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Theorem pcexp 16918

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 7418	. . . . 5 ⊢ (𝑥 = 0 → (𝐴↑𝑥) = (𝐴↑0))
2	1	oveq2d 7426	. . . 4 ⊢ (𝑥 = 0 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑0)))
3		oveq1 7417	. . . 4 ⊢ (𝑥 = 0 → (𝑥 · (𝑃 pCnt 𝐴)) = (0 · (𝑃 pCnt 𝐴)))
4	2, 3	eqeq12d 2777	. . 3 ⊢ (𝑥 = 0 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑0)) = (0 · (𝑃 pCnt 𝐴))))
5		oveq2 7418	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴↑𝑥) = (𝐴↑𝑦))
6	5	oveq2d 7426	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑𝑦)))
7		oveq1 7417	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 · (𝑃 pCnt 𝐴)) = (𝑦 · (𝑃 pCnt 𝐴)))
8	6, 7	eqeq12d 2777	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑𝑦)) = (𝑦 · (𝑃 pCnt 𝐴))))
9		oveq2 7418	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝐴↑𝑥) = (𝐴↑(𝑦 + 1)))
10	9	oveq2d 7426	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑(𝑦 + 1))))
11		oveq1 7417	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 · (𝑃 pCnt 𝐴)) = ((𝑦 + 1) · (𝑃 pCnt 𝐴)))
12	10, 11	eqeq12d 2777	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑(𝑦 + 1))) = ((𝑦 + 1) · (𝑃 pCnt 𝐴))))
13		oveq2 7418	. . . . 5 ⊢ (𝑥 = -𝑦 → (𝐴↑𝑥) = (𝐴↑-𝑦))
14	13	oveq2d 7426	. . . 4 ⊢ (𝑥 = -𝑦 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑-𝑦)))
15		oveq1 7417	. . . 4 ⊢ (𝑥 = -𝑦 → (𝑥 · (𝑃 pCnt 𝐴)) = (-𝑦 · (𝑃 pCnt 𝐴)))
16	14, 15	eqeq12d 2777	. . 3 ⊢ (𝑥 = -𝑦 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑-𝑦)) = (-𝑦 · (𝑃 pCnt 𝐴))))
17		oveq2 7418	. . . . 5 ⊢ (𝑥 = 𝑁 → (𝐴↑𝑥) = (𝐴↑𝑁))
18	17	oveq2d 7426	. . . 4 ⊢ (𝑥 = 𝑁 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑𝑁)))
19		oveq1 7417	. . . 4 ⊢ (𝑥 = 𝑁 → (𝑥 · (𝑃 pCnt 𝐴)) = (𝑁 · (𝑃 pCnt 𝐴)))
20	18, 19	eqeq12d 2777	. . 3 ⊢ (𝑥 = 𝑁 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑𝑁)) = (𝑁 · (𝑃 pCnt 𝐴))))
21		pc1 16914	. . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
22	21	adantr 485	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 1) = 0)
23		qcn 12986	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
24	23	ad2antrl 740	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → 𝐴 ∈ ℂ)
25	24	exp0d 14176	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝐴↑0) = 1)
26	25	oveq2d 7426	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt (𝐴↑0)) = (𝑃 pCnt 1))
27		pcqcl 16915	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℤ)
28	27	zcnd 12700	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℂ)
29	28	mul02d 11407	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (0 · (𝑃 pCnt 𝐴)) = 0)
30	22, 26, 29	3eqtr4d 2806	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt (𝐴↑0)) = (0 · (𝑃 pCnt 𝐴)))
31		oveq1 7417	. . . . 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 14104	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℕ₀) → (𝐴↑(𝑦 + 1)) = ((𝐴↑𝑦) · 𝐴))
33	24, 32	sylan 591	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝐴↑(𝑦 + 1)) = ((𝐴↑𝑦) · 𝐴))
34	33	oveq2d 7426	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝑃 pCnt (𝐴↑(𝑦 + 1))) = (𝑃 pCnt ((𝐴↑𝑦) · 𝐴)))
35		simpll 778	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝑃 ∈ ℙ)
36		simplrl 788	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝐴 ∈ ℚ)
37		simplrr 789	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝐴 ≠ 0)
38		nn0z 12614	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℤ)
39	38	adantl 486	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝑦 ∈ ℤ)
40		qexpclz 14117	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0 ∧ 𝑦 ∈ ℤ) → (𝐴↑𝑦) ∈ ℚ)
