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Theorem pcexp 16925

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 7420	. . . . 5 ⊢ (𝑥 = 0 → (𝐴↑𝑥) = (𝐴↑0))
2	1	oveq2d 7428	. . . 4 ⊢ (𝑥 = 0 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑0)))
3		oveq1 7419	. . . 4 ⊢ (𝑥 = 0 → (𝑥 · (𝑃 pCnt 𝐴)) = (0 · (𝑃 pCnt 𝐴)))
4	2, 3	eqeq12d 2778	. . 3 ⊢ (𝑥 = 0 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑0)) = (0 · (𝑃 pCnt 𝐴))))
5		oveq2 7420	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐴↑𝑥) = (𝐴↑𝑦))
6	5	oveq2d 7428	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑𝑦)))
7		oveq1 7419	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 · (𝑃 pCnt 𝐴)) = (𝑦 · (𝑃 pCnt 𝐴)))
8	6, 7	eqeq12d 2778	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑𝑦)) = (𝑦 · (𝑃 pCnt 𝐴))))
9		oveq2 7420	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝐴↑𝑥) = (𝐴↑(𝑦 + 1)))
10	9	oveq2d 7428	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑(𝑦 + 1))))
11		oveq1 7419	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 · (𝑃 pCnt 𝐴)) = ((𝑦 + 1) · (𝑃 pCnt 𝐴)))
12	10, 11	eqeq12d 2778	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑(𝑦 + 1))) = ((𝑦 + 1) · (𝑃 pCnt 𝐴))))
13		oveq2 7420	. . . . 5 ⊢ (𝑥 = -𝑦 → (𝐴↑𝑥) = (𝐴↑-𝑦))
14	13	oveq2d 7428	. . . 4 ⊢ (𝑥 = -𝑦 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑-𝑦)))
15		oveq1 7419	. . . 4 ⊢ (𝑥 = -𝑦 → (𝑥 · (𝑃 pCnt 𝐴)) = (-𝑦 · (𝑃 pCnt 𝐴)))
16	14, 15	eqeq12d 2778	. . 3 ⊢ (𝑥 = -𝑦 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑-𝑦)) = (-𝑦 · (𝑃 pCnt 𝐴))))
17		oveq2 7420	. . . . 5 ⊢ (𝑥 = 𝑁 → (𝐴↑𝑥) = (𝐴↑𝑁))
18	17	oveq2d 7428	. . . 4 ⊢ (𝑥 = 𝑁 → (𝑃 pCnt (𝐴↑𝑥)) = (𝑃 pCnt (𝐴↑𝑁)))
19		oveq1 7419	. . . 4 ⊢ (𝑥 = 𝑁 → (𝑥 · (𝑃 pCnt 𝐴)) = (𝑁 · (𝑃 pCnt 𝐴)))
20	18, 19	eqeq12d 2778	. . 3 ⊢ (𝑥 = 𝑁 → ((𝑃 pCnt (𝐴↑𝑥)) = (𝑥 · (𝑃 pCnt 𝐴)) ↔ (𝑃 pCnt (𝐴↑𝑁)) = (𝑁 · (𝑃 pCnt 𝐴))))
21		pc1 16921	. . . . 5 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
22	21	adantr 485	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 1) = 0)
23		qcn 12993	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
24	23	ad2antrl 740	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → 𝐴 ∈ ℂ)
25	24	exp0d 14183	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝐴↑0) = 1)
26	25	oveq2d 7428	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt (𝐴↑0)) = (𝑃 pCnt 1))
27		pcqcl 16922	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℤ)
28	27	zcnd 12707	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt 𝐴) ∈ ℂ)
29	28	mul02d 11414	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (0 · (𝑃 pCnt 𝐴)) = 0)
30	22, 26, 29	3eqtr4d 2807	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑃 pCnt (𝐴↑0)) = (0 · (𝑃 pCnt 𝐴)))
31		oveq1 7419	. . . . 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 14111	. . . . . . . . 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 7428	. . . . . . 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 12621	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℤ)
39	38	adantl 486	. . . . . . . . 9 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → 𝑦 ∈ ℤ)
40		qexpclz 14124	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℚ ∧ 𝐴 ≠ 0 ∧ 𝑦 ∈ ℤ) → (𝐴↑𝑦) ∈ ℚ)
41	36, 37, 39, 40	syl3anc 1397	. . . . . . . 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 14195	. . . . . . . 8 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝐴↑𝑦) ≠ 0)
44		pcqmul 16919	. . . . . . . 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 1405	. . . . . . 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 2797	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ₀) → (𝑃 pCnt (𝐴↑(𝑦 + 1))) = ((𝑃 pCnt (𝐴↑𝑦)) + (𝑃 pCnt 𝐴)))
47		nn0cn 12520	. . . . . . . 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 11241	. . . . . 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 2778	. . . . 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 11455	. . . . 5 ⊢ ((𝑃 pCnt (𝐴↑𝑦)) = (𝑦 · (𝑃 pCnt 𝐴)) → -(𝑃 pCnt (𝐴↑𝑦)) = -(𝑦 · (𝑃 pCnt 𝐴)))
55		nnnn0 12517	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ₀)
56		expneg 14112	. . . . . . . . 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 7428	. . . . . . 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 16924	. . . . . . . 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 2797	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) ∧ 𝑦 ∈ ℕ) → (𝑃 pCnt (𝐴↑-𝑦)) = -(𝑃 pCnt (𝐴↑𝑦)))
65		nncn 12247	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
66		mulneg1 11656	. . . . . . 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 2778	. . . . 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 12703	. 2 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0)) → (𝑁 ∈ ℤ → (𝑃 pCnt (𝐴↑𝑁)) = (𝑁 · (𝑃 pCnt 𝐴))))
72	71	3impia 1134	1 ⊢ ((𝑃 ∈ ℙ ∧ (𝐴 ∈ ℚ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ) → (𝑃 pCnt (𝐴↑𝑁)) = (𝑁 · (𝑃 pCnt 𝐴)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 400 ∧ w3a 1102 = wceq 1569 ∈ wcel 2142 ≠ wne 2957 (class class class)co 7412 ℂcc 11104 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 -cneg 11448 / cdiv 11877 ℕcn 12239 ℕ₀cn0 12510 ℤcz 12597 ℚcq 12978 ↑cexp 14104 ℙcprime 16735 pCnt cpc 16902

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 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 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 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 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-er 8692 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9400 df-inf 9401 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-n0 12511 df-z 12598 df-uz 12869 df-q 12979 df-rp 13023 df-fl 13832 df-mod 13910 df-seq 14045 df-exp 14105 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-dvds 16317 df-gcd 16559 df-prm 16736 df-pc 16903

This theorem is used by: qexpz 16967 expnprm 16968 dchrisum0flblem1 27683 dchrisum0flblem2 27684 aks4d1p7d1 42877 aks6d1c2p2 42914 aks6d1c7 42979

W3C validator