Theorem fprodabs2 40465

Description: The absolute value of a finite product . (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
fprodabs2.a	⊢ (𝜑 → 𝐴 ∈ Fin)
fprodabs2.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)

Assertion

Ref	Expression
fprodabs2	⊢ (𝜑 → (abs‘∏𝑘 ∈ 𝐴 𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵))

Distinct variable groups:   𝐴,𝑘   𝜑,𝑘

Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem fprodabs2

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prodeq1 14922	. . . 4 ⊢ (𝑥 = ∅ → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
2	1	fveq2d 6379	. . 3 ⊢ (𝑥 = ∅ → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ ∅ 𝐵))
3		prodeq1 14922	. . 3 ⊢ (𝑥 = ∅ → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
4	2, 3	eqeq12d 2780	. 2 ⊢ (𝑥 = ∅ → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)))
5		prodeq1 14922	. . . 4 ⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ 𝑦 𝐵)
6	5	fveq2d 6379	. . 3 ⊢ (𝑥 = 𝑦 → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ 𝑦 𝐵))
7		prodeq1 14922	. . 3 ⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵))
8	6, 7	eqeq12d 2780	. 2 ⊢ (𝑥 = 𝑦 → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)))
9		prodeq1 14922	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
10	9	fveq2d 6379	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
11		prodeq1 14922	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵))
12	10, 11	eqeq12d 2780	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵)))
13		prodeq1 14922	. . . 4 ⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ 𝐴 𝐵)
14	13	fveq2d 6379	. . 3 ⊢ (𝑥 = 𝐴 → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ 𝐴 𝐵))
15		prodeq1 14922	. . 3 ⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵))
16	14, 15	eqeq12d 2780	. 2 ⊢ (𝑥 = 𝐴 → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ 𝐴 𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵)))
17		abs1 14322	. . . 4 ⊢ (abs‘1) = 1
18		prod0 14956	. . . . 5 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
19	18	fveq2i 6378	. . . 4 ⊢ (abs‘∏𝑘 ∈ ∅ 𝐵) = (abs‘1)
20		prod0 14956	. . . 4 ⊢ ∏𝑘 ∈ ∅ (abs‘𝐵) = 1
21	17, 19, 20	3eqtr4i 2797	. . 3 ⊢ (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)
22	21	a1i 11	. 2 ⊢ (𝜑 → (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
23		eqidd 2766	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)) → (∏𝑘 ∈ 𝑦 (abs‘𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵)) = (∏𝑘 ∈ 𝑦 (abs‘𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵)))
24		nfv 2009	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)))
25		nfcsb1v 3707	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
26		fprodabs2.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ Fin)
27	26	adantr 472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝐴 ∈ Fin)
28		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ 𝐴)
29		ssfi 8387	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin)
30	27, 28, 29	syl2anc 579	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin)
31	30	adantrr 708	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
32		simprr 789	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
33	32	eldifbd 3745	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
34		simpll 783	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝜑)
35	28	sselda 3761	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝐴)
36	35	adantlrr 712	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝐴)
37		fprodabs2.b	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
38	34, 36, 37	syl2anc 579	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℂ)
39		csbeq1a 3700	. . . . . . . 8 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
40		simpl 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝜑)
41	32	eldifad 3744	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
42		nfv 2009	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑧 ∈ 𝐴)
43	25	nfel1 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ
44	42, 43	nfim 1995	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
45		eleq1w 2827	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑧 → (𝑘 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
46	45	anbi2d 622	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑧 ∈ 𝐴)))
47	39	eleq1d 2829	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))
48	46, 47	imbi12d 335	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)))
49	44, 48, 37	chvar 2368	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
50	40, 41, 49	syl2anc 579	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
51	24, 25, 31, 32, 33, 38, 39, 50	fprodsplitsn 15002	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
52	51	adantr 472	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
53	52	fveq2d 6379	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (abs‘(∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)))
54	24, 31, 38	fprodclf 15005	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ 𝑦 𝐵 ∈ ℂ)
55	54, 50	absmuld 14478	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (abs‘(∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)) = ((abs‘∏𝑘 ∈ 𝑦 𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵)))
56	55	adantr 472	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)) → (abs‘(∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵)) = ((abs‘∏𝑘 ∈ 𝑦 𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵)))
57		oveq1 6849	. . . . . 6 ⊢ ((abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵) → ((abs‘∏𝑘 ∈ 𝑦 𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵)) = (∏𝑘 ∈ 𝑦 (abs‘𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵)))
58	57	adantl 473	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)) → ((abs‘∏𝑘 ∈ 𝑦 𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵)) = (∏𝑘 ∈ 𝑦 (abs‘𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵)))
59	53, 56, 58	3eqtrd 2803	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (∏𝑘 ∈ 𝑦 (abs‘𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵)))
60		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑘abs
61	60, 25	nffv 6385	. . . . . 6 ⊢ Ⅎ𝑘(abs‘⦋𝑧 / 𝑘⦌𝐵)
62	38	abscld 14460	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → (abs‘𝐵) ∈ ℝ)
63	62	recnd 10322	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → (abs‘𝐵) ∈ ℂ)
64	39	fveq2d 6379	. . . . . 6 ⊢ (𝑘 = 𝑧 → (abs‘𝐵) = (abs‘⦋𝑧 / 𝑘⦌𝐵))
65	50	abscld 14460	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (abs‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)
66	65	recnd 10322	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (abs‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ)
67	24, 61, 31, 32, 33, 63, 64, 66	fprodsplitsn 15002	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵) = (∏𝑘 ∈ 𝑦 (abs‘𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵)))
68	67	adantr 472	. . . 4 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵) = (∏𝑘 ∈ 𝑦 (abs‘𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵)))
69	23, 59, 68	3eqtr4d 2809	. . 3 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵))
70	69	ex 401	. 2 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ((abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵) → (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵)))
71	4, 8, 12, 16, 22, 70, 26	findcard2d 8409	1 ⊢ (𝜑 → (abs‘∏𝑘 ∈ 𝐴 𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1652   ∈ wcel 2155  ⦋csb 3691   ∖ cdif 3729   ∪ cun 3730   ⊆ wss 3732  ∅c0 4079  {csn 4334  ‘cfv 6068  (class class class)co 6842  Fincfn 8160  ℂcc 10187  1c1 10190   · cmul 10194  abscabs 14259  ∏cprod 14918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-rp 12029  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-clim 14504  df-prod 14919

This theorem is referenced by:  etransclem41  41129