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Theorem pcexp 16837

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 7398	. . . . 5 ⊢ (𝑥 = 0 → (𝐴↑𝑥) = (𝐴↑0))
2	1	oveq2d 7406	. . . 4 ⊢ (𝑥 = 0 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑0)))
3		oveq1 7397	. . . 4 ⊢ (𝑥 = 0 → (𝑥 · (𝑃 pCnt 𝐴)) = (0 · (𝑃 pCnt 𝐴)))
4	2, 3	eqeq12d 2746	. . 3 ⊢ (𝑥 = 0 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑0)) = (0 · (𝑃 pCnt 𝐴))))
5		oveq2 7398	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴↑𝑥) = (𝐴↑𝑦))
6	5	oveq2d 7406	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑𝑦)))
7		oveq1 7397	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 · (𝑃 pCnt 𝐴)) = (𝑦 · (𝑃 pCnt 𝐴)))
8	6, 7	eqeq12d 2746	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑𝑦)) = (𝑦 · (𝑃 pCnt 𝐴))))
9		oveq2 7398	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝐴↑𝑥) = (𝐴↑(𝑦 + 1)))
10	9	oveq2d 7406	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑(𝑦 + 1))))
11		oveq1 7397	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 · (𝑃 pCnt 𝐴)) = ((𝑦 + 1) · (𝑃 pCnt 𝐴)))
12	10, 11	eqeq12d 2746	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑(𝑦 + 1))) = ((𝑦 + 1) · (𝑃 pCnt 𝐴))))
13		oveq2 7398	. . . . 5 ⊢ (𝑥 = -𝑦 → (𝐴↑𝑥) = (𝐴↑-𝑦))
14	13	oveq2d 7406	. . . 4 ⊢ (𝑥 = -𝑦 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑-𝑦)))
15		oveq1 7397	. . . 4 ⊢ (𝑥 = -𝑦 → (𝑥 · (𝑃 pCnt 𝐴)) = (-𝑦 · (𝑃 pCnt 𝐴)))
16	14, 15	eqeq12d 2746	. . 3 ⊢ (𝑥 = -𝑦 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑-𝑦)) = (-𝑦 · (𝑃 pCnt 𝐴))))
17		oveq2 7398	. . . . 5 ⊢ (𝑥 = 𝑁 → (𝐴↑𝑥) = (𝐴↑𝑁))
18	17	oveq2d 7406	. . . 4 ⊢ (𝑥 = 𝑁 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑𝑁)))
19		oveq1 7397	. . . 4 ⊢ (𝑥 = 𝑁 → (𝑥 · (𝑃 pCnt 𝐴)) = (𝑁 · (𝑃 pCnt 𝐴)))
20	18, 19	eqeq12d 2746	. . 3 ⊢ (𝑥 = 𝑁 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑𝑁)) = (𝑁 · (𝑃 pCnt 𝐴))))
21		pc1 16833	. . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
22	21	adantr 480	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 1) = 0)
23		qcn 12929	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
24	23	ad2antrl 728	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → 𝐴 ∈ ℂ)
25	24	exp0d 14112	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝐴↑0) = 1)
26	25	oveq2d 7406	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt (𝐴↑0)) = (𝑃 pCnt 1))
27		pcqcl 16834	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℤ)
28	27	zcnd 12646	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℂ)
29	28	mul02d 11379	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (0 · (𝑃 pCnt 𝐴)) = 0)
30	22, 26, 29	3eqtr4d 2775	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt (𝐴↑0)) = (0 · (𝑃 pCnt 𝐴)))
31		oveq1 7397	. . . . 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 14040	. . . . . . . . 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 7406	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝑃 pCnt (𝐴↑(𝑦 + 1))) = (𝑃 pCnt ((𝐴↑𝑦) · 𝐴)))
35		simpll 766	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝑃 ∈ ℙ)
36		simplrl 776	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝐴 ∈ ℚ)
37		simplrr 777	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝐴 ≠ 0)
38		nn0z 12561	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℤ)
39	38	adantl 481	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝑦 ∈ ℤ)
40		qexpclz 14053	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0 ∧ 𝑦 ∈ ℤ) → (𝐴↑𝑦) ∈ ℚ)
41	36, 37, 39, 40	syl3anc 1373	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝐴↑𝑦) ∈ ℚ)
42	24	adantr 480	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝐴 ∈ ℂ)
43	42, 37, 39	expne0d 14124	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝐴↑𝑦) ≠ 0)
44		pcqmul 16831	. . . . . . . 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 1381	. . . . . . 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 2765	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝑃 pCnt (𝐴↑(𝑦 + 1))) = ((𝑃 pCnt (𝐴↑𝑦)) + (𝑃 pCnt 𝐴)))
47		nn0cn 12459	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℂ)
48	47	adantl 481	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝑦 ∈ ℂ)
49	28	adantr 480	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝑃 pCnt 𝐴) ∈ ℂ)
50	48, 49	adddirp1d 11207	. . . . . 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 2746	. . . . 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 246	. . . 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 412	. . 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 11420	. . . . 5 ⊢ ((𝑃 pCnt (𝐴↑𝑦)) = (𝑦 · (𝑃 pCnt 𝐴)) → -(𝑃 pCnt (𝐴↑𝑦)) = -(𝑦 · (𝑃 pCnt 𝐴)))
55		nnnn0 12456	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ₀)
56		expneg 14041	. . . . . . . . 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 7406	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝑃 pCnt (𝐴↑-𝑦)) = (𝑃 pCnt (1 / (𝐴↑𝑦))))
59		simpll 766	. . . . . . . 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 16836	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ ((𝐴↑𝑦) ∈ ℚ ∧ (𝐴↑𝑦) ≠ 0)) → (𝑃 pCnt (1 / (𝐴↑𝑦))) = -(𝑃 pCnt (𝐴↑𝑦)))
63	59, 60, 61, 62	syl12anc 836	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝑃 pCnt (1 / (𝐴↑𝑦))) = -(𝑃 pCnt (𝐴↑𝑦)))
64	58, 63	eqtrd 2765	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝑃 pCnt (𝐴↑-𝑦)) = -(𝑃 pCnt (𝐴↑𝑦)))
65		nncn 12201	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
66		mulneg1 11621	. . . . . . 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 2746	. . . . 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 246	. . . 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 412	. . 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 12642	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑁 ∈ ℤ → (𝑃 pCnt (𝐴↑𝑁)) = (𝑁 · (𝑃 pCnt 𝐴))))
72	71	3impia 1117	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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 (class class class)co 7390 ℂcc 11073 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 -cneg 11413 / cdiv 11842 ℕcn 12193 ℕ₀cn0 12449 ℤcz 12536 ℚcq 12914 ↑cexp 14033 ℙcprime 16648 pCnt cpc 16814

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-q 12915 df-rp 12959 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-dvds 16230 df-gcd 16472 df-prm 16649 df-pc 16815

This theorem is referenced by: qexpz 16879 expnprm 16880 dchrisum0flblem1 27426 dchrisum0flblem2 27427 aks4d1p7d1 42077 aks6d1c2p2 42114 aks6d1c7 42179

W3C validator