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Theorem prmind2 15621
Description: A variation on prmind 15622 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 6823 . . . 4 (𝑛 = 1 → (1...𝑛) = (1...1))
32raleqdv 3284 . . 3 (𝑛 = 1 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...1)𝜑))
4 oveq2 6823 . . . 4 (𝑛 = 𝑘 → (1...𝑛) = (1...𝑘))
54raleqdv 3284 . . 3 (𝑛 = 𝑘 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝑘)𝜑))
6 oveq2 6823 . . . 4 (𝑛 = (𝑘 + 1) → (1...𝑛) = (1...(𝑘 + 1)))
76raleqdv 3284 . . 3 (𝑛 = (𝑘 + 1) → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...(𝑘 + 1))𝜑))
8 oveq2 6823 . . . 4 (𝑛 = 𝐴 → (1...𝑛) = (1...𝐴))
98raleqdv 3284 . . 3 (𝑛 = 𝐴 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝐴)𝜑))
10 prmind.6 . . . . 5 𝜓
11 elfz1eq 12566 . . . . . 6 (𝑥 ∈ (1...1) → 𝑥 = 1)
12 prmind.1 . . . . . 6 (𝑥 = 1 → (𝜑𝜓))
1311, 12syl 17 . . . . 5 (𝑥 ∈ (1...1) → (𝜑𝜓))
1410, 13mpbiri 248 . . . 4 (𝑥 ∈ (1...1) → 𝜑)
1514rgen 3061 . . 3 𝑥 ∈ (1...1)𝜑
16 peano2nn 11245 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
1716ad2antrr 764 . . . . . . . . . . . 12 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℕ)
1817nncnd 11249 . . . . . . . . . . 11 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℂ)
19 elfzuz 12552 . . . . . . . . . . . . . 14 (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ∈ (ℤ‘2))
2019ad2antrl 766 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ (ℤ‘2))
21 eluz2nn 11940 . . . . . . . . . . . . 13 (𝑦 ∈ (ℤ‘2) → 𝑦 ∈ ℕ)
2220, 21syl 17 . . . . . . . . . . . 12 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℕ)
2322nncnd 11249 . . . . . . . . . . 11 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℂ)
2422nnne0d 11278 . . . . . . . . . . 11 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≠ 0)
2518, 23, 24divcan2d 11016 . . . . . . . . . 10 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 · ((𝑘 + 1) / 𝑦)) = (𝑘 + 1))
26 simprr 813 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∥ (𝑘 + 1))
2722nnzd 11694 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℤ)
2817nnzd 11694 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℤ)
29 dvdsval2 15206 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℤ ∧ 𝑦 ≠ 0 ∧ (𝑘 + 1) ∈ ℤ) → (𝑦 ∥ (𝑘 + 1) ↔ ((𝑘 + 1) / 𝑦) ∈ ℤ))
3027, 24, 28, 29syl3anc 1477 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ∥ (𝑘 + 1) ↔ ((𝑘 + 1) / 𝑦) ∈ ℤ))
3126, 30mpbid 222 . . . . . . . . . . . 12 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℤ)
3223mulid2d 10271 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) = 𝑦)
33 elfzle2 12559 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ≤ ((𝑘 + 1) − 1))
3433ad2antrl 766 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≤ ((𝑘 + 1) − 1))
35 nncn 11241 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
3635ad2antrr 764 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℂ)
37 ax-1cn 10207 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
38 pncan 10500 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 1) = 𝑘)
3936, 37, 38sylancl 697 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) − 1) = 𝑘)
4034, 39breqtrd 4831 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦𝑘)
41 nnz 11612 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
4241ad2antrr 764 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℤ)
43 zleltp1 11641 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑦𝑘𝑦 < (𝑘 + 1)))
4427, 42, 43syl2anc 696 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦𝑘𝑦 < (𝑘 + 1)))
4540, 44mpbid 222 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 < (𝑘 + 1))
4632, 45eqbrtrd 4827 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) < (𝑘 + 1))
47 1red 10268 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 ∈ ℝ)
4817nnred 11248 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℝ)
4922nnred 11248 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ)