41	36, 37, 39, 40	syl3anc 1396	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝐴↑𝑦) ∈ ℚ)
42	24	adantr 485	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝐴 ∈ ℂ)
43	42, 37, 39	expne0d 14188	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝐴↑𝑦) ≠ 0)
44		pcqmul 16912	. . . . . . . 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 1404	. . . . . . 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 2796	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝑃 pCnt (𝐴↑(𝑦 + 1))) = ((𝑃 pCnt (𝐴↑𝑦)) + (𝑃 pCnt 𝐴)))
47		nn0cn 12513	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℂ)
48	47	adantl 486	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝑦 ∈ ℂ)
49	28	adantr 485	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝑃 pCnt 𝐴) ∈ ℂ)
50	48, 49	adddirp1d 11234	. . . . . 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 2777	. . . . 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	imbitrrid 249	. . . 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 417	. . 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 11448	. . . . 5 ⊢ ((𝑃 pCnt (𝐴↑𝑦)) = (𝑦 · (𝑃 pCnt 𝐴)) → -(𝑃 pCnt (𝐴↑𝑦)) = -(𝑦 · (𝑃 pCnt 𝐴)))
55		nnnn0 12510	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ₀)
56		expneg 14105	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ ℕ₀) → (𝐴↑-𝑦) = (1 / (𝐴↑𝑦)))
57	24, 55, 56	syl2an 607	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝐴↑-𝑦) = (1 / (𝐴↑𝑦)))
58	57	oveq2d 7426	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝑃 pCnt (𝐴↑-𝑦)) = (𝑃 pCnt (1 / (𝐴↑𝑦))))
59		simpll 778	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → 𝑃 ∈ ℙ)
60	55, 41	sylan2 604	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝐴↑𝑦) ∈ ℚ)
61	55, 43	sylan2 604	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝐴↑𝑦) ≠ 0)
62		pcrec 16917	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ ((𝐴↑𝑦) ∈ ℚ ∧ (𝐴↑𝑦) ≠ 0)) → (𝑃 pCnt (1 / (𝐴↑𝑦))) = -(𝑃 pCnt (𝐴↑𝑦)))
63	59, 60, 61, 62	syl12anc 849	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝑃 pCnt (1 / (𝐴↑𝑦))) = -(𝑃 pCnt (𝐴↑𝑦)))
64	58, 63	eqtrd 2796	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝑃 pCnt (𝐴↑-𝑦)) = -(𝑃 pCnt (𝐴↑𝑦)))
65		nncn 12240	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
66		mulneg1 11649	. . . . . . 7 ⊢ ((𝑦 ∈ ℂ ∧ (𝑃 pCnt 𝐴) ∈ ℂ) → (-𝑦 · (𝑃 pCnt 𝐴)) = -(𝑦 · (𝑃 pCnt 𝐴)))
67	65, 28, 66	syl2anr 608	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (-𝑦 · (𝑃 pCnt 𝐴)) = -(𝑦 · (𝑃 pCnt 𝐴)))
68	64, 67	eqeq12d 2777	. . . . 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	imbitrrid 249	. . . 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 417	. . 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 12696	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑁 ∈ ℤ → (𝑃 pCnt (𝐴↑𝑁)) = (𝑁 · (𝑃 pCnt 𝐴))))
72	71	3impia 1133	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 400 ∧ w3a 1101 = wceq 1568 ∈ wcel 2141 ≠ wne 2956 (class class class)co 7410 ℂcc 11097 0cc0 11099 1c1 11100 + caddc 11102 · cmul 11104 -cneg 11441 / cdiv 11870 ℕcn 12232 ℕ₀cn0 12503 ℤcz 12590 ℚcq 12971 ↑cexp 14097 ℙcprime 16728 pCnt cpc 16895

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9401 df-inf 9402 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-n0 12504 df-z 12591 df-uz 12862 df-q 12972 df-rp 13016 df-fl 13825 df-mod 13903 df-seq 14038 df-exp 14098 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-dvds 16310 df-gcd 16552 df-prm 16729 df-pc 16896

This theorem is referenced by: qexpz 16960 expnprm 16961 dchrisum0flblem1 27648 dchrisum0flblem2 27649 aks4d1p7d1 42817 aks6d1c2p2 42854 aks6d1c7 42919

W3C validator