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 16570

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 7283	. . . 4 ⊢ (𝑁 = 0 → (𝑃 pCnt 𝑁) = (𝑃 pCnt 0))
2	1	breq2d 5086	. . 3 ⊢ (𝑁 = 0 → (𝐴 ≤ (𝑃 pCnt 𝑁) ↔ 𝐴 ≤ (𝑃 pCnt 0)))
3		breq2 5078	. . 3 ⊢ (𝑁 = 0 → ((𝑃↑𝐴) ∥ 𝑁 ↔ (𝑃↑𝐴) ∥ 0))
4	2, 3	bibi12d 346	. 2 ⊢ (𝑁 = 0 → ((𝐴 ≤ (𝑃 pCnt 𝑁) ↔ (𝑃↑𝐴) ∥ 𝑁) ↔ (𝐴 ≤ (𝑃 pCnt 0) ↔ (𝑃↑𝐴) ∥ 0)))
5		simpl3 1192	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝐴 ∈ ℕ₀)
6	5	nn0zd 12424	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝐴 ∈ ℤ)
7		simpl1 1190	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑃 ∈ ℙ)
8		simpl2 1191	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑁 ∈ ℤ)
9		simpr 485	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑁 ≠ 0)
10		pczcl 16549	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃 pCnt 𝑁) ∈ ℕ₀)
11	7, 8, 9, 10	syl12anc 834	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃 pCnt 𝑁) ∈ ℕ₀)
12	11	nn0zd 12424	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃 pCnt 𝑁) ∈ ℤ)
13		eluz 12596	. . . . . 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 16379	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
16	7, 15	syl 17	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑃 ∈ ℕ)
17	16	nnzd 12425	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → 𝑃 ∈ ℤ)
18		dvdsexp 16037	. . . . . . 7 ⊢ ((𝑃 ∈ ℤ ∧ 𝐴 ∈ ℕ₀ ∧ (𝑃 pCnt 𝑁) ∈ (ℤ_≥‘𝐴)) → (𝑃↑𝐴) ∥ (𝑃↑(𝑃 pCnt 𝑁)))
19	18	3expia 1120	. . . . . 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 259	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝐴 ≤ (𝑃 pCnt 𝑁) → (𝑃↑𝐴) ∥ (𝑃↑(𝑃 pCnt 𝑁))))
22		pczdvds 16564	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃↑(𝑃 pCnt 𝑁)) ∥ 𝑁)
23	7, 8, 9, 22	syl12anc 834	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑(𝑃 pCnt 𝑁)) ∥ 𝑁)
24		nnexpcl 13795	. . . . . . . . . 10 ⊢ ((𝑃 ∈ ℕ ∧ 𝐴 ∈ ℕ₀) → (𝑃↑𝐴) ∈ ℕ)
25	15, 24	sylan 580	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℕ₀) → (𝑃↑𝐴) ∈ ℕ)
26	25	3adant2 1130	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝑃↑𝐴) ∈ ℕ)
27	26	nnzd 12425	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝑃↑𝐴) ∈ ℤ)
28	27	adantr 481	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑𝐴) ∈ ℤ)
29	16, 11	nnexpcld 13960	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑(𝑃 pCnt 𝑁)) ∈ ℕ)
30	29	nnzd 12425	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑(𝑃 pCnt 𝑁)) ∈ ℤ)
31		dvdstr 16003	. . . . . 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 1370	. . . . 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 691	. . . 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 12242	. . . . . . 7 ⊢ ((𝑃 pCnt 𝑁) ∈ ℕ₀ → (𝑃 pCnt 𝑁) ∈ ℝ)
36		nn0re 12242	. . . . . . 7 ⊢ (𝐴 ∈ ℕ₀ → 𝐴 ∈ ℝ)
37		ltnle 11054	. . . . . . 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 12378	. . . . . 6 ⊢ (((𝑃 pCnt 𝑁) ∈ ℕ₀ ∧ 𝐴 ∈ ℕ₀) → ((𝑃 pCnt 𝑁) < 𝐴 ↔ ((𝑃 pCnt 𝑁) + 1) ≤ 𝐴))
40	38, 39	bitr3d 280	. . . . 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 12273	. . . . . . . . 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 12424	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ((𝑃 pCnt 𝑁) + 1) ∈ ℤ)
45		eluz 12596	. . . . . . 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 16037	. . . . . . . 8 ⊢ ((𝑃 ∈ ℤ ∧ ((𝑃 pCnt 𝑁) + 1) ∈ ℕ₀ ∧ 𝐴 ∈ (ℤ_≥‘((𝑃 pCnt 𝑁) + 1))) → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ (𝑃↑𝐴))
48	47	3expia 1120	. . . . . . 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 259	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (((𝑃 pCnt 𝑁) + 1) ≤ 𝐴 → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ (𝑃↑𝐴)))
51		pczndvds 16566	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ¬ (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ 𝑁)
52	7, 8, 9, 51	syl12anc 834	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ¬ (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ 𝑁)
53	16, 43	nnexpcld 13960	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∈ ℕ)
54	53	nnzd 12425	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝑃↑((𝑃 pCnt 𝑁) + 1)) ∈ ℤ)
55		dvdstr 16003	. . . . . . . 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 1370	. . . . . . 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 197	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → ¬ ((𝑃↑((𝑃 pCnt 𝑁) + 1)) ∥ (𝑃↑𝐴) ∧ (𝑃↑𝐴) ∥ 𝑁))
58		imnan 400	. . . . . 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 233	. . . . 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 239	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (¬ 𝐴 ≤ (𝑃 pCnt 𝑁) → ¬ (𝑃↑𝐴) ∥ 𝑁))
62	34, 61	impcon4bid 226	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) ∧ 𝑁 ≠ 0) → (𝐴 ≤ (𝑃 pCnt 𝑁) ↔ (𝑃↑𝐴) ∥ 𝑁))
63	36	3ad2ant3 1134	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → 𝐴 ∈ ℝ)
64	63	rexrd 11025	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → 𝐴 ∈ ℝ^*)
65		pnfge 12866	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ +∞)
66	64, 65	syl 17	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → 𝐴 ≤ +∞)
67		pc0 16555	. . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 0) = +∞)
68	67	3ad2ant1 1132	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝑃 pCnt 0) = +∞)
69	66, 68	breqtrrd 5102	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → 𝐴 ≤ (𝑃 pCnt 0))
70		dvds0 15981	. . . 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 264	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℕ₀) → (𝐴 ≤ (𝑃 pCnt 0) ↔ (𝑃↑𝐴) ∥ 0))
73	4, 62, 72	pm2.61ne 3030	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 +∞cpnf 11006 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ↑cexp 13782 ∥ cdvds 15963 ℙcprime 16376 pCnt cpc 16537

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689 df-rp 12731 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-dvds 15964 df-gcd 16202 df-prm 16377 df-pc 16538

This theorem is referenced by: pcelnn 16571 pcidlem 16573 pcdvdstr 16577 pcgcd1 16578 pcfac 16600 pockthlem 16606 pockthg 16607 prmreclem2 16618 sylow1lem1 19203 sylow1lem3 19205 sylow1lem5 19207 ablfac1c 19674 ablfac1eu 19676 issqf 26285 vmasum 26364

W3C validator