Theorem stoweidlem19 45940

Description: If a set of real functions is closed under multiplication and it contains constants, then it is closed under finite exponentiation. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem19.1	⊢ Ⅎ𝑡𝐹
stoweidlem19.2	⊢ Ⅎ𝑡𝜑
stoweidlem19.3	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem19.4	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem19.5	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
stoweidlem19.6	⊢ (𝜑 → 𝐹 ∈ 𝐴)
stoweidlem19.7	⊢ (𝜑 → 𝑁 ∈ ℕ0)

Assertion

Ref	Expression
stoweidlem19	⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑁)) ∈ 𝐴)

Distinct variable groups:   𝑓,𝑔,𝑡,𝐴   𝑓,𝐹,𝑔   𝑇,𝑓,𝑔,𝑡   𝜑,𝑓,𝑔   𝑡,𝑁   𝑥,𝑡,𝐴   𝑥,𝑇   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑡)   𝐹(𝑥,𝑡)   𝑁(𝑥,𝑓,𝑔)

Proof of Theorem stoweidlem19

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem19.7	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
2		oveq2 7456	. . . . . 6 ⊢ (𝑛 = 0 → ((𝐹‘𝑡)↑𝑛) = ((𝐹‘𝑡)↑0))
3	2	mpteq2dv 5268	. . . . 5 ⊢ (𝑛 = 0 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑0)))
4	3	eleq1d 2829	. . . 4 ⊢ (𝑛 = 0 → ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑0)) ∈ 𝐴))
5	4	imbi2d 340	. . 3 ⊢ (𝑛 = 0 → ((𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑0)) ∈ 𝐴)))
6		oveq2 7456	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑡)↑𝑛) = ((𝐹‘𝑡)↑𝑚))
7	6	mpteq2dv 5268	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)))
8	7	eleq1d 2829	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴))
9	8	imbi2d 340	. . 3 ⊢ (𝑛 = 𝑚 → ((𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)))
10		oveq2 7456	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((𝐹‘𝑡)↑𝑛) = ((𝐹‘𝑡)↑(𝑚 + 1)))
11	10	mpteq2dv 5268	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))))
12	11	eleq1d 2829	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))) ∈ 𝐴))
13	12	imbi2d 340	. . 3 ⊢ (𝑛 = (𝑚 + 1) → ((𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))) ∈ 𝐴)))
14		oveq2 7456	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝐹‘𝑡)↑𝑛) = ((𝐹‘𝑡)↑𝑁))
15	14	mpteq2dv 5268	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑁)))
16	15	eleq1d 2829	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑁)) ∈ 𝐴))
17	16	imbi2d 340	. . 3 ⊢ (𝑛 = 𝑁 → ((𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑁)) ∈ 𝐴)))
18		stoweidlem19.2	. . . . 5 ⊢ Ⅎ𝑡𝜑
19		stoweidlem19.6	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
20	19	ancli 548	. . . . . . . . 9 ⊢ (𝜑 → (𝜑 ∧ 𝐹 ∈ 𝐴))
21		eleq1 2832	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ 𝐴 ↔ 𝐹 ∈ 𝐴))
22	21	anbi2d 629	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐹 ∈ 𝐴)))
23		feq1 6728	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑓:𝑇⟶ℝ ↔ 𝐹:𝑇⟶ℝ))
24	22, 23	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ)))
25		stoweidlem19.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
26	24, 25	vtoclg 3566	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐴 → ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ))
27	19, 20, 26	sylc 65	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ)
28	27	ffvelcdmda 7118	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
29		recn 11274	. . . . . . 7 ⊢ ((𝐹‘𝑡) ∈ ℝ → (𝐹‘𝑡) ∈ ℂ)
30		exp0 14116	. . . . . . 7 ⊢ ((𝐹‘𝑡) ∈ ℂ → ((𝐹‘𝑡)↑0) = 1)
31	28, 29, 30	3syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)↑0) = 1)
32	31	eqcomd 2746	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 = ((𝐹‘𝑡)↑0))
33	18, 32	mpteq2da 5264	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑0)))
34		1re 11290	. . . . 5 ⊢ 1 ∈ ℝ
35		stoweidlem19.5	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
36	35	stoweidlem4 45925	. . . . 5 ⊢ ((𝜑 ∧ 1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
37	34, 36	mpan2 690	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
38	33, 37	eqeltrrd 2845	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑0)) ∈ 𝐴)
39		simpr 484	. . . . 5 ⊢ (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → 𝜑)
