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Theorem pceu 16175
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 767 . . . 4 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℚ)
2 elq 12342 . . . 4 (𝑁 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦))
31, 2sylib 219 . . 3 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦))
4 ovex 7184 . . . . . . . . 9 (𝑆𝑇) ∈ V
5 biidd 263 . . . . . . . . 9 (𝑧 = (𝑆𝑇) → (𝑁 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑥 / 𝑦)))
64, 5ceqsexv 3546 . . . . . . . 8 (∃𝑧(𝑧 = (𝑆𝑇) ∧ 𝑁 = (𝑥 / 𝑦)) ↔ 𝑁 = (𝑥 / 𝑦))
7 exancom 1854 . . . . . . . 8 (∃𝑧(𝑧 = (𝑆𝑇) ∧ 𝑁 = (𝑥 / 𝑦)) ↔ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
86, 7bitr3i 278 . . . . . . 7 (𝑁 = (𝑥 / 𝑦) ↔ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
98rexbii 3251 . . . . . 6 (∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑦 ∈ ℕ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
10 rexcom4 3253 . . . . . 6 (∃𝑦 ∈ ℕ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ↔ ∃𝑧𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
119, 10bitri 276 . . . . 5 (∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑧𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
1211rexbii 3251 . . . 4 (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑥 ∈ ℤ ∃𝑧𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
13 rexcom4 3253 . . . 4 (∃𝑥 ∈ ℤ ∃𝑧𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ↔ ∃𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
1412, 13bitri 276 . . 3 (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
153, 14sylib 219 . 2 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∃𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
16 pcval.1 . . . . . . . . . . 11 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < )
17 pcval.2 . . . . . . . . . . 11 𝑇 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < )
18 eqid 2824 . . . . . . . . . . 11 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < )
19 eqid 2824 . . . . . . . . . . 11 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )
20 simp11l 1278 . . . . . . . . . . 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)
21 simp11r 1279 . . . . . . . . . . 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)
22 simp12 1198 . . . . . . . . . . 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))
23 simp13l 1282 . . . . . . . . . . 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))
24 simp2 1131 . . . . . . . . . . 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))
25 simp3l 1195 . . . . . . . . . . 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))
2616, 17, 18, 19, 20, 21, 22, 23, 24, 25pceulem 16174 . . . . . . . . . 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, < )))
27 simp13r 1283 . . . . . . . . . 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))
28 simp3r 1196 . . . . . . . . . 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, < )))
2926, 27, 283eqtr4d 2870 . . . . . . . . 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)
30293exp 1113 . . . . . . . 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)))
3130rexlimdvv 3297 . . . . . . 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))
32313exp 1113 . . . . . 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))))
3332adantrl 712 . . . . 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))))
3433rexlimdvv 3297 . . . 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)))
3534impd 411 . . 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))
3635alrimivv 1922 . 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))
37 eqeq1 2828 . . . . . 6 (𝑧 = 𝑤 → (𝑧 = (𝑆𝑇) ↔ 𝑤 = (𝑆𝑇)))
3837anbi2d 628 . . . . 5 (𝑧 = 𝑤 → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ↔ (𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆𝑇))))
39382rexbidv 3304 . . . 4 (𝑧 = 𝑤 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆𝑇))))
40 oveq1 7158 . . . . . . . . 9 (𝑥 = 𝑠 → (𝑥 / 𝑦) = (𝑠 / 𝑦))
4140eqeq2d 2835 . . . . . . . 8 (𝑥 = 𝑠 → (𝑁 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑠 / 𝑦)))
42 breq2 5066 . . . . . . . . . . . . 13 (𝑥 = 𝑠 → ((𝑃𝑛) ∥ 𝑥 ↔ (𝑃𝑛) ∥ 𝑠))
4342rabbidv 3485 . . . . . . . . . . . 12 (𝑥 = 𝑠 → {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠})
4443supeq1d 8902 . . . . . . . . . . 11 (𝑥 = 𝑠 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ))
4516, 44syl5eq 2872 . . . . . . . . . 10 (𝑥 = 𝑠𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ))
4645oveq1d 7166 . . . . . . . . 9 (𝑥 = 𝑠 → (𝑆𝑇) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇))
4746eqeq2d 2835 . . . . . . . 8 (𝑥 = 𝑠 → (𝑤 = (𝑆𝑇) ↔ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇)))
4841, 47anbi12d 630 . . . . . . 7 (𝑥 = 𝑠 → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆𝑇)) ↔ (𝑁 = (𝑠 / 𝑦) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇))))
4948rexbidv 3301 . . . . . 6 (𝑥 = 𝑠 → (∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆𝑇)) ↔ ∃𝑦 ∈ ℕ (𝑁 = (𝑠 / 𝑦) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇))))
50 oveq2 7159 . . . . . . . . 9 (𝑦 = 𝑡 → (𝑠 / 𝑦) = (𝑠 / 𝑡))
5150eqeq2d 2835 . . . . . . . 8 (𝑦 = 𝑡 → (𝑁 = (𝑠 / 𝑦) ↔ 𝑁 = (𝑠 / 𝑡)))
52 breq2 5066 . . . . . . . . . . . . 13 (𝑦 = 𝑡 → ((𝑃𝑛) ∥ 𝑦 ↔ (𝑃𝑛) ∥ 𝑡))
5352rabbidv 3485 . . . . . . . . . . . 12 (𝑦 = 𝑡 → {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡})
5453supeq1d 8902 . . . . . . . . . . 11 (𝑦 = 𝑡 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ))
5517, 54syl5eq 2872 . . . . . . . . . 10 (𝑦 = 𝑡𝑇 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ))
5655oveq2d 7167 . . . . . . . . 9 (𝑦 = 𝑡 → (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))
5756eqeq2d 2835 . . . . . . . 8 (𝑦 = 𝑡 → (𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇) ↔ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ))))
5851, 57anbi12d 630 . . . . . . 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, < )))))
5958cbvrexvw 3455 . . . . . 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, < ))))
6049, 59syl6bb 288 . . . . 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, < )))))
6160cbvrexvw 3455 . . . 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, < ))))
6239, 61syl6bb 288 . . 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, < )))))
6362eu4 2696 . 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)))
6415, 36, 63sylanbrc 583 1 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∃!𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081  wal 1528   = wceq 1530  wex 1773  wcel 2106  ∃!weu 2648  wne 3020  wrex 3143  {crab 3146   class class class wbr 5062  (class class class)co 7151  supcsup 8896  cr 10528  0cc0 10529   < clt 10667  cmin 10862   / cdiv 11289  cn 11630  0cn0 11889  cz 11973  cq 12340  cexp 13422  cdvds 15599  cprime 16007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-inf 8899  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12383  df-fl 13155  df-mod 13231  df-seq 13363  df-exp 13423  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-abs 14588  df-dvds 15600  df-gcd 15836  df-prm 16008
This theorem is referenced by:  pczpre  16176  pcdiv  16181
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