Theorem exp3val 10011

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 7558	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ 𝑁 = 0) → 1 ∈ ℂ)
2		simp1 944	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → 𝐴 ∈ ℂ)
3		nnuz 9108	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
4		1zzd 8831	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℤ)
5		fvconst2g 5525	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → ((ℕ × {𝐴})‘𝑥) = 𝐴)
6		simpl 108	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → 𝐴 ∈ ℂ)
7	5, 6	eqeltrd 2165	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ) → ((ℕ × {𝐴})‘𝑥) ∈ ℂ)
8		mulcl 7523	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
9	8	adantl 272	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑥 · 𝑦) ∈ ℂ)
10	3, 4, 7, 9	seqf 9934	. . . . . . 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 473	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → seq1( · , (ℕ × {𝐴})):ℕ⟶ℂ)
13		simp2 945	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → 𝑁 ∈ ℤ)
14	13	ad2antrr 473	. . . . . 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 8814	. . . . . 6 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℤ ∧ 0 < 𝑁))
17	14, 15, 16	sylanbrc 409	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → 𝑁 ∈ ℕ)
18	12, 17	ffvelrnd 5449	. . . 4 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ 0 < 𝑁) → (seq1( · , (ℕ × {𝐴}))‘𝑁) ∈ ℂ)
19	11	ad2antrr 473	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → seq1( · , (ℕ × {𝐴})):ℕ⟶ℂ)
20	13	ad2antrr 473	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ∈ ℤ)
21	20	znegcld 8924	. . . . . . 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 498	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 𝑁 = 0)
24		eqcom 2091	. . . . . . . . . . . 12 ⊢ (𝑁 = 0 ↔ 0 = 𝑁)
25	23, 24	sylnib 637	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 0 = 𝑁)
26		ioran 705	. . . . . . . . . . 11 ⊢ (¬ (0 < 𝑁 ∨ 0 = 𝑁) ↔ (¬ 0 < 𝑁 ∧ ¬ 0 = 𝑁))
27	22, 25, 26	sylanbrc 409	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ (0 < 𝑁 ∨ 0 = 𝑁))
28		0zd 8816	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 0 ∈ ℤ)
29		zleloe 8851	. . . . . . . . . . 11 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁)))
30	28, 20, 29	syl2anc 404	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (0 ≤ 𝑁 ↔ (0 < 𝑁 ∨ 0 = 𝑁)))
31	27, 30	mtbird 634	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → ¬ 0 ≤ 𝑁)
32		zltnle 8850	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑁 < 0 ↔ ¬ 0 ≤ 𝑁))
33	20, 28, 32	syl2anc 404	. . . . . . . . 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 8922	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝑁 ∈ ℝ)
36	35	lt0neg1d 8047	. . . . . . . 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 8814	. . . . . . 7 ⊢ (-𝑁 ∈ ℕ ↔ (-𝑁 ∈ ℤ ∧ 0 < -𝑁))
39	21, 37, 38	sylanbrc 409	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → -𝑁 ∈ ℕ)
40	19, 39	ffvelrnd 5449	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (seq1( · , (ℕ × {𝐴}))‘-𝑁) ∈ ℂ)
41	2	ad2antrr 473	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝐴 ∈ ℂ)
42		simpll3 985	. . . . . . 7 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (𝐴 # 0 ∨ 0 ≤ 𝑁))
43	31, 42	ecased 1286	. . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → 𝐴 # 0)
44	41, 43, 39	exp3vallem 10010	. . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (seq1( · , (ℕ × {𝐴}))‘-𝑁) # 0)