5022nngt0d 11277 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < 𝑦)
51 ltmuldiv 11109 . . . . . . . . . . . . . 14 ((1 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) → ((1 · 𝑦) < (𝑘 + 1) ↔ 1 < ((𝑘 + 1) / 𝑦)))
5247, 48, 49, 50, 51syl112anc 1481 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((1 · 𝑦) < (𝑘 + 1) ↔ 1 < ((𝑘 + 1) / 𝑦)))
5346, 52mpbid 222 . . . . . . . . . . . 12 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 < ((𝑘 + 1) / 𝑦))
54 eluz2b1 11973 . . . . . . . . . . . 12 (((𝑘 + 1) / 𝑦) ∈ (ℤ‘2) ↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 1 < ((𝑘 + 1) / 𝑦)))
5531, 53, 54sylanbrc 701 . . . . . . . . . . 11 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ (ℤ‘2))
56 prmind.2 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝜑𝜒))
57 simplr 809 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ∀𝑥 ∈ (1...𝑘)𝜑)
58 fznn 12622 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℤ → (𝑦 ∈ (1...𝑘) ↔ (𝑦 ∈ ℕ ∧ 𝑦𝑘)))
5942, 58syl 17 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ∈ (1...𝑘) ↔ (𝑦 ∈ ℕ ∧ 𝑦𝑘)))
6022, 40, 59mpbir2and 995 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ (1...𝑘))
6156, 57, 60rspcdva 3456 . . . . . . . . . . . 12 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝜒)
62 vex 3344 . . . . . . . . . . . . . . 15 𝑧 ∈ V
63 prmind.3 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝜑𝜃))
6462, 63sbcie 3612 . . . . . . . . . . . . . 14 ([𝑧 / 𝑥]𝜑𝜃)
65 dfsbcq 3579 . . . . . . . . . . . . . 14 (𝑧 = ((𝑘 + 1) / 𝑦) → ([𝑧 / 𝑥]𝜑[((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
6664, 65syl5bbr 274 . . . . . . . . . . . . 13 (𝑧 = ((𝑘 + 1) / 𝑦) → (𝜃[((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
6763cbvralv 3311 . . . . . . . . . . . . . 14 (∀𝑥 ∈ (1...𝑘)𝜑 ↔ ∀𝑧 ∈ (1...𝑘)𝜃)
6857, 67sylib 208 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ∀𝑧 ∈ (1...𝑘)𝜃)
6917nnrpd 12084 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℝ+)
7022nnrpd 12084 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ+)
7169, 70rpdivcld 12103 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℝ+)
7271rpgt0d 12089 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < ((𝑘 + 1) / 𝑦))
73 elnnz 11600 . . . . . . . . . . . . . . 15 (((𝑘 + 1) / 𝑦) ∈ ℕ ↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 0 < ((𝑘 + 1) / 𝑦)))
7431, 72, 73sylanbrc 701 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℕ)
7517nnne0d 11278 . . . . . . . . . . . . . . . . . 18 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ≠ 0)
7618, 75dividd 11012 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) = 1)
77 eluz2b2 11975 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (ℤ‘2) ↔ (𝑦 ∈ ℕ ∧ 1 < 𝑦))
7877simprbi 483 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (ℤ‘2) → 1 < 𝑦)
7920, 78syl 17 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 < 𝑦)
8076, 79eqbrtrd 4827 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) < 𝑦)
8117nngt0d 11277 . . . . . . . . . . . . . . . . 17 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < (𝑘 + 1))
82 ltdiv23 11127 . . . . . . . . . . . . . . . . 17 (((𝑘 + 1) ∈ ℝ ∧ ((𝑘 + 1) ∈ ℝ ∧ 0 < (𝑘 + 1)) ∧ (𝑦 ∈ ℝ ∧ 0 < 𝑦)) → (((𝑘 + 1) / (𝑘 + 1)) < 𝑦 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1)))
8348, 48, 81, 49, 50, 82syl122anc 1486 . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) < (𝑘 + 1))
85 zleltp1 11641 . . . . . . . . . . . . . . . 16 ((((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (((𝑘 + 1) / 𝑦) ≤ 𝑘 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1)))
8631, 42, 85syl2anc 696 . . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . . 14 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ≤ 𝑘)
88 fznn 12622 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℤ → (((𝑘 + 1) / 𝑦) ∈ (1...𝑘) ↔ (((𝑘 + 1) / 𝑦) ∈ ℕ ∧ ((𝑘 + 1) / 𝑦) ≤ 𝑘)))
8942, 88syl 17 . . . . . . . . . . . . . 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)))
