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Theorem pcdvdsb 16954

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 7431	. . . 4 ⊢ (𝑁 = 0 → (𝑃 pCnt 𝑁) = (𝑃 pCnt 0))
2	1	breq2d 5126	. . 3 ⊢ (𝑁 = 0 → (𝐴 ≤ (𝑃 pCnt 𝑁) ↔ 𝐴 ≤ (𝑃 pCnt 0)))
3		breq2 5118	. . 3 ⊢ (𝑁 = 0 → ((𝑃↑𝐴) ∥ 𝑁 ↔ (𝑃↑𝐴) ∥ 0))
4	2, 3	bibi12d 348	. 2 ⊢ (𝑁 = 0 → ((𝐴 ≤ (𝑃 pCnt 𝑁) ↔ (𝑃↑𝐴) ∥ 𝑁) ↔ (𝐴 ≤ (𝑃 pCnt 0) ↔ (𝑃↑𝐴) ∥ 0)))
5		simpl3 1212	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝐴 ∈ ℕ₀)
6	5	nn0zd 12634	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝐴 ∈ ℤ)
7		simpl1 1210	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑃 ∈ ℙ)
8		simpl2 1211	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℤ)
9		simpr 490	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑁 ≠ 0)
10		pczcl 16933	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) ∈ ℕ₀)
11	7, 8, 9, 10	syl12anc 850	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃 pCnt 𝑁) ∈ ℕ₀)
12	11	nn0zd 12634	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃 pCnt 𝑁) ∈ ℤ)
13		eluz 12894	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝑃 pCnt 𝑁) ∈ ℤ) → ((𝑃 pCnt 𝑁) ∈ (ℤ_≥‘𝐴) ↔ 𝐴 ≤ (𝑃 pCnt 𝑁)))
14	6, 12, 13	syl2anc 596	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ((𝑃 pCnt 𝑁) ∈ (ℤ_≥‘𝐴) ↔ 𝐴 ≤ (𝑃 pCnt 𝑁)))
15		prmnn 16757	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
16	7, 15	syl 18	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑃 ∈ ℕ)
17	16	nnzd 12635	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑃 ∈ ℤ)
18		dvdsexp 16411	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ 𝐴 ∈ ℕ₀ ∧ (𝑃 pCnt 𝑁) ∈ (ℤ_≥‘𝐴)) → (𝑃↑𝐴) ∥ (𝑃↑(𝑃 pCnt 𝑁)))
19	18	3expia 1139	. . . . . 6 ⊢ ((𝑃 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → ((𝑃 pCnt 𝑁) ∈ (ℤ_≥‘𝐴) → (𝑃↑𝐴) ∥ (𝑃↑(𝑃 pCnt 𝑁))))
20	17, 5, 19	syl2anc 596	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ((𝑃 pCnt 𝑁) ∈ (ℤ_≥‘𝐴) → (𝑃↑𝐴) ∥ (𝑃↑(𝑃 pCnt 𝑁))))
21	14, 20	sylbird 263	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝐴 ≤ (𝑃 pCnt 𝑁) → (𝑃↑𝐴) ∥ (𝑃↑(𝑃 pCnt 𝑁))))
22		pczdvds 16948	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝑁)) ∥ 𝑁)
23	7, 8, 9, 22	syl12anc 850	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑(𝑃 pCnt 𝑁)) ∥ 𝑁)
24		nnexpcl 14130	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℕ ∧ 𝐴 ∈ ℕ₀) → (𝑃↑𝐴) ∈ ℕ)
25	15, 24	sylan 592	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ₀) → (𝑃↑𝐴) ∈ ℕ)
26	25	3adant2 1149	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝑃↑𝐴) ∈ ℕ)
27	26	nnzd 12635	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝑃↑𝐴) ∈ ℤ)
28	27	adantr 486	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑𝐴) ∈ ℤ)
29	16, 11	nnexpcld 14301	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑(𝑃 pCnt 𝑁)) ∈ ℕ)
30	29	nnzd 12635	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑(𝑃 pCnt 𝑁)) ∈ ℤ)
31		dvdstr 16377	. . . . . 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 1398	. . . . 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 707	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ((𝑃↑𝐴) ∥ (𝑃↑(𝑃 pCnt 𝑁)) → (𝑃↑𝐴) ∥ 𝑁))
34	21, 33	syld 48	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝐴 ≤ (𝑃 pCnt 𝑁) → (𝑃↑𝐴) ∥ 𝑁))
35		nn0re 12531	. . . . . . 7 ⊢ ((𝑃 pCnt 𝑁) ∈ ℕ₀ → (𝑃 pCnt 𝑁) ∈ ℝ)
36		nn0re 12531	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ → 𝐴 ∈ ℝ)
37		ltnle 11307	. . . . . . 7 ⊢ (((𝑃 pCnt 𝑁) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑃 pCnt 𝑁) < 𝐴 ↔ ¬ 𝐴 ≤ (𝑃 pCnt 𝑁)))
38	35, 36, 37	syl2an 608	. . . . . 6 ⊢ (((𝑃 pCnt 𝑁) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀) → ((𝑃 pCnt 𝑁) < 𝐴 ↔ ¬ 𝐴 ≤ (𝑃 pCnt 𝑁)))
39		nn0ltp1le 12672	. . . . . 6 ⊢ (((𝑃 pCnt 𝑁) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀) → ((𝑃 pCnt 𝑁) < 𝐴 ↔ ((𝑃 pCnt 𝑁) + 1) ≤ 𝐴))
40	38, 39	bitr3d 284	. . . . 5 ⊢ (((𝑃 pCnt 𝑁) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀) → (¬ 𝐴 ≤ (𝑃 pCnt 𝑁) ↔ ((𝑃 pCnt 𝑁) + 1) ≤ 𝐴))
41	11, 5, 40	syl2anc 596	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (¬ 𝐴 ≤ (𝑃 pCnt 𝑁) ↔ ((𝑃 pCnt 𝑁) + 1) ≤ 𝐴))
