pcdvdsb - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pcdvdsb		Structured version Visualization version GIF version

Theorem pcdvdsb 16831

Description: 𝑃↑𝐴 divides 𝑁 if and only if 𝐴 is at most the count of 𝑃. (Contributed by Mario Carneiro, 3-Oct-2014.)

**Assertion**
Ref	Expression
pcdvdsb	⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝐴 ≤ (𝑃 pCnt 𝑁) ↔ (𝑃↑𝐴) ∥ 𝑁))

**Proof of Theorem pcdvdsb**
Step	Hyp	Ref	Expression
1		oveq2 7368	. . . 4 ⊢ (𝑁 = 0 → (𝑃 pCnt 𝑁) = (𝑃 pCnt 0))
2	1	breq2d 5098	. . 3 ⊢ (𝑁 = 0 → (𝐴 ≤ (𝑃 pCnt 𝑁) ↔ 𝐴 ≤ (𝑃 pCnt 0)))
3		breq2 5090	. . 3 ⊢ (𝑁 = 0 → ((𝑃↑𝐴) ∥ 𝑁 ↔ (𝑃↑𝐴) ∥ 0))
4	2, 3	bibi12d 345	. 2 ⊢ (𝑁 = 0 → ((𝐴 ≤ (𝑃 pCnt 𝑁) ↔ (𝑃↑𝐴) ∥ 𝑁) ↔ (𝐴 ≤ (𝑃 pCnt 0) ↔ (𝑃↑𝐴) ∥ 0)))
5		simpl3 1195	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝐴 ∈ ℕ₀)
6	5	nn0zd 12540	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝐴 ∈ ℤ)
7		simpl1 1193	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑃 ∈ ℙ)
8		simpl2 1194	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℤ)
9		simpr 484	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑁 ≠ 0)
10		pczcl 16810	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) ∈ ℕ₀)
11	7, 8, 9, 10	syl12anc 837	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃 pCnt 𝑁) ∈ ℕ₀)
12	11	nn0zd 12540	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃 pCnt 𝑁) ∈ ℤ)
13		eluz 12793	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝑃 pCnt 𝑁) ∈ ℤ) → ((𝑃 pCnt 𝑁) ∈ (ℤ_≥‘𝐴) ↔ 𝐴 ≤ (𝑃 pCnt 𝑁)))
14	6, 12, 13	syl2anc 585	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ((𝑃 pCnt 𝑁) ∈ (ℤ_≥‘𝐴) ↔ 𝐴 ≤ (𝑃 pCnt 𝑁)))
15		prmnn 16634	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
16	7, 15	syl 17	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑃 ∈ ℕ)
17	16	nnzd 12541	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑃 ∈ ℤ)
18		dvdsexp 16288	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ 𝐴 ∈ ℕ₀ ∧ (𝑃 pCnt 𝑁) ∈ (ℤ_≥‘𝐴)) → (𝑃↑𝐴) ∥ (𝑃↑(𝑃 pCnt 𝑁)))
19	18	3expia 1122	. . . . . 6 ⊢ ((𝑃 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → ((𝑃 pCnt 𝑁) ∈ (ℤ_≥‘𝐴) → (𝑃↑𝐴) ∥ (𝑃↑(𝑃 pCnt 𝑁))))
20	17, 5, 19	syl2anc 585	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ((𝑃 pCnt 𝑁) ∈ (ℤ_≥‘𝐴) → (𝑃↑𝐴) ∥ (𝑃↑(𝑃 pCnt 𝑁))))
21	14, 20	sylbird 260	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝐴 ≤ (𝑃 pCnt 𝑁) → (𝑃↑𝐴) ∥ (𝑃↑(𝑃 pCnt 𝑁))))
22		pczdvds 16825	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝑁)) ∥ 𝑁)
23	7, 8, 9, 22	syl12anc 837	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑(𝑃 pCnt 𝑁)) ∥ 𝑁)
24		nnexpcl 14027	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℕ ∧ 𝐴 ∈ ℕ₀) → (𝑃↑𝐴) ∈ ℕ)
25	15, 24	sylan 581	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ₀) → (𝑃↑𝐴) ∈ ℕ)
26	25	3adant2 1132	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝑃↑𝐴) ∈ ℕ)
27	26	nnzd 12541	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝑃↑𝐴) ∈ ℤ)
28	27	adantr 480	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑𝐴) ∈ ℤ)
29	16, 11	nnexpcld 14198	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑(𝑃 pCnt 𝑁)) ∈ ℕ)
