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Theorem pcexp 16560

Description: Prime power of an exponential. (Contributed by Mario Carneiro, 10-Aug-2015.)

**Assertion**
Ref	Expression
pcexp	⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ) → (𝑃 pCnt (𝐴↑𝑁)) = (𝑁 · (𝑃 pCnt 𝐴)))

Proof of Theorem pcexp Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7283	. . . . 5 ⊢ (𝑥 = 0 → (𝐴↑𝑥) = (𝐴↑0))
2	1	oveq2d 7291	. . . 4 ⊢ (𝑥 = 0 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑0)))
3		oveq1 7282	. . . 4 ⊢ (𝑥 = 0 → (𝑥 · (𝑃 pCnt 𝐴)) = (0 · (𝑃 pCnt 𝐴)))
4	2, 3	eqeq12d 2754	. . 3 ⊢ (𝑥 = 0 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑0)) = (0 · (𝑃 pCnt 𝐴))))
5		oveq2 7283	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴↑𝑥) = (𝐴↑𝑦))
6	5	oveq2d 7291	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑𝑦)))
7		oveq1 7282	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 · (𝑃 pCnt 𝐴)) = (𝑦 · (𝑃 pCnt 𝐴)))
8	6, 7	eqeq12d 2754	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑𝑦)) = (𝑦 · (𝑃 pCnt 𝐴))))
9		oveq2 7283	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝐴↑𝑥) = (𝐴↑(𝑦 + 1)))
10	9	oveq2d 7291	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑(𝑦 + 1))))
11		oveq1 7282	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 · (𝑃 pCnt 𝐴)) = ((𝑦 + 1) · (𝑃 pCnt 𝐴)))
12	10, 11	eqeq12d 2754	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑(𝑦 + 1))) = ((𝑦 + 1) · (𝑃 pCnt 𝐴))))
13		oveq2 7283	. . . . 5 ⊢ (𝑥 = -𝑦 → (𝐴↑𝑥) = (𝐴↑-𝑦))
14	13	oveq2d 7291	. . . 4 ⊢ (𝑥 = -𝑦 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑-𝑦)))
15		oveq1 7282	. . . 4 ⊢ (𝑥 = -𝑦 → (𝑥 · (𝑃 pCnt 𝐴)) = (-𝑦 · (𝑃 pCnt 𝐴)))
16	14, 15	eqeq12d 2754	. . 3 ⊢ (𝑥 = -𝑦 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑-𝑦)) = (-𝑦 · (𝑃 pCnt 𝐴))))
17		oveq2 7283	. . . . 5 ⊢ (𝑥 = 𝑁 → (𝐴↑𝑥) = (𝐴↑𝑁))
18	17	oveq2d 7291	. . . 4 ⊢ (𝑥 = 𝑁 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑𝑁)))
19		oveq1 7282	. . . 4 ⊢ (𝑥 = 𝑁 → (𝑥 · (𝑃 pCnt 𝐴)) = (𝑁 · (𝑃 pCnt 𝐴)))
20	18, 19	eqeq12d 2754	. . 3 ⊢ (𝑥 = 𝑁 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑𝑁)) = (𝑁 · (𝑃 pCnt 𝐴))))
21		pc1 16556	. . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
22	21	adantr 481	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 1) = 0)
23		qcn 12703	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
24	23	ad2antrl 725	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → 𝐴 ∈ ℂ)
25	24	exp0d 13858	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝐴↑0) = 1)
26	25	oveq2d 7291	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt (𝐴↑0)) = (𝑃 pCnt 1))
27		pcqcl 16557	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℤ)
28	27	zcnd 12427	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℂ)
29	28	mul02d 11173	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (0 · (𝑃 pCnt 𝐴)) = 0)
30	22, 26, 29	3eqtr4d 2788	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt (𝐴↑0)) = (0 · (𝑃 pCnt 𝐴)))
31		oveq1 7282	. . . . 5 ⊢ ((𝑃 pCnt (𝐴↑𝑦)) = (𝑦 · (𝑃 pCnt 𝐴)) → ((𝑃 pCnt (𝐴↑𝑦)) + (𝑃 pCnt 𝐴)) = ((𝑦 · (𝑃 pCnt 𝐴)) + (𝑃 pCnt 𝐴)))
32		expp1 13789	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℕ₀) → (𝐴↑(𝑦 + 1)) = ((𝐴↑𝑦) · 𝐴))
33	24, 32	sylan 580	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝐴↑(𝑦 + 1)) = ((𝐴↑𝑦) · 𝐴))
34	33	oveq2d 7291	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝑃 pCnt (𝐴↑(𝑦 + 1))) = (𝑃 pCnt ((𝐴↑𝑦) · 𝐴)))
35		simpll 764	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝑃 ∈ ℙ)
36		simplrl 774	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝐴 ∈ ℚ)
37		simplrr 775	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝐴 ≠ 0)
38		nn0z 12343	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℤ)
39	38	adantl 482	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝑦 ∈ ℤ)
40		qexpclz 13803	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0 ∧ 𝑦 ∈ ℤ) → (𝐴↑𝑦) ∈ ℚ)
41	36, 37, 39, 40	syl3anc 1370	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝐴↑𝑦) ∈ ℚ)
42	24	adantr 481	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝐴 ∈ ℂ)