42		peano2nn0 12562	. . . . . . . . 9 ⊢ ((𝑃 pCnt 𝑁) ∈ ℕ₀ → ((𝑃 pCnt 𝑁) + 1) ∈ ℕ₀)
43	11, 42	syl 18	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ((𝑃 pCnt 𝑁) + 1) ∈ ℕ₀)
44	43	nn0zd 12634	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ((𝑃 pCnt 𝑁) + 1) ∈ ℤ)
45		eluz 12894	. . . . . . 7 ⊢ ((((𝑃 pCnt 𝑁) + 1) ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ (ℤ_≥‘((𝑃 pCnt 𝑁) + 1)) ↔ ((𝑃 pCnt 𝑁) + 1) ≤ 𝐴))
46	44, 6, 45	syl2anc 596	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝐴 ∈ (ℤ_≥‘((𝑃 pCnt 𝑁) + 1)) ↔ ((𝑃 pCnt 𝑁) + 1) ≤ 𝐴))
47		dvdsexp 16411	. . . . . . . 8 ⊢ ((𝑃 ∈ ℤ ∧ ((𝑃 pCnt 𝑁) + 1) ∈ ℕ₀ ∧ 𝐴 ∈ (ℤ_≥‘((𝑃 pCnt 𝑁) + 1))) → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ (𝑃↑𝐴))
48	47	3expia 1139	. . . . . . 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 596	. . . . . 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 263	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (((𝑃 pCnt 𝑁) + 1) ≤ 𝐴 → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ (𝑃↑𝐴)))
51		pczndvds 16950	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ¬ (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ 𝑁)
52	7, 8, 9, 51	syl12anc 850	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ¬ (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ 𝑁)
53	16, 43	nnexpcld 14301	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∈ ℕ)
54	53	nnzd 12635	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∈ ℤ)
55		dvdstr 16377	. . . . . . . 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 1398	. . . . . . 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 201	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ¬ ((𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ (𝑃↑𝐴) ∧ (𝑃↑𝐴) ∥ 𝑁))
58		imnan 405	. . . . . 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 237	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ((𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ (𝑃↑𝐴) → ¬ (𝑃↑𝐴) ∥ 𝑁))
60	50, 59	syld 48	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (((𝑃 pCnt 𝑁) + 1) ≤ 𝐴 → ¬ (𝑃↑𝐴) ∥ 𝑁))
61	41, 60	sylbid 243	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (¬ 𝐴 ≤ (𝑃 pCnt 𝑁) → ¬ (𝑃↑𝐴) ∥ 𝑁))
62	34, 61	impcon4bid 230	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝐴 ≤ (𝑃 pCnt 𝑁) ↔ (𝑃↑𝐴) ∥ 𝑁))
63	36	3ad2ant3 1153	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → 𝐴 ∈ ℝ)
64	63	rexrd 11277	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → 𝐴 ∈ ℝ^*)
65		pnfge 13173	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
66	64, 65	syl 18	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → 𝐴 ≤ +∞)
67		pc0 16939	. . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 0) = +∞)
68	67	3ad2ant1 1151	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝑃 pCnt 0) = +∞)
69	66, 68	breqtrrd 5144	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → 𝐴 ≤ (𝑃 pCnt 0))
70		dvds0 16354	. . . 4 ⊢ ((𝑃↑𝐴) ∈ ℤ → (𝑃↑𝐴) ∥ 0)
71	27, 70	syl 18	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝑃↑𝐴) ∥ 0)
72	69, 71	2thd 268	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝐴 ≤ (𝑃 pCnt 0) ↔ (𝑃↑𝐴) ∥ 0))
73	4, 62, 72	pm2.61ne 3046	1 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝐴 ≤ (𝑃 pCnt 𝑁) ↔ (𝑃↑𝐴) ∥ 𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 class class class wbr 5114 ‘cfv 6543 (class class class)co 7423 ℝcr 11117 0cc0 11118 1c1 11119 + caddc 11121 +∞cpnf 11258 ℝ^*cxr 11260 < clt 11261 ≤ cle 11262 ℕcn 12251 ℕ₀cn0 12522 ℤcz 12609 ℤ_≥cuz 12880 ↑cexp 14117 ∥ cdvds 16335 ℙcprime 16754 pCnt cpc 16921

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-n0 12523 df-z 12610 df-uz 12881 df-q 12991 df-rp 13035 df-fl 13845 df-mod 13923 df-seq 14058 df-exp 14118 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-dvds 16336 df-gcd 16578 df-prm 16755 df-pc 16922

This theorem is used by: pcelnn 16955 pcidlem 16957 pcdvdstr 16961 pcgcd1 16962 pcfac 16984 pockthlem 16990 pockthg 16991 prmreclem2 17002 sylow1lem1 19699 sylow1lem3 19701 sylow1lem5 19703 ablfac1c 20174 ablfac1eu 20176 issqf 27337 vmasum 27417

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