . . . . . 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 8962 . . . . . . . . . . 11 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℂ)
2422nnap0d 8994 . . . . . . . . . . 11 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 # 0)
2518, 23, 24divcanap2d 8778 . . . . . . . . . 10 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 · ((𝑘 + 1) / 𝑦)) = (𝑘 + 1))
26 simprr 531 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∥ (𝑘 + 1))
2722nnzd 9403 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℤ)
2822nnne0d 8993 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≠ 0)
2917nnzd 9403 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℤ)
30 dvdsval2 11828 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℤ ∧ 𝑦 ≠ 0 ∧ (𝑘 + 1) ∈ ℤ) → (𝑦 ∥ (𝑘 + 1) ↔ ((𝑘 + 1) / 𝑦) ∈ ℤ))
3127, 28, 29, 30syl3anc 1249 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ∥ (𝑘 + 1) ↔ ((𝑘 + 1) / 𝑦) ∈ ℤ))
3226, 31mpbid 147 . . . . . . . . . . . 12 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℤ)
3323mulid2d 8005 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) = 𝑦)
34 elfzle2 10057 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ≤ ((𝑘 + 1) − 1))
3534ad2antrl 490 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≤ ((𝑘 + 1) − 1))
36 nncn 8956 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
3736ad2antrr 488 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℂ)
38 ax-1cn 7933 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
39 pncan 8192 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
4037, 38, 39sylancl 413 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) − 1) = 𝑘)
4135, 40breqtrd 4044 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦𝑘)
42 nnz 9301 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
4342ad2antrr 488 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℤ)
44 zleltp1 9337 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑦𝑘𝑦 < (𝑘 + 1)))
4527, 43, 44syl2anc 411 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦𝑘𝑦 < (𝑘 + 1)))
4641, 45mpbid 147 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 < (𝑘 + 1))
4733, 46eqbrtrd 4040 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) < (𝑘 + 1))
48 1red 8001 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 ∈ ℝ)
4917nnred 8961 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℝ)
5022nnred 8961 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ)
5122nngt0d 8992 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < 𝑦)
52 ltmuldiv 8860 . . . . . . . . . . . . . 14 ((1 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) → ((1 · 𝑦) < (𝑘 + 1) ↔ 1 < ((𝑘 + 1) / 𝑦)))
5348, 49, 50, 51, 52syl112anc 1253 . . . . . . . . . . . . 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 147 . . . . . . . . . . . 12 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 < ((𝑘 + 1) / 𝑦))
55 eluz2b1 9630 . . . . . . . . . . . 12 (((𝑘 + 1) / 𝑦) ∈ (ℤ‘2) ↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 1 < ((𝑘 + 1) / 𝑦)))
5632, 54, 55sylanbrc 417 . . . . . . . . . . 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 528 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ∀𝑥 ∈ (1...𝑘)𝜑)
59 fznn 10118 . . . . . . . . . . . . . . 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 946 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ (1...𝑘))
6257, 58, 61rspcdva 2861 . . . . . . . . . . . 12 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝜒)
63 vex 2755 . . . . . . . . . . . . . . 15 𝑧 ∈ V
64 prmind.3 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝜑𝜃))
