fprodexp - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fprodexp		Structured version Visualization version GIF version

Theorem fprodexp 44297

Description: Positive integer exponentiation of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
fprodexp.kph	⊢ Ⅎ𝑘𝜑
fprodexp.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
fprodexp.a	⊢ (𝜑 → 𝐴 ∈ Fin)
fprodexp.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)

**Assertion**
Ref	Expression
fprodexp	⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (𝐵↑𝑁) = (∏𝑘 ∈ 𝐴 𝐵↑𝑁))

Distinct variable groups: 𝐴,𝑘 𝑘,𝑁 Allowed substitution hints: 𝜑(𝑘) 𝐵(𝑘)

Proof of Theorem fprodexp Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prodeq1 15850	. . 3 ⊢ (𝑥 = ∅ → ∏𝑘 ∈ 𝑥 (𝐵↑𝑁) = ∏𝑘 ∈ ∅ (𝐵↑𝑁))
2		prodeq1 15850	. . . 4 ⊢ (𝑥 = ∅ → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
3	2	oveq1d 7421	. . 3 ⊢ (𝑥 = ∅ → (∏𝑘 ∈ 𝑥 𝐵↑𝑁) = (∏𝑘 ∈ ∅ 𝐵↑𝑁))
4	1, 3	eqeq12d 2749	. 2 ⊢ (𝑥 = ∅ → (∏𝑘 ∈ 𝑥 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑥 𝐵↑𝑁) ↔ ∏𝑘 ∈ ∅ (𝐵↑𝑁) = (∏𝑘 ∈ ∅ 𝐵↑𝑁)))
5		prodeq1 15850	. . 3 ⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 (𝐵↑𝑁) = ∏𝑘 ∈ 𝑦 (𝐵↑𝑁))
6		prodeq1 15850	. . . 4 ⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ 𝑦 𝐵)
7	6	oveq1d 7421	. . 3 ⊢ (𝑥 = 𝑦 → (∏𝑘 ∈ 𝑥 𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁))
8	5, 7	eqeq12d 2749	. 2 ⊢ (𝑥 = 𝑦 → (∏𝑘 ∈ 𝑥 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑥 𝐵↑𝑁) ↔ ∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁)))
9		prodeq1 15850	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑥 (𝐵↑𝑁) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵↑𝑁))
10		prodeq1 15850	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
11	10	oveq1d 7421	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∏𝑘 ∈ 𝑥 𝐵↑𝑁) = (∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵↑𝑁))
12	9, 11	eqeq12d 2749	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∏𝑘 ∈ 𝑥 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑥 𝐵↑𝑁) ↔ ∏𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵↑𝑁) = (∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵↑𝑁)))
13		prodeq1 15850	. . 3 ⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ 𝑥 (𝐵↑𝑁) = ∏𝑘 ∈ 𝐴 (𝐵↑𝑁))
14		prodeq1 15850	. . . 4 ⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ 𝐴 𝐵)
15	14	oveq1d 7421	. . 3 ⊢ (𝑥 = 𝐴 → (∏𝑘 ∈ 𝑥 𝐵↑𝑁) = (∏𝑘 ∈ 𝐴 𝐵↑𝑁))
16	13, 15	eqeq12d 2749	. 2 ⊢ (𝑥 = 𝐴 → (∏𝑘 ∈ 𝑥 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑥 𝐵↑𝑁) ↔ ∏𝑘 ∈ 𝐴 (𝐵↑𝑁) = (∏𝑘 ∈ 𝐴 𝐵↑𝑁)))
17		fprodexp.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
18	17	nn0zd 12581	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
19		1exp 14054	. . . . 5 ⊢ (𝑁 ∈ ℤ → (1↑𝑁) = 1)
20	18, 19	syl 17	. . . 4 ⊢ (𝜑 → (1↑𝑁) = 1)
21	20	eqcomd 2739	. . 3 ⊢ (𝜑 → 1 = (1↑𝑁))
22		prod0 15884	. . . 4 ⊢ ∏𝑘 ∈ ∅ (𝐵↑𝑁) = 1
23	22	a1i 11	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ ∅ (𝐵↑𝑁) = 1)
24		prod0 15884	. . . . 5 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
25	24	oveq1i 7416	. . . 4 ⊢ (∏𝑘 ∈ ∅ 𝐵↑𝑁) = (1↑𝑁)
26	25	a1i 11	. . 3 ⊢ (𝜑 → (∏𝑘 ∈ ∅ 𝐵↑𝑁) = (1↑𝑁))
27	21, 23, 26	3eqtr4d 2783	. 2 ⊢ (𝜑 → ∏𝑘 ∈ ∅ (𝐵↑𝑁) = (∏𝑘 ∈ ∅ 𝐵↑𝑁))
28		fprodexp.kph	. . . . . . . . 9 ⊢ Ⅎ𝑘𝜑
29		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))
30	28, 29	nfan 1903	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)))
31		fprodexp.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ Fin)
32	31	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝐴 ∈ Fin)
33		simpr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ 𝐴)
34		ssfi 9170	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin)
35	32, 33, 34	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin)
36	35	adantrr 716	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
37		simpll 766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑘 ∈ 𝑦) → 𝜑)
38	33	sselda 3982	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝐴)
39		fprodexp.b	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
40	37, 38, 39	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℂ)
41	40	adantlrr 720	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℂ)
42	30, 36, 41	fprodclf 15933	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ 𝑦 𝐵 ∈ ℂ)
43		simpl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝜑)
44		simprr 772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
45	44	eldifad 3960	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
46		nfv 1918	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 𝑧 ∈ 𝐴
47	28, 46	nfan 1903	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑧 ∈ 𝐴)