43	42, 37, 39	expne0d 13870	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝐴↑𝑦) ≠ 0)
44		pcqmul 16554	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ ((𝐴↑𝑦) ∈ ℚ ∧ (𝐴↑𝑦) ≠ 0) ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt ((𝐴↑𝑦) · 𝐴)) = ((𝑃 pCnt (𝐴↑𝑦)) + (𝑃 pCnt 𝐴)))
45	35, 41, 43, 36, 37, 44	syl122anc 1378	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝑃 pCnt ((𝐴↑𝑦) · 𝐴)) = ((𝑃 pCnt (𝐴↑𝑦)) + (𝑃 pCnt 𝐴)))
46	34, 45	eqtrd 2778	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝑃 pCnt (𝐴↑(𝑦 + 1))) = ((𝑃 pCnt (𝐴↑𝑦)) + (𝑃 pCnt 𝐴)))
47		nn0cn 12243	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℂ)
48	47	adantl 482	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝑦 ∈ ℂ)
49	28	adantr 481	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝑃 pCnt 𝐴) ∈ ℂ)
50	48, 49	adddirp1d 11001	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → ((𝑦 + 1) · (𝑃 pCnt 𝐴)) = ((𝑦 · (𝑃 pCnt 𝐴)) + (𝑃 pCnt 𝐴)))
51	46, 50	eqeq12d 2754	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → ((𝑃 pCnt (𝐴↑(𝑦 + 1))) = ((𝑦 + 1) · (𝑃 pCnt 𝐴)) ↔ ((𝑃 pCnt (𝐴↑𝑦)) + (𝑃 pCnt 𝐴)) = ((𝑦 · (𝑃 pCnt 𝐴)) + (𝑃 pCnt 𝐴))))
52	31, 51	syl5ibr 245	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → ((𝑃 pCnt (𝐴↑𝑦)) = (𝑦 · (𝑃 pCnt 𝐴)) → (𝑃 pCnt (𝐴↑(𝑦 + 1))) = ((𝑦 + 1) · (𝑃 pCnt 𝐴))))
53	52	ex 413	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑦 ∈ ℕ₀ → ((𝑃 pCnt (𝐴↑𝑦)) = (𝑦 · (𝑃 pCnt 𝐴)) → (𝑃 pCnt (𝐴↑(𝑦 + 1))) = ((𝑦 + 1) · (𝑃 pCnt 𝐴)))))
54		negeq 11213	. . . . 5 ⊢ ((𝑃 pCnt (𝐴↑𝑦)) = (𝑦 · (𝑃 pCnt 𝐴)) → -(𝑃 pCnt (𝐴↑𝑦)) = -(𝑦 · (𝑃 pCnt 𝐴)))
55		nnnn0 12240	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ₀)
56		expneg 13790	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℕ₀) → (𝐴↑-𝑦) = (1 / (𝐴↑𝑦)))
57	24, 55, 56	syl2an 596	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝐴↑-𝑦) = (1 / (𝐴↑𝑦)))
58	57	oveq2d 7291	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝑃 pCnt (𝐴↑-𝑦)) = (𝑃 pCnt (1 / (𝐴↑𝑦))))
59		simpll 764	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → 𝑃 ∈ ℙ)
60	55, 41	sylan2 593	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝐴↑𝑦) ∈ ℚ)
61	55, 43	sylan2 593	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝐴↑𝑦) ≠ 0)
62		pcrec 16559	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ ((𝐴↑𝑦) ∈ ℚ ∧ (𝐴↑𝑦) ≠ 0)) → (𝑃 pCnt (1 / (𝐴↑𝑦))) = -(𝑃 pCnt (𝐴↑𝑦)))
63	59, 60, 61, 62	syl12anc 834	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝑃 pCnt (1 / (𝐴↑𝑦))) = -(𝑃 pCnt (𝐴↑𝑦)))
64	58, 63	eqtrd 2778	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝑃 pCnt (𝐴↑-𝑦)) = -(𝑃 pCnt (𝐴↑𝑦)))
65		nncn 11981	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
66		mulneg1 11411	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ (𝑃 pCnt 𝐴) ∈ ℂ) → (-𝑦 · (𝑃 pCnt 𝐴)) = -(𝑦 · (𝑃 pCnt 𝐴)))
67	65, 28, 66	syl2anr 597	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (-𝑦 · (𝑃 pCnt 𝐴)) = -(𝑦 · (𝑃 pCnt 𝐴)))
68	64, 67	eqeq12d 2754	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → ((𝑃 pCnt (𝐴↑-𝑦)) = (-𝑦 · (𝑃 pCnt 𝐴)) ↔ -(𝑃 pCnt (𝐴↑𝑦)) = -(𝑦 · (𝑃 pCnt 𝐴))))
69	54, 68	syl5ibr 245	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → ((𝑃 pCnt (𝐴↑𝑦)) = (𝑦 · (𝑃 pCnt 𝐴)) → (𝑃 pCnt (𝐴↑-𝑦)) = (-𝑦 · (𝑃 pCnt 𝐴))))
70	69	ex 413	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑦 ∈ ℕ → ((𝑃 pCnt (𝐴↑𝑦)) = (𝑦 · (𝑃 pCnt 𝐴)) → (𝑃 pCnt (𝐴↑-𝑦)) = (-𝑦 · (𝑃 pCnt 𝐴)))))
71	4, 8, 12, 16, 20, 30, 53, 70	zindd 12421	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑁 ∈ ℤ → (𝑃 pCnt (𝐴↑𝑁)) = (𝑁 · (𝑃 pCnt 𝐴))))
72	71	3impia 1116	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ) → (𝑃 pCnt (𝐴↑𝑁)) = (𝑁 · (𝑃 pCnt 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 (class class class)co 7275 ℂcc 10869 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 -cneg 11206 / cdiv 11632 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ℚcq 12688 ↑cexp 13782 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: qexpz 16602 expnprm 16603 dchrisum0flblem1 26656 dchrisum0flblem2 26657 aks4d1p7d1 40090

W3C validator