ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  pceu GIF version

Theorem pceu 12436
Description: Uniqueness for the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)
Hypotheses
Ref Expression
pcval.1 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < )
pcval.2 𝑇 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )
Assertion
Ref Expression
pceu ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∃!𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
Distinct variable groups:   𝑥,𝑛,𝑦,𝑧,𝑁   𝑃,𝑛,𝑥,𝑦,𝑧   𝑧,𝑆   𝑧,𝑇
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑛)   𝑇(𝑥,𝑦,𝑛)

Proof of Theorem pceu
Dummy variables 𝑠 𝑡 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 529 . . . 4 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℚ)
2 elq 9690 . . . 4 (𝑁 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦))
31, 2sylib 122 . . 3 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦))
4 simpr 110 . . . . . . . . . . 11 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑁 = (𝑥 / 𝑦))
5 simprr 531 . . . . . . . . . . . . 13 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → 𝑁 ≠ 0)
65ad3antrrr 492 . . . . . . . . . . . 12 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑁 ≠ 0)
71ad3antrrr 492 . . . . . . . . . . . . 13 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑁 ∈ ℚ)
8 0z 9331 . . . . . . . . . . . . . 14 0 ∈ ℤ
9 zq 9694 . . . . . . . . . . . . . 14 (0 ∈ ℤ → 0 ∈ ℚ)
108, 9mp1i 10 . . . . . . . . . . . . 13 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 0 ∈ ℚ)
11 qapne 9707 . . . . . . . . . . . . 13 ((𝑁 ∈ ℚ ∧ 0 ∈ ℚ) → (𝑁 # 0 ↔ 𝑁 ≠ 0))
127, 10, 11syl2anc 411 . . . . . . . . . . . 12 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → (𝑁 # 0 ↔ 𝑁 ≠ 0))
136, 12mpbird 167 . . . . . . . . . . 11 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑁 # 0)
144, 13eqbrtrrd 4054 . . . . . . . . . 10 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → (𝑥 / 𝑦) # 0)
15 simpllr 534 . . . . . . . . . . . 12 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑥 ∈ ℤ)
1615zcnd 9443 . . . . . . . . . . 11 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑥 ∈ ℂ)
17 nnz 9339 . . . . . . . . . . . . . 14 (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
1817adantl 277 . . . . . . . . . . . . 13 ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℤ)
1918adantr 276 . . . . . . . . . . . 12 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑦 ∈ ℤ)
2019zcnd 9443 . . . . . . . . . . 11 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑦 ∈ ℂ)
21 simplr 528 . . . . . . . . . . . 12 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑦 ∈ ℕ)
2221nnap0d 9030 . . . . . . . . . . 11 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑦 # 0)
2316, 20, 22divap0bd 8823 . . . . . . . . . 10 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → (𝑥 # 0 ↔ (𝑥 / 𝑦) # 0))
2414, 23mpbird 167 . . . . . . . . 9 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑥 # 0)
25 0zd 9332 . . . . . . . . . 10 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 0 ∈ ℤ)
26 zapne 9394 . . . . . . . . . 10 ((𝑥 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑥 # 0 ↔ 𝑥 ≠ 0))
2715, 25, 26syl2anc 411 . . . . . . . . 9 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → (𝑥 # 0 ↔ 𝑥 ≠ 0))
2824, 27mpbid 147 . . . . . . . 8 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑥 ≠ 0)
2928ex 115 . . . . . . 7 ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → (𝑁 = (𝑥 / 𝑦) → 𝑥 ≠ 0))
3029adantrd 279 . . . . . . . 8 ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) → 𝑥 ≠ 0))
3130exlimdv 1830 . . . . . . 7 ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → (∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) → 𝑥 ≠ 0))
32 prmuz2 12272 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ‘2))
3332ad3antrrr 492 . . . . . . . . . . . . . 14 ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → 𝑃 ∈ (ℤ‘2))
3433adantr 276 . . . . . . . . . . . . 13 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≠ 0) → 𝑃 ∈ (ℤ‘2))
35 simpllr 534 . . . . . . . . . . . . 13 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≠ 0) → 𝑥 ∈ ℤ)
36 simpr 110 . . . . . . . . . . . . 13 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≠ 0) → 𝑥 ≠ 0)
37 eqid 2193 . . . . . . . . . . . . . . 15 {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}
38 pcval.1 . . . . . . . . . . . . . . 15 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < )
3937, 38pcprecl 12430 . . . . . . . . . . . . . 14 ((𝑃 ∈ (ℤ‘2) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃𝑆) ∥ 𝑥))
4039simpld 112 . . . . . . . . . . . . 13 ((𝑃 ∈ (ℤ‘2) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → 𝑆 ∈ ℕ0)
4134, 35, 36, 40syl12anc 1247 . . . . . . . . . . . 12 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≠ 0) → 𝑆 ∈ ℕ0)
4241nn0zd 9440 . . . . . . . . . . 11 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≠ 0) → 𝑆 ∈ ℤ)
43 nnne0 9012 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ℕ → 𝑦 ≠ 0)
