Theorem pceu 16879

Description: Uniqueness for the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Hypotheses

Ref	Expression
pcval.1	⊢ 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < )
pcval.2	⊢ 𝑇 = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )

Assertion

Ref	Expression
pceu	⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∃!𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))

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.

Step	Hyp	Ref	Expression
1		simprl 771	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℚ)
2		elq 12989	. . . 4 ⊢ (𝑁 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦))
3	1, 2	sylib 218	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦))
4		ovex 7463	. . . . . . . . 9 ⊢ (𝑆 − 𝑇) ∈ V
5		biidd 262	. . . . . . . . 9 ⊢ (𝑧 = (𝑆 − 𝑇) → (𝑁 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑥 / 𝑦)))
6	4, 5	ceqsexv 3529	. . . . . . . 8 ⊢ (∃𝑧(𝑧 = (𝑆 − 𝑇) ∧ 𝑁 = (𝑥 / 𝑦)) ↔ 𝑁 = (𝑥 / 𝑦))
7		exancom 1858	. . . . . . . 8 ⊢ (∃𝑧(𝑧 = (𝑆 − 𝑇) ∧ 𝑁 = (𝑥 / 𝑦)) ↔ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))
8	6, 7	bitr3i 277	. . . . . . 7 ⊢ (𝑁 = (𝑥 / 𝑦) ↔ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))
9	8	rexbii 3091	. . . . . 6 ⊢ (∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑦 ∈ ℕ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))
10		rexcom4 3285	. . . . . 6 ⊢ (∃𝑦 ∈ ℕ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) ↔ ∃𝑧∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))
11	9, 10	bitri 275	. . . . 5 ⊢ (∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑧∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))
12	11	rexbii 3091	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑥 ∈ ℤ ∃𝑧∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))
13		rexcom4 3285	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑧∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) ↔ ∃𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))
14	12, 13	bitri 275	. . 3 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))
15	3, 14	sylib 218	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∃𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))
16		pcval.1	. . . . . . . . . . 11 ⊢ 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < )
17		pcval.2	. . . . . . . . . . 11 ⊢ 𝑇 = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )
18		eqid 2734	. . . . . . . . . . 11 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < )
19		eqid 2734	. . . . . . . . . . 11 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )
20		simp11l 1283	. . . . . . . . . . 11 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑃 ∈ ℙ)
21		simp11r 1284	. . . . . . . . . . 11 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑁 ≠ 0)
22		simp12 1203	. . . . . . . . . . 11 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))) → (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ))
23		simp13l 1287	. . . . . . . . . . 11 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑁 = (𝑥 / 𝑦))
24		simp2 1136	. . . . . . . . . . 11 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))) → (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ))
25		simp3l 1200	. . . . . . . . . . 11 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑁 = (𝑠 / 𝑡))
26	16, 17, 18, 19, 20, 21, 22, 23, 24, 25	pceulem 16878	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))) → (𝑆 − 𝑇) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))
27		simp13r 1288	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑧 = (𝑆 − 𝑇))
28		simp3r 1201	. . . . . . . . . 10 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))
29	26, 27, 28	3eqtr4d 2784	. . . . . . . . 9 ⊢ ((((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))) ∧ (𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) ∧ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑧 = 𝑤)
30	29	3exp 1118	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))) → ((𝑠 ∈ ℤ ∧ 𝑡 ∈ ℕ) → ((𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < ))) → 𝑧 = 𝑤)))
31	30	rexlimdvv 3209	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) ∧ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))) → (∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < ))) → 𝑧 = 𝑤))
32	31	3exp 1118	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ≠ 0) → ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) → (∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < ))) → 𝑧 = 𝑤))))
33	32	adantrl 716	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) → (∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < ))) → 𝑧 = 𝑤))))