48		nfcsb1v 3918	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
49	48	nfel1 2920	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ
50	47, 49	nfim 1900	. . . . . . . . 9 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
51		eleq1w 2817	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → (𝑘 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
52	51	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑧 ∈ 𝐴)))
53		csbeq1a 3907	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
54	53	eleq1d 2819	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))
55	52, 54	imbi12d 345	. . . . . . . . 9 ⊢ (𝑘 = 𝑧 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)))
56	50, 55, 39	chvarfv 2234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
57	43, 45, 56	syl2anc 585	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
58	17	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑁 ∈ ℕ₀)
59		mulexp 14064	. . . . . . 7 ⊢ ((∏𝑘 ∈ 𝑦 𝐵 ∈ ℂ ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ ∧ 𝑁 ∈ ℕ₀) → ((∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)↑𝑁) = ((∏𝑘 ∈ 𝑦 𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁)))
60	42, 57, 58, 59	syl3anc 1372	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)↑𝑁) = ((∏𝑘 ∈ 𝑦 𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁)))
61	60	eqcomd 2739	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((∏𝑘 ∈ 𝑦 𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁)) = ((∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)↑𝑁))
62	61	adantr 482	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁)) → ((∏𝑘 ∈ 𝑦 𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁)) = ((∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)↑𝑁))
63		nfcv 2904	. . . . . . . 8 ⊢ Ⅎ𝑘↑
64		nfcv 2904	. . . . . . . 8 ⊢ Ⅎ𝑘𝑁
65	48, 63, 64	nfov 7436	. . . . . . 7 ⊢ Ⅎ𝑘(⦋𝑧 / 𝑘⦌𝐵↑𝑁)
66	44	eldifbd 3961	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
67	17	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑘 ∈ 𝑦) → 𝑁 ∈ ℕ₀)
68	40, 67	expcld 14108	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑘 ∈ 𝑦) → (𝐵↑𝑁) ∈ ℂ)
69	68	adantlrr 720	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → (𝐵↑𝑁) ∈ ℂ)
70	53	oveq1d 7421	. . . . . . 7 ⊢ (𝑘 = 𝑧 → (𝐵↑𝑁) = (⦋𝑧 / 𝑘⦌𝐵↑𝑁))
71	57, 58	expcld 14108	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (⦋𝑧 / 𝑘⦌𝐵↑𝑁) ∈ ℂ)
72	30, 65, 36, 44, 66, 69, 70, 71	fprodsplitsn 15930	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 (𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁)))
73	72	adantr 482	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 (𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁)))
74		oveq1 7413	. . . . . 6 ⊢ (∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁) → (∏𝑘 ∈ 𝑦 (𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁)) = ((∏𝑘 ∈ 𝑦 𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁)))
75	74	adantl 483	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁)) → (∏𝑘 ∈ 𝑦 (𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁)) = ((∏𝑘 ∈ 𝑦 𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁)))
76	73, 75	eqtrd 2773	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵↑𝑁) = ((∏𝑘 ∈ 𝑦 𝐵↑𝑁) · (⦋𝑧 / 𝑘⦌𝐵↑𝑁)))
77	30, 48, 36, 44, 66, 41, 53, 57	fprodsplitsn 15930	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
78	77	adantr 482	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
79	78	oveq1d 7421	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁)) → (∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵↑𝑁) = ((∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)↑𝑁))
80	62, 76, 79	3eqtr4d 2783	. . 3 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ ∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵↑𝑁) = (∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵↑𝑁))
81	80	ex 414	. 2 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (∏𝑘 ∈ 𝑦 (𝐵↑𝑁) = (∏𝑘 ∈ 𝑦 𝐵↑𝑁) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵↑𝑁) = (∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵↑𝑁)))
82	4, 8, 12, 16, 27, 81, 31	findcard2d 9163	1 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 (𝐵↑𝑁) = (∏𝑘 ∈ 𝐴 𝐵↑𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 ⦋csb 3893 ∖ cdif 3945 ∪ cun 3946 ⊆ wss 3948 ∅c0 4322 {csn 4628 (class class class)co 7406 Fincfn 8936 ℂcc 11105 1c1 11108 · cmul 11112 ℕ₀cn0 12469 ℤcz 12555 ↑cexp 14024 ∏cprod 15846

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-oi 9502 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-fz 13482 df-fzo 13625 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-prod 15847

This theorem is referenced by: etransclem35 44972

W3C validator