Theorem stoweidlem19 46659

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 7419	. . . . . 6 ⊢ (𝑛 = 0 → ((𝐹‘𝑡)↑𝑛) = ((𝐹‘𝑡)↑0))
3	2	mpteq2dv 5209	. . . . 5 ⊢ (𝑛 = 0 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑0)))
4	3	eleq1d 2854	. . . 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 7419	. . . . . 6 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑡)↑𝑛) = ((𝐹‘𝑡)↑𝑚))
7	6	mpteq2dv 5209	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)))
8	7	eleq1d 2854	. . . 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 7419	. . . . . 6 ⊢ (𝑛 = (𝑚 + 1) → ((𝐹‘𝑡)↑𝑛) = ((𝐹‘𝑡)↑(𝑚 + 1)))
11	10	mpteq2dv 5209	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))))
12	11	eleq1d 2854	. . . 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 7419	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((𝐹‘𝑡)↑𝑛) = ((𝐹‘𝑡)↑𝑁))
15	14	mpteq2dv 5209	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑛)) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑁)))
16	15	eleq1d 2854	. . . 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 557	. . . . . . . . 9 ⊢ (𝜑 → (𝜑 ∧ 𝐹 ∈ 𝐴))
21		eleq1 2857	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (𝑓 ∈ 𝐴 ↔ 𝐹 ∈ 𝐴))
22	21	anbi2d 641	. . . . . . . . . . 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 3531	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐴 → ((𝜑 ∧ 𝐹 ∈ 𝐴) → 𝐹:𝑇⟶ℝ))
27	19, 20, 26	sylc 66	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ)
28	27	ffvelcdmda 7080	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
29		recn 11190	. . . . . . 7 ⊢ ((𝐹‘𝑡) ∈ ℝ → (𝐹‘𝑡) ∈ ℂ)
30		exp0 14101	. . . . . . 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 2775	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 = ((𝐹‘𝑡)↑0))
33	18, 32	mpteq2da 5207	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑0)))
34		1re 11208	. . . . 5 ⊢ 1 ∈ ℝ
35		stoweidlem19.5	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
36	35	stoweidlem4 46644	. . . . 5 ⊢ ((𝜑 ∧ 1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
37	34, 36	mpan2 703	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
38	33, 37	eqeltrrd 2870	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑0)) ∈ 𝐴)
39		simpr 489	. . . . 5 ⊢ (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → 𝜑)
40		simpll 778	. . . . 5 ⊢ (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → 𝑚 ∈ ℕ0)
41		simplr 780	. . . . . 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 1941	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ0
44		nfmpt1 5214	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))
45	44	nfel1 2947	. . . . . . . 8 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴
46	18, 43, 45	nf3an 1928	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)
47		simpl1 1208	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝜑)
48		simpr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
49	28	recnd 11237	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ)
50	47, 48, 49	syl2anc 595	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℂ)
51		simpl2 1209	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℕ0)
52	50, 51	expp1d 14183	. . . . . . 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 5207	. . . . . 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 1147	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
55		simp2 1153	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℕ0)
56	54, 55	reexpcld 14199	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)↑𝑚) ∈ ℝ)
57	47, 51, 48, 56	syl3anc 1396	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)↑𝑚) ∈ ℝ)
58		eqid 2769	. . . . . . . . . . . 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 2775	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐹‘𝑡)↑𝑚) ∈ ℝ) → ((𝐹‘𝑡)↑𝑚) = ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡))
61	48, 57, 60	syl2anc 595	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡)↑𝑚) = ((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡))
62	61	oveq1d 7426	. . . . . . . 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 5207	. . . . . . 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 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → 𝐹 ∈ 𝐴)
65	44	nfeq2 2948	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑓 = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))
66		stoweidlem19.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝐹
67	66	nfeq2 2948	. . . . . . . . . 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 46646	. . . . . . . . 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 1488	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚))‘𝑡) · (𝐹‘𝑡))) ∈ 𝐴)
71	70	3adant2 1147	. . . . . . 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 2869	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ (((𝐹‘𝑡)↑𝑚) · (𝐹‘𝑡))) ∈ 𝐴)
73	53, 72	eqeltrd 2869	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0 ∧ (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))) ∈ 𝐴)
74	39, 40, 42, 73	syl3anc 1396	. . . 4 ⊢ (((𝑚 ∈ ℕ0 ∧ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑚)) ∈ 𝐴)) ∧ 𝜑) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑(𝑚 + 1))) ∈ 𝐴)
75	74	exp31 424	. . 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 12691	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑁)) ∈ 𝐴))
77	1, 76	mpcom 39	1 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡)↑𝑁)) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567  Ⅎwnf 1810   ∈ wcel 2149  Ⅎwnfc 2916   ↦ cmpt 5196  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  ℂcc 11098  ℝcr 11099  0cc0 11100  1c1 11101   + caddc 11103   · cmul 11105  ℕ₀cn0 12504  ↑cexp 14097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  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 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-n0 12505  df-z 12592  df-uz 12863  df-seq 14038  df-exp 14098

This theorem is referenced by:  stoweidlem40  46680