9074, 87, 89mpbir2and 995 . . . . . . . . . . . . 13 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ (1...𝑘))
9166, 68, 90rspcdva 3456 . . . . . . . . . . . 12 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)
9261, 91jca 555 . . . . . . . . . . 11 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝜒[((𝑘 + 1) / 𝑦) / 𝑥]𝜑))
9366anbi2d 742 . . . . . . . . . . . . . 14 (𝑧 = ((𝑘 + 1) / 𝑦) → ((𝜒𝜃) ↔ (𝜒[((𝑘 + 1) / 𝑦) / 𝑥]𝜑)))
94 ovex 6843 . . . . . . . . . . . . . . . 16 (𝑦 · 𝑧) ∈ V
95 prmind.4 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 · 𝑧) → (𝜑𝜏))
9694, 95sbcie 3612 . . . . . . . . . . . . . . 15 ([(𝑦 · 𝑧) / 𝑥]𝜑𝜏)
97 oveq2 6823 . . . . . . . . . . . . . . . 16 (𝑧 = ((𝑘 + 1) / 𝑦) → (𝑦 · 𝑧) = (𝑦 · ((𝑘 + 1) / 𝑦)))
9897sbceq1d 3582 . . . . . . . . . . . . . . 15 (𝑧 = ((𝑘 + 1) / 𝑦) → ([(𝑦 · 𝑧) / 𝑥]𝜑[(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑))
9996, 98syl5bbr 274 . . . . . . . . . . . . . 14 (𝑧 = ((𝑘 + 1) / 𝑦) → (𝜏[(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑))
10093, 99imbi12d 333 . . . . . . . . . . . . 13 (𝑧 = ((𝑘 + 1) / 𝑦) → (((𝜒𝜃) → 𝜏) ↔ ((𝜒[((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)))
101100imbi2d 329 . . . . . . . . . . . 12 (𝑧 = ((𝑘 + 1) / 𝑦) → ((𝑦 ∈ (ℤ‘2) → ((𝜒𝜃) → 𝜏)) ↔ (𝑦 ∈ (ℤ‘2) → ((𝜒[((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑))))
102 prmind2.8 . . . . . . . . . . . . 13 ((𝑦 ∈ (ℤ‘2) ∧ 𝑧 ∈ (ℤ‘2)) → ((𝜒𝜃) → 𝜏))
103102expcom 450 . . . . . . . . . . . 12 (𝑧 ∈ (ℤ‘2) → (𝑦 ∈ (ℤ‘2) → ((𝜒𝜃) → 𝜏)))
104101, 103vtoclga 3413 . . . . . . . . . . 11 (((𝑘 + 1) / 𝑦) ∈ (ℤ‘2) → (𝑦 ∈ (ℤ‘2) → ((𝜒[((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)))
10555, 20, 92, 104syl3c 66 . . . . . . . . . 10 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)
10625, 105sbceq1dd 3583 . . . . . . . . 9 (((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [(𝑘 + 1) / 𝑥]𝜑)
107106rexlimdvaa 3171 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑))
108 ralnex 3131 . . . . . . . . 9 (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ ¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1))
109 simpl 474 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℕ)
110 elnnuz 11938 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ‘1))
111109, 110sylib 208 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ (ℤ‘1))
112 eluzp1p1 11926 . . . . . . . . . . . . 13 (𝑘 ∈ (ℤ‘1) → (𝑘 + 1) ∈ (ℤ‘(1 + 1)))
113111, 112syl 17 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ‘(1 + 1)))
114 df-2 11292 . . . . . . . . . . . . 13 2 = (1 + 1)
115114fveq2i 6357 . . . . . . . . . . . 12 (ℤ‘2) = (ℤ‘(1 + 1))
116113, 115syl6eleqr 2851 . . . . . . . . . . 11 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ‘2))
117 isprm3 15619 . . . . . . . . . . . 12 ((𝑘 + 1) ∈ ℙ ↔ ((𝑘 + 1) ∈ (ℤ‘2) ∧ ∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1)))
118117baibr 983 . . . . . . . . . . 11 ((𝑘 + 1) ∈ (ℤ‘2) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ (𝑘 + 1) ∈ ℙ))
119116, 118syl 17 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ (𝑘 + 1) ∈ ℙ))
120 simpr 479 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑥 ∈ (1...𝑘)𝜑)
12156cbvralv 3311 . . . . . . . . . . . . 13 (∀𝑥 ∈ (1...𝑘)𝜑 ↔ ∀𝑦 ∈ (1...𝑘)𝜒)
122120, 121sylib 208 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑦 ∈ (1...𝑘)𝜒)
123109nncnd 11249 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℂ)
124123, 37, 38sylancl 697 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘)
125124oveq2d 6831 . . . . . . . . . . . . 13 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (1...((𝑘 + 1) − 1)) = (1...𝑘))
126125raleqdv 3284 . . . . . . . . . . . 12 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 ↔ ∀𝑦 ∈ (1...𝑘)𝜒))
127122, 126mpbird 247 . . . . . . . . . . 11 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒)
128 nfcv 2903 . . . . . . . . . . . 12 𝑥(𝑘 + 1)
129 nfv 1993 . . . . . . . . . . . . 13 𝑥𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒
