Theorem stoweidlem19 46855

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 7425	. . . . . 6 ⊢ (𝑛 = 0 → ((𝐹‘𝑡)↑𝑛) = ((𝐹‘𝑡)↑0))
3	2	mpteq2dv 5203	. . . . 5 ⊢ (𝑛 = 0 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑0)))
4	3	eleq1d 2847	. . . 4 ⊢ (𝑛 = 0 → ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑0)) ∈ 𝐴))
5	4	imbi2d 343	. . 3 ⊢ (𝑛 = 0 → ((𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑0)) ∈ 𝐴)))
6		oveq2 7425	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑡)↑𝑛) = ((𝐹‘𝑡)↑𝑚))
7	6	mpteq2dv 5203	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)))
8	7	eleq1d 2847	. . . 4 ⊢ (𝑛 = 𝑚 → ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴))
9	8	imbi2d 343	. . 3 ⊢ (𝑛 = 𝑚 → ((𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)))
10		oveq2 7425	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((𝐹‘𝑡)↑𝑛) = ((𝐹‘𝑡)↑(𝑚 + 1)))
11	10	mpteq2dv 5203	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))))
12	11	eleq1d 2847	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))) ∈ 𝐴))
13	12	imbi2d 343	. . 3 ⊢ (𝑛 = (𝑚 + 1) → ((𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) ∈ 𝐴) ↔ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))) ∈ 𝐴)))
14		oveq2 7425	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝐹‘𝑡)↑𝑛) = ((𝐹‘𝑡)↑𝑁))
15	14	mpteq2dv 5203	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑁)))
16	15	eleq1d 2847	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑁)) ∈ 𝐴))
17	16	imbi2d 343	. . 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 558	. . . . . . . . 9 ⊢ (𝜑 → (𝜑 ∧ 𝐹 ∈ 𝐴))
21		eleq1 2850	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ 𝐴 ↔ 𝐹 ∈ 𝐴))
22	21	anbi2d 642	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐹 ∈ 𝐴)))
23		feq1 6684	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → (𝑓:𝑇⟶ℝ ↔ 𝐹:𝑇⟶ℝ))
24	22, 23	imbi12d 347	. . . . . . . . . 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 3520	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐴 → ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ))
27	19, 20, 26	sylc 66	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ)
28	27	ffvelcdmda 7081	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
29		recn 11218	. . . . . . 7 ⊢ ((𝐹‘𝑡) ∈ ℝ → (𝐹‘𝑡) ∈ ℂ)
30		exp0 14133	. . . . . . 7 ⊢ ((𝐹‘𝑡) ∈ ℂ → ((𝐹‘𝑡)↑0) = 1)
31	28, 29, 30	3syl 19	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)↑0) = 1)
32	31	eqcomd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 = ((𝐹‘𝑡)↑0))
33	18, 32	mpteq2da 5201	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑0)))
34		1re 11236	. . . . 5 ⊢ 1 ∈ ℝ
35		stoweidlem19.5	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
36	35	stoweidlem4 46840	. . . . 5 ⊢ ((𝜑 ∧ 1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
37	34, 36	mpan2 704	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
38	33, 37	eqeltrrd 2863	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑0)) ∈ 𝐴)
39		simpr 490	. . . . 5 ⊢ (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → 𝜑)
40		simpll 779	. . . . 5 ⊢ (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → 𝑚 ∈ ℕ0)
41		simplr 781	. . . . . 6 ⊢ (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴))
42	39, 41	mpd 16	. . . . 5 ⊢ (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)
43		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ0
44		nfmpt1 5208	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))
45	44	nfel1 2940	. . . . . . . 8 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴
46	18, 43, 45	nf3an 1934	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)
47		simpl1 1210	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝜑)
48		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
49	28	recnd 11265	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ)
50	47, 48, 49	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ)
51		simpl2 1211	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℕ0)
52	50, 51	expp1d 14215	. . . . . . 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 5201	. . . . . 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 1149	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
55		simp2 1155	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℕ0)
56	54, 55	reexpcld 14231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)↑𝑚) ∈ ℝ)
57	47, 51, 48, 56	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)↑𝑚) ∈ ℝ)
58		eqid 2762	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))
59	58	fvmpt2 7002	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐹‘𝑡)↑𝑚) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡) = ((𝐹‘𝑡)↑𝑚))
60	59	eqcomd 2768	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐹‘𝑡)↑𝑚) ∈ ℝ) → ((𝐹‘𝑡)↑𝑚) = ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡))
61	48, 57, 60	syl2anc 596	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)↑𝑚) = ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡))
62	61	oveq1d 7432	. . . . . . . 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 5201	. . . . . . 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 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → 𝐹 ∈ 𝐴)
65	44	nfeq2 2941	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑓 = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))
66		stoweidlem19.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝐹
67	66	nfeq2 2941	. . . . . . . . . 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 46842	. . . . . . . . 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 1491	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴)
71	70	3adant2 1149	. . . . . . 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 2862	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝐹‘𝑡)↑𝑚) · (𝐹‘𝑡))) ∈ 𝐴)
73	53, 72	eqeltrd 2862	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))) ∈ 𝐴)
74	39, 40, 42, 73	syl3anc 1398	. . . 4 ⊢ (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))) ∈ 𝐴)
75	74	exp31 425	. . 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 12720	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑁)) ∈ 𝐴))
77	1, 76	mpcom 39	1 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑁)) ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2145  Ⅎwnfc 2909   ↦ cmpt 5190  ⟶wf 6533  ‘cfv 6537  (class class class)co 7417  ℂcc 11126  ℝcr 11127  0cc0 11128  1c1 11129   + caddc 11131   · cmul 11133  ℕ₀cn0 12532  ↑cexp 14129

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-en 8957  df-dom 8958  df-sdom 8959  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-n0 12533  df-z 12620  df-uz 12892  df-seq 14070  df-exp 14130

This theorem is used by:  stoweidlem40  46876