40		simpll 766	. . . . 5 ⊢ (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → 𝑚 ∈ ℕ0)
41		simplr 768	. . . . . 6 ⊢ (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴))
42	39, 41	mpd 15	. . . . 5 ⊢ (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)
43		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ0
44		nfmpt1 5274	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))
45	44	nfel1 2925	. . . . . . . 8 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴
46	18, 43, 45	nf3an 1900	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)
47		simpl1 1191	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝜑)
48		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
49	28	recnd 11318	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ)
50	47, 48, 49	syl2anc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ)
51		simpl2 1192	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℕ0)
52	50, 51	expp1d 14197	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)↑(𝑚 + 1)) = (((𝐹‘𝑡)↑𝑚) · (𝐹‘𝑡)))
53	46, 52	mpteq2da 5264	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))) = (𝑡 ∈ 𝑇 ↦ (((𝐹‘𝑡)↑𝑚) · (𝐹‘𝑡))))
54	28	3adant2 1131	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
55		simp2 1137	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℕ0)
56	54, 55	reexpcld 14213	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)↑𝑚) ∈ ℝ)
57	47, 51, 48, 56	syl3anc 1371	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)↑𝑚) ∈ ℝ)
58		eqid 2740	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))
59	58	fvmpt2 7040	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐹‘𝑡)↑𝑚) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡) = ((𝐹‘𝑡)↑𝑚))
60	59	eqcomd 2746	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐹‘𝑡)↑𝑚) ∈ ℝ) → ((𝐹‘𝑡)↑𝑚) = ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡))
61	48, 57, 60	syl2anc 583	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)↑𝑚) = ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡))
62	61	oveq1d 7463	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (((𝐹‘𝑡)↑𝑚) · (𝐹‘𝑡)) = (((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡) · (𝐹‘𝑡)))
63	46, 62	mpteq2da 5264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝐹‘𝑡)↑𝑚) · (𝐹‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡) · (𝐹‘𝑡))))
64	19	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → 𝐹 ∈ 𝐴)
65	44	nfeq2 2926	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑓 = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))
66		stoweidlem19.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝐹
67	66	nfeq2 2926	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑔 = 𝐹
68		stoweidlem19.4	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
69	65, 67, 68	stoweidlem6 45927	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴)
70	64, 69	mpd3an3 1462	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴)
71	70	3adant2 1131	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴)
72	63, 71	eqeltrd 2844	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝐹‘𝑡)↑𝑚) · (𝐹‘𝑡))) ∈ 𝐴)
73	53, 72	eqeltrd 2844	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))) ∈ 𝐴)
74	39, 40, 42, 73	syl3anc 1371	. . . 4 ⊢ (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))) ∈ 𝐴)
75	74	exp31 419	. . 3 ⊢ (𝑚 ∈ ℕ0 → ((𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))) ∈ 𝐴)))
76	5, 9, 13, 17, 38, 75	nn0ind 12738	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑁)) ∈ 𝐴))
77	1, 76	mpcom 38	1 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑁)) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2108  Ⅎwnfc 2893   ↦ cmpt 5249  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  ℂcc 11182  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187   · cmul 11189  ℕ₀cn0 12553  ↑cexp 14112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-seq 14053  df-exp 14113

This theorem is referenced by:  stoweidlem40  45961