4443adantl 277 . . . . . . . . . . . . . . 15 ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → 𝑦 ≠ 0)
45 eqid 2193 . . . . . . . . . . . . . . . 16 {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}
46 pcval.2 . . . . . . . . . . . . . . . 16 𝑇 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )
4745, 46pcprecl 12430 . . . . . . . . . . . . . . 15 ((𝑃 ∈ (ℤ‘2) ∧ (𝑦 ∈ ℤ ∧ 𝑦 ≠ 0)) → (𝑇 ∈ ℕ0 ∧ (𝑃𝑇) ∥ 𝑦))
4833, 18, 44, 47syl12anc 1247 . . . . . . . . . . . . . 14 ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → (𝑇 ∈ ℕ0 ∧ (𝑃𝑇) ∥ 𝑦))
4948simpld 112 . . . . . . . . . . . . 13 ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → 𝑇 ∈ ℕ0)
5049adantr 276 . . . . . . . . . . . 12 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≠ 0) → 𝑇 ∈ ℕ0)
5150nn0zd 9440 . . . . . . . . . . 11 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≠ 0) → 𝑇 ∈ ℤ)
5242, 51zsubcld 9447 . . . . . . . . . 10 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≠ 0) → (𝑆𝑇) ∈ ℤ)
53 biidd 172 . . . . . . . . . . 11 (𝑧 = (𝑆𝑇) → (𝑁 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑥 / 𝑦)))
5453ceqsexgv 2890 . . . . . . . . . 10 ((𝑆𝑇) ∈ ℤ → (∃𝑧(𝑧 = (𝑆𝑇) ∧ 𝑁 = (𝑥 / 𝑦)) ↔ 𝑁 = (𝑥 / 𝑦)))
5552, 54syl 14 . . . . . . . . 9 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≠ 0) → (∃𝑧(𝑧 = (𝑆𝑇) ∧ 𝑁 = (𝑥 / 𝑦)) ↔ 𝑁 = (𝑥 / 𝑦)))
56 exancom 1619 . . . . . . . . 9 (∃𝑧(𝑧 = (𝑆𝑇) ∧ 𝑁 = (𝑥 / 𝑦)) ↔ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
5755, 56bitr3di 195 . . . . . . . 8 (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≠ 0) → (𝑁 = (𝑥 / 𝑦) ↔ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))))
5857ex 115 . . . . . . 7 ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → (𝑥 ≠ 0 → (𝑁 = (𝑥 / 𝑦) ↔ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))))
5929, 31, 58pm5.21ndd 706 . . . . . 6 ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → (𝑁 = (𝑥 / 𝑦) ↔ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))))
6059rexbidva 2491 . . . . 5 (((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) → (∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑦 ∈ ℕ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))))
6160rexbidva 2491 . . . 4 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))))
62 rexcom4 2783 . . . . . 6 (∃𝑦 ∈ ℕ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ↔ ∃𝑧𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
6362rexbii 2501 . . . . 5 (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ↔ ∃𝑥 ∈ ℤ ∃𝑧𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
64 rexcom4 2783 . . . . 5 (∃𝑥 ∈ ℤ ∃𝑧𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ↔ ∃𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
6563, 64bitri 184 . . . 4 (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ↔ ∃𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
6661, 65bitrdi 196 . . 3 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))))
673, 66mpbid 147 . 2 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∃𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
68 eqid 2193 . . . . . . . . . . 11 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < )
69 eqid 2193 . . . . . . . . . . 11 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )
70 simp11l 1110 . . . . . . . . . . 11 ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑃 ∈ ℙ)
71 simp11r 1111 . . . . . . . . . . 11 ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑁 ≠ 0)
72 simp12 1030 . . . . . . . . . . 11 ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))) → (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ))
73 simp13l 1114 . . . . . . . . . . 11 ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑁 = (𝑥 / 𝑦))
74 simp2 1000 . . . . . . . . . . 11 ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))) → (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ))
75 simp3l 1027 . . . . . . . . . . 11 ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑁 = (𝑠 / 𝑡))
7638, 46, 68, 69, 70, 71, 72, 73, 74, 75pceulem 12435 . . . . . . . . . 10 ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))) → (𝑆𝑇) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))
77 simp13r 1115 . . . . . . . . . 10 ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑧 = (𝑆𝑇))
78 simp3r 1028 . . . . . . . . . 10 ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))
7976, 77, 783eqtr4d 2236 . . . . . . . . 9 ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑧 = 𝑤)
80793exp 1204 . . . . . . . 8 (((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))) → ((𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) → ((𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ))) → 𝑧 = 𝑤)))
8180rexlimdvv 2618 . . . . . . 7 (((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇))) → (∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ))) → 𝑧 = 𝑤))
82813exp 1204 . . . . . 6 ((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) → ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) → (∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ))) → 𝑧 = 𝑤))))
