Theorem pceu 12433

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 529	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → 𝑁 ∈ ℚ)
2		elq 9687	. . . 4 ⊢ (𝑁 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦))
3	1, 2	sylib 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)
6	5	ad3antrrr 492	. . . . . . . . . . . 12 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑁 ≠ 0)
7	1	ad3antrrr 492	. . . . . . . . . . . . 13 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑁 ∈ ℚ)
8		0z 9328	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℤ
9		zq 9691	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ℤ → 0 ∈ ℚ)
10	8, 9	mp1i 10	. . . . . . . . . . . . 13 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 0 ∈ ℚ)
11		qapne 9704	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℚ ∧ 0 ∈ ℚ) → (𝑁 # 0 ↔ 𝑁 ≠ 0))
12	7, 10, 11	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → (𝑁 # 0 ↔ 𝑁 ≠ 0))
13	6, 12	mpbird 167	. . . . . . . . . . 11 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑁 # 0)
14	4, 13	eqbrtrrd 4053	. . . . . . . . . 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)
16	15	zcnd 9440	. . . . . . . . . . 11 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑥 ∈ ℂ)
17		nnz 9336	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
18	17	adantl 277	. . . . . . . . . . . . 13 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℤ)
19	18	adantr 276	. . . . . . . . . . . 12 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑦 ∈ ℤ)
20	19	zcnd 9440	. . . . . . . . . . 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)
22	21	nnap0d 9028	. . . . . . . . . . 11 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑦 # 0)
23	16, 20, 22	divap0bd 8821	. . . . . . . . . 10 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → (𝑥 # 0 ↔ (𝑥 / 𝑦) # 0))
24	14, 23	mpbird 167	. . . . . . . . 9 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑥 # 0)
25		0zd 9329	. . . . . . . . . 10 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 0 ∈ ℤ)
26		zapne 9391	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑥 # 0 ↔ 𝑥 ≠ 0))
27	15, 25, 26	syl2anc 411	. . . . . . . . 9 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → (𝑥 # 0 ↔ 𝑥 ≠ 0))
28	24, 27	mpbid 147	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑁 = (𝑥 / 𝑦)) → 𝑥 ≠ 0)
29	28	ex 115	. . . . . . 7 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → (𝑁 = (𝑥 / 𝑦) → 𝑥 ≠ 0))
30	29	adantrd 279	. . . . . . . 8 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) → 𝑥 ≠ 0))
31	30	exlimdv 1830	. . . . . . 7 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → (∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) → 𝑥 ≠ 0))
32		prmuz2 12269	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ≥‘2))
33	32	ad3antrrr 492	. . . . . . . . . . . . . 14 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → 𝑃 ∈ (ℤ≥‘2))
34	33	adantr 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 ∣ (𝑃↑𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}
38		pcval.1	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < )
39	37, 38	pcprecl 12427	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → (𝑆 ∈ ℕ0 ∧ (𝑃↑𝑆) ∥ 𝑥))
40	39	simpld 112	. . . . . . . . . . . . 13 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑥 ∈ ℤ ∧ 𝑥 ≠ 0)) → 𝑆 ∈ ℕ0)
41	34, 35, 36, 40	syl12anc 1247	. . . . . . . . . . . 12 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≠ 0) → 𝑆 ∈ ℕ0)
42	41	nn0zd 9437	. . . . . . . . . . 11 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≠ 0) → 𝑆 ∈ ℤ)
43		nnne0 9010	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0)
44	43	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → 𝑦 ≠ 0)
45		eqid 2193	. . . . . . . . . . . . . . . 16 ⊢ {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}
46		pcval.2	. . . . . . . . . . . . . . . 16 ⊢ 𝑇 = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < )
47	45, 46	pcprecl 12427	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ (ℤ≥‘2) ∧ (𝑦 ∈ ℤ ∧ 𝑦 ≠ 0)) → (𝑇 ∈ ℕ0 ∧ (𝑃↑𝑇) ∥ 𝑦))
48	33, 18, 44, 47	syl12anc 1247	. . . . . . . . . . . . . 14 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → (𝑇 ∈ ℕ0 ∧ (𝑃↑𝑇) ∥ 𝑦))