130 nfsbc1v 3597 . . . . . . . . . . . . 13 𝑥[(𝑘 + 1) / 𝑥]𝜑
131129, 130nfim 1975 . . . . . . . . . . . 12 𝑥(∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒[(𝑘 + 1) / 𝑥]𝜑)
132 oveq1 6822 . . . . . . . . . . . . . . 15 (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1))
133132oveq2d 6831 . . . . . . . . . . . . . 14 (𝑥 = (𝑘 + 1) → (1...(𝑥 − 1)) = (1...((𝑘 + 1) − 1)))
134133raleqdv 3284 . . . . . . . . . . . . 13 (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (1...(𝑥 − 1))𝜒 ↔ ∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒))
135 sbceq1a 3588 . . . . . . . . . . . . 13 (𝑥 = (𝑘 + 1) → (𝜑[(𝑘 + 1) / 𝑥]𝜑))
136134, 135imbi12d 333 . . . . . . . . . . . 12 (𝑥 = (𝑘 + 1) → ((∀𝑦 ∈ (1...(𝑥 − 1))𝜒𝜑) ↔ (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒[(𝑘 + 1) / 𝑥]𝜑)))
137 prmind2.7 . . . . . . . . . . . . 13 ((𝑥 ∈ ℙ ∧ ∀𝑦 ∈ (1...(𝑥 − 1))𝜒) → 𝜑)
138137ex 449 . . . . . . . . . . . 12 (𝑥 ∈ ℙ → (∀𝑦 ∈ (1...(𝑥 − 1))𝜒𝜑))
139128, 131, 136, 138vtoclgaf 3412 . . . . . . . . . . 11 ((𝑘 + 1) ∈ ℙ → (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒[(𝑘 + 1) / 𝑥]𝜑))
140127, 139syl5com 31 . . . . . . . . . 10 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) ∈ ℙ → [(𝑘 + 1) / 𝑥]𝜑))
141119, 140sylbid 230 . . . . . . . . 9 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑))
142108, 141syl5bir 233 . . . . . . . 8 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → (¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑))
143107, 142pm2.61d 170 . . . . . . 7 ((𝑘 ∈ ℕ ∧ ∀𝑥 ∈ (1...𝑘)𝜑) → [(𝑘 + 1) / 𝑥]𝜑)
144143ex 449 . . . . . 6 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑[(𝑘 + 1) / 𝑥]𝜑))
145 ralsnsg 4361 . . . . . . 7 ((𝑘 + 1) ∈ ℕ → (∀𝑥 ∈ {(𝑘 + 1)}𝜑[(𝑘 + 1) / 𝑥]𝜑))
14616, 145syl 17 . . . . . 6 (𝑘 ∈ ℕ → (∀𝑥 ∈ {(𝑘 + 1)}𝜑[(𝑘 + 1) / 𝑥]𝜑))
147144, 146sylibrd 249 . . . . 5 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → ∀𝑥 ∈ {(𝑘 + 1)}𝜑))
148147ancld 577 . . . 4 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑)))
149 fzsuc 12602 . . . . . . 7 (𝑘 ∈ (ℤ‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
150110, 149sylbi 207 . . . . . 6 (𝑘 ∈ ℕ → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
151150raleqdv 3284 . . . . 5 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})𝜑))
152 ralunb 3938 . . . . 5 (∀𝑥 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})𝜑 ↔ (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑))
153151, 152syl6bb 276 . . . 4 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...(𝑘 + 1))𝜑 ↔ (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑)))
154148, 153sylibrd 249 . . 3 (𝑘 ∈ ℕ → (∀𝑥 ∈ (1...𝑘)𝜑 → ∀𝑥 ∈ (1...(𝑘 + 1))𝜑))
1553, 5, 7, 9, 15, 154nnind 11251 . 2 (𝐴 ∈ ℕ → ∀𝑥 ∈ (1...𝐴)𝜑)
156 elfz1end 12585 . . 3 (𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴))
157156biimpi 206 . 2 (𝐴 ∈ ℕ → 𝐴 ∈ (1...𝐴))
1581, 155, 157rspcdva 3456 1 (𝐴 ∈ ℕ → 𝜂)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1632  wcel 2140  wne 2933  wral 3051  wrex 3052  [wsbc 3577  cun 3714  {csn 4322   class class class wbr 4805  cfv 6050  (class class class)co 6815  cc 10147  cr 10148  0cc0 10149  1c1 10150   + caddc 10152   · cmul 10154   < clt 10287  cle 10288  cmin 10479   / cdiv 10897  cn 11233  2c2 11283  cz 11590  cuz 11900  ...cfz 12540  cdvds 15203  cprime 15608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226  ax-pre-sup 10227
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-1o 7731  df-2o 7732  df-er 7914  df-en 8125  df-dom 8126  df-sdom 8127  df-fin 8128  df-sup 8516  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-div 10898  df-nn 11234  df-2 11292  df-3 11293  df-n0 11506  df-z 11591  df-uz 11901  df-rp 12047  df-fz 12541  df-seq 13017  df-exp 13076  df-cj 14059  df-re 14060  df-im 14061  df-sqrt 14195  df-abs 14196  df-dvds 15204  df-prm 15609
This theorem is referenced by:  prmind  15622  4sqlem19  15890
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