45	40, 44	recclapd 8302	. . . 4 ⊢ ((((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) ∧ ¬ 0 < 𝑁) → (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)) ∈ ℂ)
46		0zd 8816	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → 0 ∈ ℤ)
47		simpl2 948	. . . . 5 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → 𝑁 ∈ ℤ)
48		zdclt 8878	. . . . 5 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 0 < 𝑁)
49	46, 47, 48	syl2anc 404	. . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) ∧ ¬ 𝑁 = 0) → DECID 0 < 𝑁)
50	18, 45, 49	ifcldadc 3424	. . 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 8816	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → 0 ∈ ℤ)
52		zdceq 8876	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 0 ∈ ℤ) → DECID 𝑁 = 0)
53	13, 51, 52	syl2anc 404	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℤ ∧ (𝐴 # 0 ∨ 0 ≤ 𝑁)) → DECID 𝑁 = 0)
54	1, 50, 53	ifcldadc 3424	. 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 3461	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
56	55	xpeq2d 4475	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (ℕ × {𝑥}) = (ℕ × {𝐴}))
57	56	seqeq3d 9920	. . . . . 6 ⊢ (𝑥 = 𝐴 → seq1( · , (ℕ × {𝑥})) = seq1( · , (ℕ × {𝐴})))
58	57	fveq1d 5320	. . . . 5 ⊢ (𝑥 = 𝐴 → (seq1( · , (ℕ × {𝑥}))‘𝑦) = (seq1( · , (ℕ × {𝐴}))‘𝑦))
59	57	fveq1d 5320	. . . . . 6 ⊢ (𝑥 = 𝐴 → (seq1( · , (ℕ × {𝑥}))‘-𝑦) = (seq1( · , (ℕ × {𝐴}))‘-𝑦))
60	59	oveq2d 5682	. . . . 5 ⊢ (𝑥 = 𝐴 → (1 / (seq1( · , (ℕ × {𝑥}))‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑦)))
61	58, 60	ifeq12d 3414	. . . 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 3413	. . 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 2095	. . . 4 ⊢ (𝑦 = 𝑁 → (𝑦 = 0 ↔ 𝑁 = 0))
64		breq2 3855	. . . . 5 ⊢ (𝑦 = 𝑁 → (0 < 𝑦 ↔ 0 < 𝑁))
65		fveq2 5318	. . . . 5 ⊢ (𝑦 = 𝑁 → (seq1( · , (ℕ × {𝐴}))‘𝑦) = (seq1( · , (ℕ × {𝐴}))‘𝑁))
66		negeq 7729	. . . . . . 7 ⊢ (𝑦 = 𝑁 → -𝑦 = -𝑁)
67	66	fveq2d 5322	. . . . . 6 ⊢ (𝑦 = 𝑁 → (seq1( · , (ℕ × {𝐴}))‘-𝑦) = (seq1( · , (ℕ × {𝐴}))‘-𝑁))
68	67	oveq2d 5682	. . . . 5 ⊢ (𝑦 = 𝑁 → (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑦)) = (1 / (seq1( · , (ℕ × {𝐴}))‘-𝑁)))
69	64, 65, 68	ifbieq12d 3421	. . . 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 3419	. . 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 10009	. . 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	ovmpt2g 5793	. 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 1220	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 665  DECID wdc 781   ∧ w3a 925   = wceq 1290   ∈ wcel 1439  ifcif 3397  {csn 3450   class class class wbr 3851   × cxp 4449  ⟶wf 5024  ‘cfv 5028  (class class class)co 5666  ℂcc 7402  0cc0 7404  1c1 7405   · cmul 7409   < clt 7576   ≤ cle 7577  -cneg 7708   # cap 8112   / cdiv 8193  ℕcn 8476  ℤcz 8804  seqcseq 9906  ↑cexp 10008

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-mulrcl 7498  ax-addcom 7499  ax-mulcom 7500  ax-addass 7501  ax-mulass 7502  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-1rid 7506  ax-0id 7507  ax-rnegex 7508  ax-precex 7509  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-apti 7514  ax-pre-ltadd 7515  ax-pre-mulgt0 7516  ax-pre-mulext 7517

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-frec 6170  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-reap 8106  df-ap 8113  df-div 8194  df-inn 8477  df-n0 8728  df-z 8805  df-uz 9074  df-iseq 9907  df-seq3 9908  df-exp 10009

This theorem is referenced by:  expnnval  10012  exp0  10013  expnegap0  10017