Theorem exp3val 10930

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 8306	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ 𝑁 = 0) → 1 ∈ ℂ)
2		simp1 1024	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → 𝐴 ∈ ℂ)
3		nnuz 9911	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
4		1zzd 9624	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℤ)
5		fvconst2g 5903	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → ((ℕ × {𝐴})‘𝑥) = 𝐴)
6		simpl 109	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝐴 ∈ ℂ)
7	5, 6	eqeltrd 2311	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → ((ℕ × {𝐴})‘𝑥) ∈ ℂ)
8		mulcl 8270	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
9	8	adantl 277	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) ∈ ℂ)
10	3, 4, 7, 9	seqf 10853	. . . . . . 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 488	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → seq1( · , (ℕ × {𝐴})):ℕ⟶ℂ)
13		simp2 1025	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → 𝑁 ∈ ℤ)
14	13	ad2antrr 488	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → 𝑁 ∈ ℤ)
15		simpr 110	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → 0 < 𝑁)
16		elnnz 9607	. . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
17	14, 15, 16	sylanbrc 417	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → 𝑁 ∈ ℕ)
18	12, 17	ffvelcdmd 5818	. . . 4 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → (seq1( · , (ℕ × {𝐴}))‘𝑁) ∈ ℂ)
19	11	ad2antrr 488	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → seq1( · , (ℕ × {𝐴})):ℕ⟶ℂ)
20	13	ad2antrr 488	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ∈ ℤ)
21	20	znegcld 9723	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ ℤ)
22		simpr 110	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 0 < 𝑁)
23		simplr 529	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 𝑁 = 0)
24		eqcom 2236	. . . . . . . . . . . 12 ⊢ (𝑁 = 0 ↔ 0 = 𝑁)
25	23, 24	sylnib 683	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 0 = 𝑁)
26		ioran 760	. . . . . . . . . . 11 ⊢ (¬ (0 < 𝑁 ∨ 0 = 𝑁) ↔ (¬ 0 < 𝑁 ∧ ¬ 0 = 𝑁))
27	22, 25, 26	sylanbrc 417	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ (0 < 𝑁 ∨ 0 = 𝑁))
28		0zd 9609	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 0 ∈ ℤ)
29		zleloe 9644	. . . . . . . . . . 11 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁)))
30	28, 20, 29	syl2anc 411	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁)))
31	27, 30	mtbird 680	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 0 ≤ 𝑁)
32		zltnle 9643	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑁 < 0 ↔ ¬ 0 ≤ 𝑁))
33	20, 28, 32	syl2anc 411	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 < 0 ↔ ¬ 0 ≤ 𝑁))
34	31, 33	mpbird 167	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 < 0)
35	20	zred 9721	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ∈ ℝ)
36	35	lt0neg1d 8807	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝑁 < 0 ↔ 0 < -𝑁))
37	34, 36	mpbid 147	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 0 < -𝑁)
38		elnnz 9607	. . . . . . 7 ⊢ (-𝑁 ∈ ℕ ↔ (-𝑁 ∈ ℤ ∧ 0 < -𝑁))
39	21, 37, 38	sylanbrc 417	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ ℕ)
40	19, 39	ffvelcdmd 5818	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (seq1( · , (ℕ × {𝐴}))‘-𝑁) ∈ ℂ)
41	2	ad2antrr 488	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝐴 ∈ ℂ)
42		simpll3 1065	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝐴 # 0 ∨ 0 ≤ 𝑁))
43	31, 42	ecased 1386	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝐴 # 0)
44	41, 43, 39	exp3vallem 10929	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (seq1( · , (ℕ × {𝐴}))‘-𝑁) # 0)
45	40, 44	recclapd 9075	. . . 4 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)) ∈ ℂ)
46		0zd 9609	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → 0 ∈ ℤ)
47		simpl2 1028	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → 𝑁 ∈ ℤ)
48		zdclt 9675	. . . . 5 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 0 < 𝑁)
49	46, 47, 48	syl2anc 411	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → DECID 0 < 𝑁)
50	18, 45, 49	ifcldadc 3656	. . 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 9609	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → 0 ∈ ℤ)
52		zdceq 9673	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
53	13, 51, 52	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → DECID 𝑁 = 0)
54	1, 50, 53	ifcldadc 3656	. 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 3705	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
56	55	xpeq2d 4778	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (ℕ × {𝑥}) = (ℕ × {𝐴}))
57	56	seqeq3d 10844	. . . . . 6 ⊢ (𝑥 = 𝐴 → seq1( · , (ℕ × {𝑥})) = seq1( · , (ℕ × {𝐴})))
58	57	fveq1d 5677	. . . . 5 ⊢ (𝑥 = 𝐴 → (seq1( · , (ℕ × {𝑥}))‘𝑦) = (seq1( · , (ℕ × {𝐴}))‘𝑦))
59	57	fveq1d 5677	. . . . . 6 ⊢ (𝑥 = 𝐴 → (seq1( · , (ℕ × {𝑥}))‘-𝑦) = (seq1( · , (ℕ × {𝐴}))‘-𝑦))
60	59	oveq2d 6074	. . . . 5 ⊢ (𝑥 = 𝐴 → (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑦)))
61	58, 60	ifeq12d 3646	. . . 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 3645	. . 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 2241	. . . 4 ⊢ (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0))
64		breq2 4118	. . . . 5 ⊢ (𝑦 = 𝑁 → (0 < 𝑦 ↔ 0 < 𝑁))
65		fveq2 5675	. . . . 5 ⊢ (𝑦 = 𝑁 → (seq1( · , (ℕ × {𝐴}))‘𝑦) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
66		negeq 8483	. . . . . . 7 ⊢ (𝑦 = 𝑁 → -𝑦 = -𝑁)
67	66	fveq2d 5679	. . . . . 6 ⊢ (𝑦 = 𝑁 → (seq1( · , (ℕ × {𝐴}))‘-𝑦) = (seq1( · , (ℕ × {𝐴}))‘-𝑁))
68	67	oveq2d 6074	. . . . 5 ⊢ (𝑦 = 𝑁 → (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))
69	64, 65, 68	ifbieq12d 3653	. . . 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 3651	. . 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 10928	. . 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 6196	. 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 1319	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 104   ↔ wb 105   ∨ wo 716  DECID wdc 842   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  ifcif 3624  {csn 3694   class class class wbr 4114   × cxp 4752  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058  ℂcc 8141  0cc0 8143  1c1 8144   · cmul 8148   < clt 8324   ≤ cle 8325  -cneg 8462   # cap 8873   / cdiv 8966  ℕcn 9257  ℤcz 9597  seqcseq 10836  ↑cexp 10927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8463  df-neg 8464  df-reap 8867  df-ap 8874  df-div 8967  df-inn 9258  df-n0 9517  df-z 9598  df-uz 9875  df-seqfrec 10837  df-exp 10928

This theorem is referenced by:  expnnval  10931  exp0  10932  expnegap0  10936