6563, 64sbcie 3012 . . . . . . . . . . . . . 14 ([𝑧 / 𝑥]𝜑𝜃)
66 dfsbcq 2979 . . . . . . . . . . . . . 14 (𝑧 = ((𝑘 + 1) / 𝑦) → ([𝑧 / 𝑥]𝜑[((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
6765, 66bitr3id 194 . . . . . . . . . . . . 13 (𝑧 = ((𝑘 + 1) / 𝑦) → (𝜃[((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
6864cbvralv 2718 . . . . . . . . . . . . . 14 (∀𝑥 ∈ (1...𝑘)𝜑 ↔ ∀𝑧 ∈ (1...𝑘)𝜃)
6958, 68sylib 122 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ∀𝑧 ∈ (1...𝑘)𝜃)
7017nnrpd 9723 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℝ+)
7122nnrpd 9723 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ+)
7270, 71rpdivcld 9743 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℝ+)
7372rpgt0d 9728 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < ((𝑘 + 1) / 𝑦))
74 elnnz 9292 . . . . . . . . . . . . . . 15 (((𝑘 + 1) / 𝑦) ∈ ℕ ↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 0 < ((𝑘 + 1) / 𝑦)))
7532, 73, 74sylanbrc 417 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℕ)
7617nnap0d 8994 . . . . . . . . . . . . . . . . . 18 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) # 0)
7718, 76dividapd 8772 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) = 1)
78 eluz2gt1 9631 . . . . . . . . . . . . . . . . . 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 4040 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) < 𝑦)
8117nngt0d 8992 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < (𝑘 + 1))
82 ltdiv23 8878 . . . . . . . . . . . . . . . . 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 1258 . . . . . . . . . . . . . . . 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 147 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) < (𝑘 + 1))
85 zleltp1 9337 . . . . . . . . . . . . . . . 16 ((((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (((𝑘 + 1) / 𝑦) ≤ 𝑘 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1)))
8632, 43, 85syl2anc 411 . . . . . . . . . . . . . . 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 167 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ≤ 𝑘)
88 fznn 10118 . . . . . . . . . . . . . . 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 946 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ (1...𝑘))
9167, 69, 90rspcdva 2861 . . . . . . . . . . . 12 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)
9262, 91jca 306 . . . . . . . . . . 11 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝜒[((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
9367anbi2d 464 . . . . . . . . . . . . . 14 (𝑧 = ((𝑘 + 1) / 𝑦) → ((𝜒𝜃) ↔ (𝜒[((𝑘 + 1) / 𝑦) / 𝑥]𝜑)))
94 oveq2 5903 . . . . . . . . . . . . . . 15 (𝑧 = ((𝑘 + 1) / 𝑦) → (𝑦 · 𝑧) = (𝑦 · ((𝑘 + 1) / 𝑦)))
9594sbceq1d 2982 . . . . . . . . . . . . . 14 (𝑧 = ((𝑘 + 1) / 𝑦) → ([(𝑦 · 𝑧) / 𝑥]𝜑[(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑))
9693, 95imbi12d 234 . . . . . . . . . . . . 13 (𝑧 = ((𝑘 + 1) / 𝑦) → (((𝜒𝜃) → [(𝑦 · 𝑧) / 𝑥]𝜑) ↔ ((𝜒[((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)))
9796imbi2d 230 . . . . . . . . . . . 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 268 . . . . . . . . . . . . . 14 ((𝑧 ∈ (ℤ‘2) ∧ 𝑦 ∈ (ℤ‘2)) → ((𝜒𝜃) → 𝜏))
100 eluzelz 9566 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (ℤ‘2) → 𝑦 ∈ ℤ)
101100adantl 277 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ (ℤ‘2) ∧ 𝑦 ∈ (ℤ‘2)) → 𝑦 ∈ ℤ)
102 eluzelz 9566 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ (ℤ‘2) → 𝑧 ∈ ℤ)
103102adantr 276 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ (ℤ‘2) ∧ 𝑦 ∈ (ℤ‘2)) → 𝑧 ∈ ℤ)
