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Theorem pcexp 16893

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 7439	. . . . 5 ⊢ (𝑥 = 0 → (𝐴↑𝑥) = (𝐴↑0))
2	1	oveq2d 7447	. . . 4 ⊢ (𝑥 = 0 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑0)))
3		oveq1 7438	. . . 4 ⊢ (𝑥 = 0 → (𝑥 · (𝑃 pCnt 𝐴)) = (0 · (𝑃 pCnt 𝐴)))
4	2, 3	eqeq12d 2751	. . 3 ⊢ (𝑥 = 0 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑0)) = (0 · (𝑃 pCnt 𝐴))))
5		oveq2 7439	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴↑𝑥) = (𝐴↑𝑦))
6	5	oveq2d 7447	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑𝑦)))
7		oveq1 7438	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 · (𝑃 pCnt 𝐴)) = (𝑦 · (𝑃 pCnt 𝐴)))
8	6, 7	eqeq12d 2751	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑𝑦)) = (𝑦 · (𝑃 pCnt 𝐴))))
9		oveq2 7439	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝐴↑𝑥) = (𝐴↑(𝑦 + 1)))
10	9	oveq2d 7447	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑(𝑦 + 1))))
11		oveq1 7438	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 · (𝑃 pCnt 𝐴)) = ((𝑦 + 1) · (𝑃 pCnt 𝐴)))
12	10, 11	eqeq12d 2751	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑(𝑦 + 1))) = ((𝑦 + 1) · (𝑃 pCnt 𝐴))))
13		oveq2 7439	. . . . 5 ⊢ (𝑥 = -𝑦 → (𝐴↑𝑥) = (𝐴↑-𝑦))
14	13	oveq2d 7447	. . . 4 ⊢ (𝑥 = -𝑦 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑-𝑦)))
15		oveq1 7438	. . . 4 ⊢ (𝑥 = -𝑦 → (𝑥 · (𝑃 pCnt 𝐴)) = (-𝑦 · (𝑃 pCnt 𝐴)))
16	14, 15	eqeq12d 2751	. . 3 ⊢ (𝑥 = -𝑦 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑-𝑦)) = (-𝑦 · (𝑃 pCnt 𝐴))))
17		oveq2 7439	. . . . 5 ⊢ (𝑥 = 𝑁 → (𝐴↑𝑥) = (𝐴↑𝑁))
18	17	oveq2d 7447	. . . 4 ⊢ (𝑥 = 𝑁 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑𝑁)))
19		oveq1 7438	. . . 4 ⊢ (𝑥 = 𝑁 → (𝑥 · (𝑃 pCnt 𝐴)) = (𝑁 · (𝑃 pCnt 𝐴)))
20	18, 19	eqeq12d 2751	. . 3 ⊢ (𝑥 = 𝑁 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑𝑁)) = (𝑁 · (𝑃 pCnt 𝐴))))
21		pc1 16889	. . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
22	21	adantr 480	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 1) = 0)
23		qcn 13003	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
24	23	ad2antrl 728	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → 𝐴 ∈ ℂ)
25	24	exp0d 14177	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝐴↑0) = 1)
26	25	oveq2d 7447	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt (𝐴↑0)) = (𝑃 pCnt 1))
27		pcqcl 16890	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℤ)
28	27	zcnd 12721	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℂ)
29	28	mul02d 11457	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (0 · (𝑃 pCnt 𝐴)) = 0)
30	22, 26, 29	3eqtr4d 2785	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt (𝐴↑0)) = (0 · (𝑃 pCnt 𝐴)))
31		oveq1 7438	. . . . 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 14106	. . . . . . . . 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 7447	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝑃 pCnt (𝐴↑(𝑦 + 1))) = (𝑃 pCnt ((𝐴↑𝑦) · 𝐴)))
35		simpll 767	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝑃 ∈ ℙ)
36		simplrl 777	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝐴 ∈ ℚ)
37		simplrr 778	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝐴 ≠ 0)
38		nn0z 12636	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℤ)
39	38	adantl 481	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝑦 ∈ ℤ)
40		qexpclz 14119	. . . . . . . . 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 480	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝐴 ∈ ℂ)
43	42, 37, 39	expne0d 14189	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝐴↑𝑦) ≠ 0)
44		pcqmul 16887	. . . . . . . 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 2775	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝑃 pCnt (𝐴↑(𝑦 + 1))) = ((𝑃 pCnt (𝐴↑𝑦)) + (𝑃 pCnt 𝐴)))
47		nn0cn 12534	. . . . . . . 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 11285	. . . . . 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 2751	. . . . 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 11498	. . . . 5 ⊢ ((𝑃 pCnt (𝐴↑𝑦)) = (𝑦 · (𝑃 pCnt 𝐴)) → -(𝑃 pCnt (𝐴↑𝑦)) = -(𝑦 · (𝑃 pCnt 𝐴)))
55		nnnn0 12531	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ₀)
56		expneg 14107	. . . . . . . . 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 7447	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝑃 pCnt (𝐴↑-𝑦)) = (𝑃 pCnt (1 / (𝐴↑𝑦))))
59		simpll 767	. . . . . . . 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 16892	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ ((𝐴↑𝑦) ∈ ℚ ∧ (𝐴↑𝑦) ≠ 0)) → (𝑃 pCnt (1 / (𝐴↑𝑦))) = -(𝑃 pCnt (𝐴↑𝑦)))
63	59, 60, 61, 62	syl12anc 837	. . . . . . 7 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝑃 pCnt (1 / (𝐴↑𝑦))) = -(𝑃 pCnt (𝐴↑𝑦)))
64	58, 63	eqtrd 2775	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝑃 pCnt (𝐴↑-𝑦)) = -(𝑃 pCnt (𝐴↑𝑦)))
65		nncn 12272	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
66		mulneg1 11697	. . . . . . 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 2751	. . . . 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 12717	. 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 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 -cneg 11491 / cdiv 11918 ℕcn 12264 ℕ₀cn0 12524 ℤcz 12611 ℚcq 12988 ↑cexp 14099 ℙcprime 16705 pCnt cpc 16870

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-gcd 16529 df-prm 16706 df-pc 16871

This theorem is referenced by: qexpz 16935 expnprm 16936 dchrisum0flblem1 27567 dchrisum0flblem2 27568 aks4d1p7d1 42064 aks6d1c2p2 42101 aks6d1c7 42166

W3C validator