Theorem exp3val 10478

Description: Value of exponentiation to integer powers. (Contributed by Jim Kingdon, 7-Jun-2020.)

Assertion

Ref	Expression
exp3val	⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))))

Proof of Theorem exp3val

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1cnd 7936	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ 𝑁 = 0) → 1 ∈ ℂ)
2		simp1 992	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → 𝐴 ∈ ℂ)
3		nnuz 9522	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
4		1zzd 9239	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℤ)
5		fvconst2g 5710	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → ((ℕ × {𝐴})‘𝑥) = 𝐴)
6		simpl 108	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝐴 ∈ ℂ)
7	5, 6	eqeltrd 2247	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → ((ℕ × {𝐴})‘𝑥) ∈ ℂ)
8		mulcl 7901	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
9	8	adantl 275	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) ∈ ℂ)
10	3, 4, 7, 9	seqf 10417	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → seq1( · , (ℕ × {𝐴})):ℕ⟶ℂ)
11	2, 10	syl 14	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → seq1( · , (ℕ × {𝐴})):ℕ⟶ℂ)
12	11	ad2antrr 485	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → seq1( · , (ℕ × {𝐴})):ℕ⟶ℂ)
13		simp2 993	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → 𝑁 ∈ ℤ)
14	13	ad2antrr 485	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → 𝑁 ∈ ℤ)
15		simpr 109	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → 0 < 𝑁)
16		elnnz 9222	. . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
17	14, 15, 16	sylanbrc 415	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → 𝑁 ∈ ℕ)
18	12, 17	ffvelrnd 5632	. . . 4 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → (seq1( · , (ℕ × {𝐴}))‘𝑁) ∈ ℂ)
19	11	ad2antrr 485	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → seq1( · , (ℕ × {𝐴})):ℕ⟶ℂ)
20	13	ad2antrr 485	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ∈ ℤ)
21	20	znegcld 9336	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ ℤ)
22		simpr 109	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 0 < 𝑁)
23		simplr 525	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 𝑁 = 0)
24		eqcom 2172	. . . . . . . . . . . 12 ⊢ (𝑁 = 0 ↔ 0 = 𝑁)
25	23, 24	sylnib 671	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 0 = 𝑁)
26		ioran 747	. . . . . . . . . . 11 ⊢ (¬ (0 < 𝑁 ∨ 0 = 𝑁) ↔ (¬ 0 < 𝑁 ∧ ¬ 0 = 𝑁))
27	22, 25, 26	sylanbrc 415	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ (0 < 𝑁 ∨ 0 = 𝑁))
28		0zd 9224	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 0 ∈ ℤ)
29		zleloe 9259	. . . . . . . . . . 11 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁)))
30	28, 20, 29	syl2anc 409	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁)))
31	27, 30	mtbird 668	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 0 ≤ 𝑁)
32		zltnle 9258	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑁 < 0 ↔ ¬ 0 ≤ 𝑁))
33	20, 28, 32	syl2anc 409	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 < 0 ↔ ¬ 0 ≤ 𝑁))
34	31, 33	mpbird 166	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 < 0)
35	20	zred 9334	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ∈ ℝ)
36	35	lt0neg1d 8434	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 < 0 ↔ 0 < -𝑁))
37	34, 36	mpbid 146	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 0 < -𝑁)
38		elnnz 9222	. . . . . . 7 ⊢ (-𝑁 ∈ ℕ ↔ (-𝑁 ∈ ℤ ∧ 0 < -𝑁))
39	21, 37, 38	sylanbrc 415	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ ℕ)
40	19, 39	ffvelrnd 5632	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (seq1( · , (ℕ × {𝐴}))‘-𝑁) ∈ ℂ)
41	2	ad2antrr 485	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝐴 ∈ ℂ)
42		simpll3 1033	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝐴 # 0 ∨ 0 ≤ 𝑁))
43	31, 42	ecased 1344	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝐴 # 0)
44	41, 43, 39	exp3vallem 10477	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (seq1( · , (ℕ × {𝐴}))‘-𝑁) # 0)
45	40, 44	recclapd 8698	. . . 4 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)) ∈ ℂ)
46		0zd 9224	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → 0 ∈ ℤ)
47		simpl2 996	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → 𝑁 ∈ ℤ)
48		zdclt 9289	. . . . 5 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 0 < 𝑁)
49	46, 47, 48	syl2anc 409	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → DECID 0 < 𝑁)
50	18, 45, 49	ifcldadc 3555	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁))) ∈ ℂ)
51		0zd 9224	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → 0 ∈ ℤ)
52		zdceq 9287	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
53	13, 51, 52	syl2anc 409	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → DECID 𝑁 = 0)
54	1, 50, 53	ifcldadc 3555	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))) ∈ ℂ)
55		sneq 3594	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
56	55	xpeq2d 4635	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (ℕ × {𝑥}) = (ℕ × {𝐴}))
57	56	seqeq3d 10409	. . . . . 6 ⊢ (𝑥 = 𝐴 → seq1( · , (ℕ × {𝑥})) = seq1( · , (ℕ × {𝐴})))
58	57	fveq1d 5498	. . . . 5 ⊢ (𝑥 = 𝐴 → (seq1( · , (ℕ × {𝑥}))‘𝑦) = (seq1( · , (ℕ × {𝐴}))‘𝑦))
59	57	fveq1d 5498	. . . . . 6 ⊢ (𝑥 = 𝐴 → (seq1( · , (ℕ × {𝑥}))‘-𝑦) = (seq1( · , (ℕ × {𝐴}))‘-𝑦))
60	59	oveq2d 5869	. . . . 5 ⊢ (𝑥 = 𝐴 → (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑦)))
61	58, 60	ifeq12d 3545	. . . 4 ⊢ (𝑥 = 𝐴 → if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦))) = if(0 < 𝑦, (seq1( · , (ℕ × {𝐴}))‘𝑦), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑦))))
62	61	ifeq2d 3544	. . 3 ⊢ (𝑥 = 𝐴 → if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))) = if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝐴}))‘𝑦), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑦)))))
63		eqeq1 2177	. . . 4 ⊢ (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0))
64		breq2 3993	. . . . 5 ⊢ (𝑦 = 𝑁 → (0 < 𝑦 ↔ 0 < 𝑁))
65		fveq2 5496	. . . . 5 ⊢ (𝑦 = 𝑁 → (seq1( · , (ℕ × {𝐴}))‘𝑦) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
66		negeq 8112	. . . . . . 7 ⊢ (𝑦 = 𝑁 → -𝑦 = -𝑁)
67	66	fveq2d 5500	. . . . . 6 ⊢ (𝑦 = 𝑁 → (seq1( · , (ℕ × {𝐴}))‘-𝑦) = (seq1( · , (ℕ × {𝐴}))‘-𝑁))
68	67	oveq2d 5869	. . . . 5 ⊢ (𝑦 = 𝑁 → (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))
69	64, 65, 68	ifbieq12d 3552	. . . 4 ⊢ (𝑦 = 𝑁 → if(0 < 𝑦, (seq1( · , (ℕ × {𝐴}))‘𝑦), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑦))) = if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁))))
70	63, 69	ifbieq2d 3550	. . 3 ⊢ (𝑦 = 𝑁 → if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝐴}))‘𝑦), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑦)))) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))))
71		df-exp 10476	. . 3 ⊢ ↑ = (𝑥 ∈ ℂ, 𝑦 ∈ ℤ ↦ if(𝑦 = 0, 1, if(0 < 𝑦, (seq1( · , (ℕ × {𝑥}))‘𝑦), (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)))))
72	62, 70, 71	ovmpog 5987	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))) ∈ ℂ) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))))
73	54, 72	syld3an3 1278	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → (𝐴↑𝑁) = if(𝑁 = 0, 1, if(0 < 𝑁, (seq1( · , (ℕ × {𝐴}))‘𝑁), (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 703  DECID wdc 829   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  ifcif 3526  {csn 3583   class class class wbr 3989   × cxp 4609  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853  ℂcc 7772  0cc0 7774  1c1 7775   · cmul 7779   < clt 7954   ≤ cle 7955  -cneg 8091   # cap 8500   / cdiv 8589  ℕcn 8878  ℤcz 9212  seqcseq 10401  ↑cexp 10475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-seqfrec 10402  df-exp 10476

This theorem is referenced by:  expnnval  10479  exp0  10480  expnegap0  10484