49	48	simpld 112	. . . . . . . . . . . . 13 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → 𝑇 ∈ ℕ0)
50	49	adantr 276	. . . . . . . . . . . 12 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≠ 0) → 𝑇 ∈ ℕ0)
51	50	nn0zd 9437	. . . . . . . . . . 11 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≠ 0) → 𝑇 ∈ ℤ)
52	42, 51	zsubcld 9444	. . . . . . . . . 10 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≠ 0) → (𝑆 − 𝑇) ∈ ℤ)
53		biidd 172	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑆 − 𝑇) → (𝑁 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑥 / 𝑦)))
54	53	ceqsexgv 2889	. . . . . . . . . 10 ⊢ ((𝑆 − 𝑇) ∈ ℤ → (∃𝑧(𝑧 = (𝑆 − 𝑇) ∧ 𝑁 = (𝑥 / 𝑦)) ↔ 𝑁 = (𝑥 / 𝑦)))
55	52, 54	syl 14	. . . . . . . . 9 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≠ 0) → (∃𝑧(𝑧 = (𝑆 − 𝑇) ∧ 𝑁 = (𝑥 / 𝑦)) ↔ 𝑁 = (𝑥 / 𝑦)))
56		exancom 1619	. . . . . . . . 9 ⊢ (∃𝑧(𝑧 = (𝑆 − 𝑇) ∧ 𝑁 = (𝑥 / 𝑦)) ↔ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))
57	55, 56	bitr3di 195	. . . . . . . 8 ⊢ (((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ≠ 0) → (𝑁 = (𝑥 / 𝑦) ↔ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))))
58	57	ex 115	. . . . . . 7 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → (𝑥 ≠ 0 → (𝑁 = (𝑥 / 𝑦) ↔ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))))
59	29, 31, 58	pm5.21ndd 706	. . . . . 6 ⊢ ((((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) → (𝑁 = (𝑥 / 𝑦) ↔ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))))
60	59	rexbidva 2491	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℤ) → (∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑦 ∈ ℕ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))))
61	60	rexbidva 2491	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))))
62		rexcom4 2783	. . . . . 6 ⊢ (∃𝑦 ∈ ℕ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) ↔ ∃𝑧∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))
63	62	rexbii 2501	. . . . 5 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) ↔ ∃𝑥 ∈ ℤ ∃𝑧∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))
64		rexcom4 2783	. . . . 5 ⊢ (∃𝑥 ∈ ℤ ∃𝑧∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) ↔ ∃𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))
65	63, 64	bitri 184	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ ∃𝑧(𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) ↔ ∃𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))
66	61, 65	bitrdi 196	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑁 = (𝑥 / 𝑦) ↔ ∃𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇))))
67	3, 66	mpbid 147	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∃𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))
68		eqid 2193	. . . . . . . . . . 11 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < )
69		eqid 2193	. . . . . . . . . . 11 ⊢ sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < )
70		simp11l 1110	. . . . . . . . . . 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)
71		simp11r 1111	. . . . . . . . . . 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)
72		simp12 1030	. . . . . . . . . . 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))
73		simp13l 1114	. . . . . . . . . . 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))
74		simp2 1000	. . . . . . . . . . 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))
75		simp3l 1027	. . . . . . . . . . 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))
76	38, 46, 68, 69, 70, 71, 72, 73, 74, 75	pceulem 12432	. . . . . . . . . 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, < )))
77		simp13r 1115	. . . . . . . . . 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))
78		simp3r 1028	. . . . . . . . . 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, < )))
79	76, 77, 78	3eqtr4d 2236	. . . . . . . . 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)
80	79	3exp 1204	. . . . . . . 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)))
81	80	rexlimdvv 2618	. . . . . . 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))
82	81	3exp 1204	. . . . . 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))))
83	82	adantrl 478	. . . . 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))))
