Theorem stoweidlem19 42311

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 7166	. . . . . 6 ⊢ (𝑛 = 0 → ((𝐹‘𝑡)↑𝑛) = ((𝐹‘𝑡)↑0))
3	2	mpteq2dv 5164	. . . . 5 ⊢ (𝑛 = 0 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑0)))
4	3	eleq1d 2899	. . . 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 7166	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑡)↑𝑛) = ((𝐹‘𝑡)↑𝑚))
7	6	mpteq2dv 5164	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)))
8	7	eleq1d 2899	. . . 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 7166	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((𝐹‘𝑡)↑𝑛) = ((𝐹‘𝑡)↑(𝑚 + 1)))
11	10	mpteq2dv 5164	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))))
12	11	eleq1d 2899	. . . 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 7166	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝐹‘𝑡)↑𝑛) = ((𝐹‘𝑡)↑𝑁))
15	14	mpteq2dv 5164	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑁)))
16	15	eleq1d 2899	. . . 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 551	. . . . . . . . 9 ⊢ (𝜑 → (𝜑 ∧ 𝐹 ∈ 𝐴))
21		eleq1 2902	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ 𝐴 ↔ 𝐹 ∈ 𝐴))
22	21	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ 𝐹 ∈ 𝐴)))
23		feq1 6497	. . . . . . . . . . 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 3569	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐴 → ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ))
27	19, 20, 26	sylc 65	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ)
28	27	ffvelrnda 6853	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
29		recn 10629	. . . . . . 7 ⊢ ((𝐹‘𝑡) ∈ ℝ → (𝐹‘𝑡) ∈ ℂ)
30		exp0 13436	. . . . . . 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 2829	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 = ((𝐹‘𝑡)↑0))
33	18, 32	mpteq2da 5162	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑0)))
34		1re 10643	. . . . 5 ⊢ 1 ∈ ℝ
35		stoweidlem19.5	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
36	35	stoweidlem4 42296	. . . . 5 ⊢ ((𝜑 ∧ 1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
37	34, 36	mpan2 689	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
38	33, 37	eqeltrrd 2916	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑0)) ∈ 𝐴)
39		simpr 487	. . . . 5 ⊢ (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → 𝜑)
40		simpll 765	. . . . 5 ⊢ (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → 𝑚 ∈ ℕ0)
41		simplr 767	. . . . . 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 1915	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ0
44		nfmpt1 5166	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))
45	44	nfel1 2996	. . . . . . . 8 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴
46	18, 43, 45	nf3an 1902	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)
47		simpl1 1187	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝜑)
48		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
49	28	recnd 10671	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ)
50	47, 48, 49	syl2anc 586	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ)
51		simpl2 1188	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℕ0)
52	50, 51	expp1d 13514	. . . . . . 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 5162	. . . . . 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 1127	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
55		simp2 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℕ0)
56	54, 55	reexpcld 13530	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)↑𝑚) ∈ ℝ)
57	47, 51, 48, 56	syl3anc 1367	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)↑𝑚) ∈ ℝ)
58		eqid 2823	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))
59	58	fvmpt2 6781	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐹‘𝑡)↑𝑚) ∈ ℝ) → ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡) = ((𝐹‘𝑡)↑𝑚))
60	59	eqcomd 2829	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐹‘𝑡)↑𝑚) ∈ ℝ) → ((𝐹‘𝑡)↑𝑚) = ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡))
61	48, 57, 60	syl2anc 586	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)↑𝑚) = ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡))
62	61	oveq1d 7173	. . . . . . . 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 5162	. . . . . . 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 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → 𝐹 ∈ 𝐴)
65	44	nfeq2 2997	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑓 = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))
66		stoweidlem19.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝐹
67	66	nfeq2 2997	. . . . . . . . . 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 42298	. . . . . . . . 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 1458	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴)
71	70	3adant2 1127	. . . . . . 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 2915	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝐹‘𝑡)↑𝑚) · (𝐹‘𝑡))) ∈ 𝐴)
73	53, 72	eqeltrd 2915	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))) ∈ 𝐴)
74	39, 40, 42, 73	syl3anc 1367	. . . 4 ⊢ (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))) ∈ 𝐴)
75	74	exp31 422	. . 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 12080	. 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 398   ∧ w3a 1083   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  Ⅎwnfc 2963   ↦ cmpt 5148  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  ℂcc 10537  ℝcr 10538  0cc0 10539  1c1 10540   + caddc 10542   · cmul 10544  ℕ₀cn0 11900  ↑cexp 13432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-n0 11901  df-z 11985  df-uz 12247  df-seq 13373  df-exp 13433

This theorem is referenced by:  stoweidlem40  42332