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Theorem pceu 15764
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 778 . . . 4 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℚ)
2 elq 12005 . . . 4 (𝑁 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦))
31, 2sylib 209 . . 3 ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦))
4 ovex 6902 . . . . . . . . 9 (𝑆𝑇) ∈ V
5 biidd 253 . . . . . . . . 9 (𝑧 = (𝑆𝑇) → (𝑁 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑥 / 𝑦)))
64, 5ceqsexv 3436 . . . . . . . 8 (∃𝑧(𝑧 = (𝑆𝑇) ∧ 𝑁 = (𝑥 / 𝑦)) ↔ 𝑁 = (𝑥 / 𝑦))
7 exancom 1947 . . . . . . . 8 (∃𝑧(𝑧 = (𝑆𝑇) ∧ 𝑁 = (𝑥 / 𝑦)) ↔ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
86, 7bitr3i 268 . . . . . . 7 (𝑁 = (𝑥 / 𝑦) ↔ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
98rexbii 3229 . . . . . 6 (∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑦 ∈ ℕ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
10 rexcom4 3419 . . . . . 6 (∃𝑦 ∈ ℕ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ↔ ∃𝑧𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
119, 10bitri 266 . . . . 5 (∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑧𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
1211rexbii 3229 . . . 4 (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑥 ∈ ℤ ∃𝑧𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
13 rexcom4 3419 . . . 4 (∃𝑥 ∈ ℤ ∃𝑧𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ↔ ∃𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
1412, 13bitri 266 . . 3 (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑧𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)))
153, 14sylib 209 . 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 2806 . . . . . . . . . . 11 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < )
19 eqid 2806 . . . . . . . . . . 11 sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )
20 simp11l 1376 . . . . . . . . . . 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 1377 . . . . . . . . . . 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 1254 . . . . . . . . . . 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 1380 . . . . . . . . . . 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 1160 . . . . . . . . . . 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 1251 . . . . . . . . . . 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 15763 . . . . . . . . . 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 1381 . . . . . . . . . 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 1252 . . . . . . . . . 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 2850 . . . . . . . . 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 1141 . . . . . . . 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 3225 . . . . . . 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 1141 . . . . . 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 698 . . . . 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 3225 . . . 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 398 . . 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 2019 . 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 2810 . . . . . 6 (𝑧 = 𝑤 → (𝑧 = (𝑆𝑇) ↔ 𝑤 = (𝑆𝑇)))
3837anbi2d 616 . . . . 5 (𝑧 = 𝑤 → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ↔ (𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆𝑇))))
39382rexbidv 3245 . . . 4 (𝑧 = 𝑤 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆𝑇)) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆𝑇))))
40 oveq1 6877 . . . . . . . . 9 (𝑥 = 𝑠 → (𝑥 / 𝑦) = (𝑠 / 𝑦))
4140eqeq2d 2816 . . . . . . . 8 (𝑥 = 𝑠 → (𝑁 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑠 / 𝑦)))
42 breq2 4848 . . . . . . . . . . . . 13 (𝑥 = 𝑠 → ((𝑃𝑛) ∥ 𝑥 ↔ (𝑃𝑛) ∥ 𝑠))
4342rabbidv 3379 . . . . . . . . . . . 12 (𝑥 = 𝑠 → {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠})
4443supeq1d 8587 . . . . . . . . . . 11 (𝑥 = 𝑠 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ))
4516, 44syl5eq 2852 . . . . . . . . . 10 (𝑥 = 𝑠𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ))
4645oveq1d 6885 . . . . . . . . 9 (𝑥 = 𝑠 → (𝑆𝑇) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇))
4746eqeq2d 2816 . . . . . . . 8 (𝑥 = 𝑠 → (𝑤 = (𝑆𝑇) ↔ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇)))
4841, 47anbi12d 618 . . . . . . 7 (𝑥 = 𝑠 → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆𝑇)) ↔ (𝑁 = (𝑠 / 𝑦) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇))))
4948rexbidv 3240 . . . . . 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 6878 . . . . . . . . 9 (𝑦 = 𝑡 → (𝑠 / 𝑦) = (𝑠 / 𝑡))
5150eqeq2d 2816 . . . . . . . 8 (𝑦 = 𝑡 → (𝑁 = (𝑠 / 𝑦) ↔ 𝑁 = (𝑠 / 𝑡)))
52 breq2 4848 . . . . . . . . . . . . 13 (𝑦 = 𝑡 → ((𝑃𝑛) ∥ 𝑦 ↔ (𝑃𝑛) ∥ 𝑡))
5352rabbidv 3379 . . . . . . . . . . . 12 (𝑦 = 𝑡 → {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡})
5453supeq1d 8587 . . . . . . . . . . 11 (𝑦 = 𝑡 → sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ))
5517, 54syl5eq 2852 . . . . . . . . . 10 (𝑦 = 𝑡𝑇 = sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < ))
5655oveq2d 6886 . . . . . . . . 9 (𝑦 = 𝑡 → (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃𝑛) ∥ 𝑡}, ℝ, < )))
5756eqeq2d 2816 . . . . . . . 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 618 . . . . . . 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, < )))))
5958cbvrexv 3361 . . . . . 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 278 . . . . 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, < )))))
6160cbvrexv 3361 . . . 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 278 . . 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 2681 . 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 574 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 384  w3a 1100  wal 1635   = wceq 1637  wex 1859  wcel 2156  ∃!weu 2630  wne 2978  wrex 3097  {crab 3100   class class class wbr 4844  (class class class)co 6870  supcsup 8581  cr 10216  0cc0 10217   < clt 10355  cmin 10547   / cdiv 10965  cn 11301  0cn0 11555  cz 11639  cq 12003  cexp 13079  cdvds 15199  cprime 15599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175  ax-cnex 10273  ax-resscn 10274  ax-1cn 10275  ax-icn 10276  ax-addcl 10277  ax-addrcl 10278  ax-mulcl 10279  ax-mulrcl 10280  ax-mulcom 10281  ax-addass 10282  ax-mulass 10283  ax-distr 10284  ax-i2m1 10285  ax-1ne0 10286  ax-1rid 10287  ax-rnegex 10288  ax-rrecex 10289  ax-cnre 10290  ax-pre-lttri 10291  ax-pre-lttrn 10292  ax-pre-ltadd 10293  ax-pre-mulgt0 10294  ax-pre-sup 10295
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-riota 6831  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-om 7292  df-1st 7394  df-2nd 7395  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-1o 7792  df-2o 7793  df-er 7975  df-en 8189  df-dom 8190  df-sdom 8191  df-fin 8192  df-sup 8583  df-inf 8584  df-pnf 10357  df-mnf 10358  df-xr 10359  df-ltxr 10360  df-le 10361  df-sub 10549  df-neg 10550  df-div 10966  df-nn 11302  df-2 11360  df-3 11361  df-n0 11556  df-z 11640  df-uz 11901  df-q 12004  df-rp 12043  df-fl 12813  df-mod 12889  df-seq 13021  df-exp 13080  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-dvds 15200  df-gcd 15432  df-prm 15600
This theorem is referenced by:  pczpre  15765  pcdiv  15770
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