104101, 103zmulcld 9410 . . . . . . . . . . . . . . 15 ((𝑧 ∈ (ℤ‘2) ∧ 𝑦 ∈ (ℤ‘2)) → (𝑦 · 𝑧) ∈ ℤ)
105 prmind.4 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 · 𝑧) → (𝜑𝜏))
106105sbcieg 3010 . . . . . . . . . . . . . . 15 ((𝑦 · 𝑧) ∈ ℤ → ([(𝑦 · 𝑧) / 𝑥]𝜑𝜏))
107104, 106syl 14 . . . . . . . . . . . . . 14 ((𝑧 ∈ (ℤ‘2) ∧ 𝑦 ∈ (ℤ‘2)) → ([(𝑦 · 𝑧) / 𝑥]𝜑𝜏))
10899, 107sylibrd 169 . . . . . . . . . . . . 13 ((𝑧 ∈ (ℤ‘2) ∧ 𝑦 ∈ (ℤ‘2)) → ((𝜒𝜃) → [(𝑦 · 𝑧) / 𝑥]𝜑))
109108ex 115 . . . . . . . . . . . 12 (𝑧 ∈ (ℤ‘2) → (𝑦 ∈ (ℤ‘2) → ((𝜒𝜃) → [(𝑦 · 𝑧) / 𝑥]𝜑)))
11097, 109vtoclga 2818 . . . . . . . . . . 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 2983 . . . . . . . . 9 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [(𝑘 + 1) / 𝑥]𝜑)
113112rexlimdvaa 2608 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑))
114 ralnex 2478 . . . . . . . . 9 (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ ¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1))
115 simpl 109 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℕ)
116 elnnuz 9593 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ‘1))
117115, 116sylib 122 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ (ℤ‘1))
118 eluzp1p1 9582 . . . . . . . . . . . . 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 9007 . . . . . . . . . . . . 13 2 = (1 + 1)
121120fveq2i 5537 . . . . . . . . . . . 12 (ℤ‘2) = (ℤ‘(1 + 1))
122119, 121eleqtrrdi 2283 . . . . . . . . . . 11 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ‘2))
123 isprm3 12149 . . . . . . . . . . . 12 ((𝑘 + 1) ∈ ℙ ↔ ((𝑘 + 1) ∈ (ℤ‘2) ∧ ∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1)))
124123baibr 921 . . . . . . . . . . 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 110 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑥 ∈ (1...𝑘)𝜑)
12757cbvralv 2718 . . . . . . . . . . . . 13 (∀𝑥 ∈ (1...𝑘)𝜑 ↔ ∀𝑦 ∈ (1...𝑘)𝜒)
128126, 127sylib 122 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑦 ∈ (1...𝑘)𝜒)
129115nncnd 8962 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℂ)
130129, 38, 39sylancl 413 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘)
131130oveq2d 5911 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (1...((𝑘 + 1) − 1)) = (1...𝑘))
132131raleqdv 2692 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 ↔ ∀𝑦 ∈ (1...𝑘)𝜒))
133128, 132mpbird 167 . . . . . . . . . . 11 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒)
134 nfcv 2332 . . . . . . . . . . . 12 𝑥(𝑘 + 1)
135 nfv 1539 . . . . . . . . . . . . 13 𝑥𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒
136 nfsbc1v 2996 . . . . . . . . . . . . 13 𝑥[(𝑘 + 1) / 𝑥]𝜑
137135, 136nfim 1583 . . . . . . . . . . . 12 𝑥(∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒[(𝑘 + 1) / 𝑥]𝜑)
138 oveq1 5902 . . . . . . . . . . . . . . 15 (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1))
139138oveq2d 5911 . . . . . . . . . . . . . 14 (𝑥 = (𝑘 + 1) → (1...(𝑥 − 1)) = (1...((𝑘 + 1) − 1)))
140139raleqdv 2692 . . . . . . . . . . . . 13 (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (1...(𝑥 − 1))𝜒 ↔ ∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒))
141 sbceq1a 2987 . . . . . . . . . . . . 13 (𝑥 = (𝑘 + 1) → (𝜑[(𝑘 + 1) / 𝑥]𝜑))
142140, 141imbi12d 234 . . . . . . . . . . . 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 115 . . . . . . . . . . . 12 (𝑥 ∈ ℙ → (∀𝑦 ∈ (1...(𝑥 − 1))𝜒𝜑))
145134, 137, 142, 144vtoclgaf 2817 . . . . . . . . . . 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 150 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑))