84	83	rexlimdvv 2618	. . . 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)))
85	84	impd 254	. . 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))
86	85	alrimivv 1886	. 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))
87		eqeq1 2200	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝑆 − 𝑇) ↔ 𝑤 = (𝑆 − 𝑇)))
88	87	anbi2d 464	. . . . 5 ⊢ (𝑧 = 𝑤 → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) ↔ (𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆 − 𝑇))))
89	88	2rexbidv 2519	. . . 4 ⊢ (𝑧 = 𝑤 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆 − 𝑇))))
90		oveq1 5925	. . . . . . . . 9 ⊢ (𝑥 = 𝑠 → (𝑥 / 𝑦) = (𝑠 / 𝑦))
91	90	eqeq2d 2205	. . . . . . . 8 ⊢ (𝑥 = 𝑠 → (𝑁 = (𝑥 / 𝑦) ↔ 𝑁 = (𝑠 / 𝑦)))
92		breq2 4033	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → ((𝑃↑𝑛) ∥ 𝑥 ↔ (𝑃↑𝑛) ∥ 𝑠))
93	92	rabbidv 2749	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠})
94	93	supeq1d 7046	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑥}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ))
95	38, 94	eqtrid 2238	. . . . . . . . . 10 ⊢ (𝑥 = 𝑠 → 𝑆 = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ))
96	95	oveq1d 5933	. . . . . . . . 9 ⊢ (𝑥 = 𝑠 → (𝑆 − 𝑇) = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇))
97	96	eqeq2d 2205	. . . . . . . 8 ⊢ (𝑥 = 𝑠 → (𝑤 = (𝑆 − 𝑇) ↔ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇)))
98	91, 97	anbi12d 473	. . . . . . 7 ⊢ (𝑥 = 𝑠 → ((𝑁 = (𝑥 / 𝑦) ∧ 𝑤 = (𝑆 − 𝑇)) ↔ (𝑁 = (𝑠 / 𝑦) ∧ 𝑤 = (sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑠}, ℝ, < ) − 𝑇))))
99	98	rexbidv 2495	. . . . . 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))))
100		oveq2 5926	. . . . . . . . 9 ⊢ (𝑦 = 𝑡 → (𝑠 / 𝑦) = (𝑠 / 𝑡))
101	100	eqeq2d 2205	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → (𝑁 = (𝑠 / 𝑦) ↔ 𝑁 = (𝑠 / 𝑡)))
102		breq2 4033	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑡 → ((𝑃↑𝑛) ∥ 𝑦 ↔ (𝑃↑𝑛) ∥ 𝑡))
103	102	rabbidv 2749	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑡 → {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦} = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡})
104	103	supeq1d 7046	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑡 → sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑦}, ℝ, < ) = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < ))
105	46, 104	eqtrid 2238	. . . . . . . . . 10 ⊢ (𝑦 = 𝑡 → 𝑇 = sup({𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑡}, ℝ, < ))
106	105	oveq2d 5934	. . . . . . . . 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, < )))
107	106	eqeq2d 2205	. . . . . . . 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, < ))))
108	101, 107	anbi12d 473	. . . . . . 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, < )))))
109	108	cbvrexvw 2731	. . . . . 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, < ))))
110	99, 109	bitrdi 196	. . . . 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, < )))))
111	110	cbvrexvw 2731	. . . 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, < ))))
112	89, 111	bitrdi 196	. . 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, < )))))
113	112	eu4 2104	. 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)))
114	67, 86, 113	sylanbrc 417	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℚ ∧ 𝑁 ≠ 0)) → ∃!𝑧∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (𝑁 = (𝑥 / 𝑦) ∧ 𝑧 = (𝑆 − 𝑇)))

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 4029  ‘cfv 5254  (class class class)co 5918  supcsup 7041  ℝcr 7871  0cc0 7872   < clt 8054   − cmin 8190   # cap 8600   / cdiv 8691  ℕcn 8982  2c2 9033  ℕ₀cn0 9240  ℤcz 9317  ℤ_≥cuz 9592  ℚcq 9684  ↑cexp 10609   ∥ cdvds 11930  ℙcprime 12245

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 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992

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 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-1o 6469  df-2o 6470  df-er 6587  df-en 6795  df-sup 7043  df-inf 7044  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-dvds 11931  df-gcd 12080  df-prm 12246

This theorem is referenced by:  pcval  12434  pczpre  12435  pcdiv  12440