8382adantrl 478 . . . . 5 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) → (∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ))) → 𝑧 = 𝑤))))
8483rexlimdvv 2618 . . . 4 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) → (∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ))) → 𝑧 = 𝑤)))
8584impd 254 . . 3 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ((∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ∧ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑧 = 𝑤))
8685alrimivv 1886 . 2 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∀𝑧𝑤((∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ∧ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑧 = 𝑤))
87 eqeq1 2200 . . . . . 6 (𝑧 = 𝑤 → (𝑧 = (𝑆𝑇) ↔ 𝑤 = (𝑆𝑇)))
8887anbi2d 464 . . . . 5 (𝑧 = 𝑤 → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ↔ (𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆𝑇))))
89882rexbidv 2519 . . . 4 (𝑧 = 𝑤 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆𝑇))))
90 oveq1 5926 . . . . . . . . 9 (𝑥 = 𝑠 → (𝑥 / 𝑦) = (𝑠 / 𝑦))
9190eqeq2d 2205 . . . . . . . 8 (𝑥 = 𝑠 → (𝑁 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑠 / 𝑦)))
92 breq2 4034 . . . . . . . . . . . . 13 (𝑥 = 𝑠 → ((𝑃𝑛) ∥ 𝑥 ↔ (𝑃𝑛) ∥ 𝑠))
9392rabbidv 2749 . . . . . . . . . . . 12 (𝑥 = 𝑠 → {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠})
9493supeq1d 7048 . . . . . . . . . . 11 (𝑥 = 𝑠 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ))
9538, 94eqtrid 2238 . . . . . . . . . 10 (𝑥 = 𝑠𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ))
9695oveq1d 5934 . . . . . . . . 9 (𝑥 = 𝑠 → (𝑆𝑇) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇))
9796eqeq2d 2205 . . . . . . . 8 (𝑥 = 𝑠 → (𝑤 = (𝑆𝑇) ↔ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇)))
9891, 97anbi12d 473 . . . . . . 7 (𝑥 = 𝑠 → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆𝑇)) ↔ (𝑁 = (𝑠 / 𝑦) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇))))
9998rexbidv 2495 . . . . . 6 (𝑥 = 𝑠 → (∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆𝑇)) ↔ ∃𝑦 ∈ ℕ (𝑁 = (𝑠 / 𝑦) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇))))
100 oveq2 5927 . . . . . . . . 9 (𝑦 = 𝑡 → (𝑠 / 𝑦) = (𝑠 / 𝑡))
101100eqeq2d 2205 . . . . . . . 8 (𝑦 = 𝑡 → (𝑁 = (𝑠 / 𝑦) ↔ 𝑁 = (𝑠 / 𝑡)))
102 breq2 4034 . . . . . . . . . . . . 13 (𝑦 = 𝑡 → ((𝑃𝑛) ∥ 𝑦 ↔ (𝑃𝑛) ∥ 𝑡))
103102rabbidv 2749 . . . . . . . . . . . 12 (𝑦 = 𝑡 → {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡})
104103supeq1d 7048 . . . . . . . . . . 11 (𝑦 = 𝑡 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ))
10546, 104eqtrid 2238 . . . . . . . . . 10 (𝑦 = 𝑡𝑇 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ))
106105oveq2d 5935 . . . . . . . . 9 (𝑦 = 𝑡 → (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))
107106eqeq2d 2205 . . . . . . . 8 (𝑦 = 𝑡 → (𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇) ↔ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ))))
108101, 107anbi12d 473 . . . . . . 7 (𝑦 = 𝑡 → ((𝑁 = (𝑠 / 𝑦) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇)) ↔ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))))
109108cbvrexvw 2731 . . . . . 6 (∃𝑦 ∈ ℕ (𝑁 = (𝑠 / 𝑦) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇)) ↔ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ))))
11099, 109bitrdi 196 . . . . 5 (𝑥 = 𝑠 → (∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆𝑇)) ↔ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))))
111110cbvrexvw 2731 . . . 4 (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆𝑇)) ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ))))
11289, 111bitrdi 196 . . 3 (𝑧 = 𝑤 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))))
113112eu4 2104 . 2 (∃!𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ↔ (∃𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ∧ ∀𝑧𝑤((∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ∧ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑧 = 𝑤)))
11467, 86, 113sylanbrc 417 1 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∃!𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980  wal 1362   = wceq 1364  wex 1503  ∃!weu 2042  wcel 2164  wne 2364  wrex 2473  {crab 2476   class class class wbr 4030  cfv 5255  (class class class)co 5919  supcsup 7043  cr 7873  0cc0 7874   < clt 8056  cmin 8192   # cap 8602   / cdiv 8693  cn 8984  2c2 9035  0cn0 9243  cz 9320  cuz 9595  cq 9687  cexp 10612  cdvds 11933  cprime 12248
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 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-1o 6471  df-2o 6472  df-er 6589  df-en 6797  df-sup 7045  df-inf 7046  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-fz 10078  df-fzo 10212  df-fl 10342  df-mod 10397  df-seqfrec 10522  df-exp 10613  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-dvds 11934  df-gcd 12083  df-prm 12249
This theorem is referenced by:  pcval  12437  pczpre  12438  pcdiv  12443
  Copyright terms: Public domain W3C validator