30	29	nnzd 12541	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑(𝑃 pCnt 𝑁)) ∈ ℤ)
31		dvdstr 16254	. . . . . 6 ⊢ (((𝑃↑𝐴) ∈ ℤ ∧ (𝑃↑(𝑃 pCnt 𝑁)) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑃↑𝐴) ∥ (𝑃↑(𝑃 pCnt 𝑁)) ∧ (𝑃↑(𝑃 pCnt 𝑁)) ∥ 𝑁) → (𝑃↑𝐴) ∥ 𝑁))
32	28, 30, 8, 31	syl3anc 1374	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (((𝑃↑𝐴) ∥ (𝑃↑(𝑃 pCnt 𝑁)) ∧ (𝑃↑(𝑃 pCnt 𝑁)) ∥ 𝑁) → (𝑃↑𝐴) ∥ 𝑁))
33	23, 32	mpan2d 695	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ((𝑃↑𝐴) ∥ (𝑃↑(𝑃 pCnt 𝑁)) → (𝑃↑𝐴) ∥ 𝑁))
34	21, 33	syld 47	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝐴 ≤ (𝑃 pCnt 𝑁) → (𝑃↑𝐴) ∥ 𝑁))
35		nn0re 12437	. . . . . . 7 ⊢ ((𝑃 pCnt 𝑁) ∈ ℕ₀ → (𝑃 pCnt 𝑁) ∈ ℝ)
36		nn0re 12437	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ → 𝐴 ∈ ℝ)
37		ltnle 11216	. . . . . . 7 ⊢ (((𝑃 pCnt 𝑁) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑃 pCnt 𝑁) < 𝐴 ↔ ¬ 𝐴 ≤ (𝑃 pCnt 𝑁)))
38	35, 36, 37	syl2an 597	. . . . . 6 ⊢ (((𝑃 pCnt 𝑁) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀) → ((𝑃 pCnt 𝑁) < 𝐴 ↔ ¬ 𝐴 ≤ (𝑃 pCnt 𝑁)))
39		nn0ltp1le 12578	. . . . . 6 ⊢ (((𝑃 pCnt 𝑁) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀) → ((𝑃 pCnt 𝑁) < 𝐴 ↔ ((𝑃 pCnt 𝑁) + 1) ≤ 𝐴))
40	38, 39	bitr3d 281	. . . . 5 ⊢ (((𝑃 pCnt 𝑁) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀) → (¬ 𝐴 ≤ (𝑃 pCnt 𝑁) ↔ ((𝑃 pCnt 𝑁) + 1) ≤ 𝐴))
41	11, 5, 40	syl2anc 585	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (¬ 𝐴 ≤ (𝑃 pCnt 𝑁) ↔ ((𝑃 pCnt 𝑁) + 1) ≤ 𝐴))
42		peano2nn0 12468	. . . . . . . . 9 ⊢ ((𝑃 pCnt 𝑁) ∈ ℕ₀ → ((𝑃 pCnt 𝑁) + 1) ∈ ℕ₀)
43	11, 42	syl 17	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ((𝑃 pCnt 𝑁) + 1) ∈ ℕ₀)
44	43	nn0zd 12540	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ((𝑃 pCnt 𝑁) + 1) ∈ ℤ)
45		eluz 12793	. . . . . . 7 ⊢ ((((𝑃 pCnt 𝑁) + 1) ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ (ℤ_≥‘((𝑃 pCnt 𝑁) + 1)) ↔ ((𝑃 pCnt 𝑁) + 1) ≤ 𝐴))
46	44, 6, 45	syl2anc 585	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝐴 ∈ (ℤ_≥‘((𝑃 pCnt 𝑁) + 1)) ↔ ((𝑃 pCnt 𝑁) + 1) ≤ 𝐴))
47		dvdsexp 16288	. . . . . . . 8 ⊢ ((𝑃 ∈ ℤ ∧ ((𝑃 pCnt 𝑁) + 1) ∈ ℕ₀ ∧ 𝐴 ∈ (ℤ_≥‘((𝑃 pCnt 𝑁) + 1))) → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ (𝑃↑𝐴))
48	47	3expia 1122	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ ((𝑃 pCnt 𝑁) + 1) ∈ ℕ₀) → (𝐴 ∈ (ℤ_≥‘((𝑃 pCnt 𝑁) + 1)) → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ (𝑃↑𝐴)))
49	17, 43, 48	syl2anc 585	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝐴 ∈ (ℤ_≥‘((𝑃 pCnt 𝑁) + 1)) → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ (𝑃↑𝐴)))
50	46, 49	sylbird 260	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (((𝑃 pCnt 𝑁) + 1) ≤ 𝐴 → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ (𝑃↑𝐴)))
51		pczndvds 16827	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ¬ (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ 𝑁)
52	7, 8, 9, 51	syl12anc 837	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ¬ (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ 𝑁)
53	16, 43	nnexpcld 14198	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∈ ℕ)
54	53	nnzd 12541	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∈ ℤ)
