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Theorem pcdvdsb 16907

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 7439	. . . 4 ⊢ (𝑁 = 0 → (𝑃 pCnt 𝑁) = (𝑃 pCnt 0))
2	1	breq2d 5155	. . 3 ⊢ (𝑁 = 0 → (𝐴 ≤ (𝑃 pCnt 𝑁) ↔ 𝐴 ≤ (𝑃 pCnt 0)))
3		breq2 5147	. . 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 1194	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝐴 ∈ ℕ₀)
6	5	nn0zd 12639	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝐴 ∈ ℤ)
7		simpl1 1192	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑃 ∈ ℙ)
8		simpl2 1193	. . . . . . . 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 16886	. . . . . . . 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 12639	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃 pCnt 𝑁) ∈ ℤ)
13		eluz 12892	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ (𝑃 pCnt 𝑁) ∈ ℤ) → ((𝑃 pCnt 𝑁) ∈ (ℤ_≥‘𝐴) ↔ 𝐴 ≤ (𝑃 pCnt 𝑁)))
14	6, 12, 13	syl2anc 584	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ((𝑃 pCnt 𝑁) ∈ (ℤ_≥‘𝐴) ↔ 𝐴 ≤ (𝑃 pCnt 𝑁)))
15		prmnn 16711	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
16	7, 15	syl 17	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑃 ∈ ℕ)
17	16	nnzd 12640	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑃 ∈ ℤ)
18		dvdsexp 16365	. . . . . . 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 584	. . . . 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 16901	. . . . . 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 14115	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℕ ∧ 𝐴 ∈ ℕ₀) → (𝑃↑𝐴) ∈ ℕ)
25	15, 24	sylan 580	. . . . . . . . 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 12640	. . . . . . 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 14284	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑(𝑃 pCnt 𝑁)) ∈ ℕ)
30	29	nnzd 12640	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑(𝑃 pCnt 𝑁)) ∈ ℤ)
31		dvdstr 16331	. . . . . 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 1373	. . . . 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 694	. . . 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 12535	. . . . . . 7 ⊢ ((𝑃 pCnt 𝑁) ∈ ℕ₀ → (𝑃 pCnt 𝑁) ∈ ℝ)
36		nn0re 12535	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ → 𝐴 ∈ ℝ)
37		ltnle 11340	. . . . . . 7 ⊢ (((𝑃 pCnt 𝑁) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑃 pCnt 𝑁) < 𝐴 ↔ ¬ 𝐴 ≤ (𝑃 pCnt 𝑁)))
38	35, 36, 37	syl2an 596	. . . . . 6 ⊢ (((𝑃 pCnt 𝑁) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀) → ((𝑃 pCnt 𝑁) < 𝐴 ↔ ¬ 𝐴 ≤ (𝑃 pCnt 𝑁)))
39		nn0ltp1le 12676	. . . . . 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 584	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (¬ 𝐴 ≤ (𝑃 pCnt 𝑁) ↔ ((𝑃 pCnt 𝑁) + 1) ≤ 𝐴))
42		peano2nn0 12566	. . . . . . . . 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 12639	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ((𝑃 pCnt 𝑁) + 1) ∈ ℤ)
45		eluz 12892	. . . . . . 7 ⊢ ((((𝑃 pCnt 𝑁) + 1) ∈ ℤ ∧ 𝐴 ∈ ℤ) → (𝐴 ∈ (ℤ_≥‘((𝑃 pCnt 𝑁) + 1)) ↔ ((𝑃 pCnt 𝑁) + 1) ≤ 𝐴))
46	44, 6, 45	syl2anc 584	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝐴 ∈ (ℤ_≥‘((𝑃 pCnt 𝑁) + 1)) ↔ ((𝑃 pCnt 𝑁) + 1) ≤ 𝐴))
47		dvdsexp 16365	. . . . . . . 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 584	. . . . . 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 16903	. . . . . . . 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 14284	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∈ ℕ)
54	53	nnzd 12640	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∈ ℤ)
55		dvdstr 16331	. . . . . . . 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 1373	. . . . . . 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 11311	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → 𝐴 ∈ ℝ^*)
65		pnfge 13172	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
66	64, 65	syl 17	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → 𝐴 ≤ +∞)
67		pc0 16892	. . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 0) = +∞)
68	67	3ad2ant1 1134	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝑃 pCnt 0) = +∞)
69	66, 68	breqtrrd 5171	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → 𝐴 ≤ (𝑃 pCnt 0))
70		dvds0 16309	. . . 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 3027	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 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 +∞cpnf 11292 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 ℕcn 12266 ℕ₀cn0 12526 ℤcz 12613 ℤ_≥cuz 12878 ↑cexp 14102 ∥ cdvds 16290 ℙcprime 16708 pCnt cpc 16874

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-prm 16709 df-pc 16875

This theorem is referenced by: pcelnn 16908 pcidlem 16910 pcdvdstr 16914 pcgcd1 16915 pcfac 16937 pockthlem 16943 pockthg 16944 prmreclem2 16955 sylow1lem1 19616 sylow1lem3 19618 sylow1lem5 19620 ablfac1c 20091 ablfac1eu 20093 issqf 27179 vmasum 27260

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