148114, 147biimtrrid 153 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑))
149 2z 9310 . . . . . . . . . . 11 2 ∈ ℤ
150149a1i 9 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 2 ∈ ℤ)
151115nnzd 9403 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℤ)
152151peano2zd 9407 . . . . . . . . . . 11 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈ ℤ)
153 1zzd 9309 . . . . . . . . . . 11 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 1 ∈ ℤ)
154152, 153zsubcld 9409 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) − 1) ∈ ℤ)
15519, 21syl 14 . . . . . . . . . . 11 (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ∈ ℕ)
156 dvdsdc 11836 . . . . . . . . . . 11 ((𝑦 ∈ ℕ ∧ (𝑘 + 1) ∈ ℤ) → DECID 𝑦 ∥ (𝑘 + 1))
157155, 152, 156syl2anr 290 . . . . . . . . . 10 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ 𝑦 ∈ (2...((𝑘 + 1) − 1))) → DECID 𝑦 ∥ (𝑘 + 1))
158150, 154, 157exfzdc 10269 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → DECID𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1))
159 exmiddc 837 . . . . . . . . 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 719 . . . . . . 7 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → [(𝑘 + 1) / 𝑥]𝜑)
162161ex 115 . . . . . 6 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑[(𝑘 + 1) / 𝑥]𝜑))
163 ralsnsg 3644 . . . . . . 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 169 . . . . 5 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → ∀𝑥 ∈ {(𝑘 + 1)}𝜑))
166165ancld 325 . . . 4 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑)))
167 fzsuc 10098 . . . . . . 7 (𝑘 ∈ (ℤ‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
168116, 167sylbi 121 . . . . . 6 (𝑘 ∈ ℕ → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
169168raleqdv 2692 . . . . 5 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})𝜑))
170 ralunb 3331 . . . . 5 (∀𝑥 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})𝜑 ↔ (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑))
171169, 170bitrdi 196 . . . 4 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...(𝑘 + 1))𝜑 ↔ (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑)))
172166, 171sylibrd 169 . . 3 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → ∀𝑥 ∈ (1...(𝑘 + 1))𝜑))
1733, 5, 7, 9, 15, 172nnind 8964 . 2 (𝐴 ∈ ℕ → ∀𝑥 ∈ (1...𝐴)𝜑)
174 elfz1end 10084 . . 3 (𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴))
175174biimpi 120 . 2 (𝐴 ∈ ℕ → 𝐴 ∈ (1...𝐴))
1761, 173, 175rspcdva 2861 1 (𝐴 ∈ ℕ → 𝜂)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709  DECID wdc 835   = wceq 1364  wcel 2160  wne 2360  wral 2468  wrex 2469  [wsbc 2977  cun 3142  {csn 3607   class class class wbr 4018  cfv 5235  (class class class)co 5895  cc 7838  cr 7839  0cc0 7840  1c1 7841   + caddc 7843   · cmul 7845   < clt 8021  cle 8022  cmin 8157   / cdiv 8658  cn 8948  2c2 8999  cz 9282  cuz 9557  ...cfz 10037  cdvds 11825  cprime 12138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958  ax-arch 7959  ax-caucvg 7960
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-frec 6415  df-1o 6440  df-2o 6441  df-er 6558  df-en 6766  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568  df-div 8659  df-inn 8949  df-2 9007  df-3 9008  df-4 9009  df-n0 9206  df-z 9283  df-uz 9558  df-q 9649  df-rp 9683  df-fz 10038  df-fzo 10172  df-fl 10300  df-mod 10353  df-seqfrec 10476  df-exp 10550  df-cj 10882  df-re 10883  df-im 10884  df-rsqrt 11038  df-abs 11039  df-dvds 11826  df-prm 12139
This theorem is referenced by:  prmind  12152  4sqlem19  12440
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