55		dvdstr 16254	. . . . . . . 8 ⊢ (((𝑃↑((𝑃 pCnt 𝑁) + 1)) ∈ ℤ ∧ (𝑃↑𝐴) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ (𝑃↑𝐴) ∧ (𝑃↑𝐴) ∥ 𝑁) → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ 𝑁))
56	54, 28, 8, 55	syl3anc 1374	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (((𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ (𝑃↑𝐴) ∧ (𝑃↑𝐴) ∥ 𝑁) → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ 𝑁))
57	52, 56	mtod 198	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ¬ ((𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ (𝑃↑𝐴) ∧ (𝑃↑𝐴) ∥ 𝑁))
58		imnan 399	. . . . . 6 ⊢ (((𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ (𝑃↑𝐴) → ¬ (𝑃↑𝐴) ∥ 𝑁) ↔ ¬ ((𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ (𝑃↑𝐴) ∧ (𝑃↑𝐴) ∥ 𝑁))
59	57, 58	sylibr 234	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ((𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ (𝑃↑𝐴) → ¬ (𝑃↑𝐴) ∥ 𝑁))
60	50, 59	syld 47	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (((𝑃 pCnt 𝑁) + 1) ≤ 𝐴 → ¬ (𝑃↑𝐴) ∥ 𝑁))
61	41, 60	sylbid 240	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (¬ 𝐴 ≤ (𝑃 pCnt 𝑁) → ¬ (𝑃↑𝐴) ∥ 𝑁))
62	34, 61	impcon4bid 227	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝐴 ≤ (𝑃 pCnt 𝑁) ↔ (𝑃↑𝐴) ∥ 𝑁))
63	36	3ad2ant3 1136	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → 𝐴 ∈ ℝ)
64	63	rexrd 11186	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → 𝐴 ∈ ℝ^*)
65		pnfge 13072	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
66	64, 65	syl 17	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → 𝐴 ≤ +∞)
67		pc0 16816	. . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 0) = +∞)
68	67	3ad2ant1 1134	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝑃 pCnt 0) = +∞)
69	66, 68	breqtrrd 5114	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → 𝐴 ≤ (𝑃 pCnt 0))
70		dvds0 16231	. . . 4 ⊢ ((𝑃↑𝐴) ∈ ℤ → (𝑃↑𝐴) ∥ 0)
71	27, 70	syl 17	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝑃↑𝐴) ∥ 0)
72	69, 71	2thd 265	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝐴 ≤ (𝑃 pCnt 0) ↔ (𝑃↑𝐴) ∥ 0))
73	4, 62, 72	pm2.61ne 3018	1 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝐴 ≤ (𝑃 pCnt 𝑁) ↔ (𝑃↑𝐴) ∥ 𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 class class class wbr 5086 ‘cfv 6492 (class class class)co 7360 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 +∞cpnf 11167 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 ℕcn 12165 ℕ₀cn0 12428 ℤcz 12515 ℤ_≥cuz 12779 ↑cexp 14014 ∥ cdvds 16212 ℙcprime 16631 pCnt cpc 16798

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9348 df-inf 9349 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-z 12516 df-uz 12780 df-q 12890 df-rp 12934 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16213 df-gcd 16455 df-prm 16632 df-pc 16799

This theorem is referenced by: pcelnn 16832 pcidlem 16834 pcdvdstr 16838 pcgcd1 16839 pcfac 16861 pockthlem 16867 pockthg 16868 prmreclem2 16879 sylow1lem1 19564 sylow1lem3 19566 sylow1lem5 19568 ablfac1c 20039 ablfac1eu 20041 issqf 27113 vmasum 27193

W3C validator