34	33	rexlimdvv 3209	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) → (∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < ))) → 𝑧 = 𝑤)))
35	34	impd 410	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ((∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) ∧ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑧 = 𝑤))
36	35	alrimivv 1925	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∀𝑧∀𝑤((∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) ∧ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑧 = 𝑤))
37		eqeq1 2738	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝑆 − 𝑇) ↔ 𝑤 = (𝑆 − 𝑇)))
38	37	anbi2d 630	. . . . 5 ⊢ (𝑧 = 𝑤 → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) ↔ (𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆 − 𝑇))))
39	38	2rexbidv 3219	. . . 4 ⊢ (𝑧 = 𝑤 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆 − 𝑇))))
40		oveq1 7437	. . . . . . . . 9 ⊢ (𝑥 = 𝑠 → (𝑥 / 𝑦) = (𝑠 / 𝑦))
41	40	eqeq2d 2745	. . . . . . . 8 ⊢ (𝑥 = 𝑠 → (𝑁 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑠 / 𝑦)))
42		breq2 5151	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → ((𝑃↑𝑛) ∥ 𝑥 ↔ (𝑃↑𝑛) ∥ 𝑠))
43	42	rabbidv 3440	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠})
44	43	supeq1d 9483	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ))
45	16, 44	eqtrid 2786	. . . . . . . . . 10 ⊢ (𝑥 = 𝑠 → 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ))
46	45	oveq1d 7445	. . . . . . . . 9 ⊢ (𝑥 = 𝑠 → (𝑆 − 𝑇) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇))
47	46	eqeq2d 2745	. . . . . . . 8 ⊢ (𝑥 = 𝑠 → (𝑤 = (𝑆 − 𝑇) ↔ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇)))
48	41, 47	anbi12d 632	. . . . . . 7 ⊢ (𝑥 = 𝑠 → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆 − 𝑇)) ↔ (𝑁 = (𝑠 / 𝑦) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇))))
49	48	rexbidv 3176	. . . . . 6 ⊢ (𝑥 = 𝑠 → (∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆 − 𝑇)) ↔ ∃𝑦 ∈ ℕ (𝑁 = (𝑠 / 𝑦) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇))))
50		oveq2 7438	. . . . . . . . 9 ⊢ (𝑦 = 𝑡 → (𝑠 / 𝑦) = (𝑠 / 𝑡))
51	50	eqeq2d 2745	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → (𝑁 = (𝑠 / 𝑦) ↔ 𝑁 = (𝑠 / 𝑡)))
52		breq2 5151	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑃↑𝑛) ∥ 𝑦 ↔ (𝑃↑𝑛) ∥ 𝑡))
53	52	rabbidv 3440	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡})
54	53	supeq1d 9483	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑡 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < ))
55	17, 54	eqtrid 2786	. . . . . . . . . 10 ⊢ (𝑦 = 𝑡 → 𝑇 = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < ))
56	55	oveq2d 7446	. . . . . . . . 9 ⊢ (𝑦 = 𝑡 → (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))
57	56	eqeq2d 2745	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → (𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇) ↔ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < ))))
58	51, 57	anbi12d 632	. . . . . . 7 ⊢ (𝑦 = 𝑡 → ((𝑁 = (𝑠 / 𝑦) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇)) ↔ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))))
59	58	cbvrexvw 3235	. . . . . 6 ⊢ (∃𝑦 ∈ ℕ (𝑁 = (𝑠 / 𝑦) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇)) ↔ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < ))))
60	49, 59	bitrdi 287	. . . . 5 ⊢ (𝑥 = 𝑠 → (∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆 − 𝑇)) ↔ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))))
61	60	cbvrexvw 3235	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆 − 𝑇)) ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < ))))
62	39, 61	bitrdi 287	. . 3 ⊢ (𝑧 = 𝑤 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))))
63	62	eu4 2612	. 2 ⊢ (∃!𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) ↔ (∃𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) ∧ ∀𝑧∀𝑤((∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) ∧ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℕ (𝑁 = (𝑠 / 𝑡) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )))) → 𝑧 = 𝑤)))
64	15, 36, 63	sylanbrc 583	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∃!𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1534   = wceq 1536  ∃wex 1775   ∈ wcel 2105  ∃!weu 2565   ≠ wne 2937  ∃wrex 3067  {crab 3432   class class class wbr 5147  (class class class)co 7430  supcsup 9477  ℝcr 11151  0cc0 11152   < clt 11292   − cmin 11489   / cdiv 11917  ℕcn 12263  ℕ₀cn0 12523  ℤcz 12610  ℚcq 12987  ↑cexp 14098   ∥ cdvds 16286  ℙcprime 16704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-dvds 16287  df-gcd 16528  df-prm 16705

This theorem is referenced by:  pczpre  16880  pcdiv  16885