Theorem fprodabs2 45567

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 15925	. . . 4 ⊢ (𝑥 = ∅ → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
2	1	fveq2d 6890	. . 3 ⊢ (𝑥 = ∅ → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ ∅ 𝐵))
3		prodeq1 15925	. . 3 ⊢ (𝑥 = ∅ → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
4	2, 3	eqeq12d 2750	. 2 ⊢ (𝑥 = ∅ → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)))
5		prodeq1 15925	. . . 4 ⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ 𝑦 𝐵)
6	5	fveq2d 6890	. . 3 ⊢ (𝑥 = 𝑦 → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ 𝑦 𝐵))
7		prodeq1 15925	. . 3 ⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵))
8	6, 7	eqeq12d 2750	. 2 ⊢ (𝑥 = 𝑦 → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)))
9		prodeq1 15925	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
10	9	fveq2d 6890	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
11		prodeq1 15925	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵))
12	10, 11	eqeq12d 2750	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵)))
13		prodeq1 15925	. . . 4 ⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ 𝐴 𝐵)
14	13	fveq2d 6890	. . 3 ⊢ (𝑥 = 𝐴 → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ 𝐴 𝐵))
15		prodeq1 15925	. . 3 ⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵))
16	14, 15	eqeq12d 2750	. 2 ⊢ (𝑥 = 𝐴 → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ 𝐴 𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵)))
17		abs1 15318	. . . 4 ⊢ (abs‘1) = 1
18		prod0 15961	. . . . 5 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
19	18	fveq2i 6889	. . . 4 ⊢ (abs‘∏𝑘 ∈ ∅ 𝐵) = (abs‘1)
20		prod0 15961	. . . 4 ⊢ ∏𝑘 ∈ ∅ (abs‘𝐵) = 1
21	17, 19, 20	3eqtr4i 2767	. . 3 ⊢ (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)
22	21	a1i 11	. 2 ⊢ (𝜑 → (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
23		eqidd 2735	. . . 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 1913	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)))
25		nfcsb1v 3903	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
26		fprodabs2.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ Fin)
27	26	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝐴 ∈ Fin)
28		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ 𝐴)
29		ssfi 9195	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin)
30	27, 28, 29	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin)
31	30	adantrr 717	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
32		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
33	32	eldifbd 3944	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
34		simpll 766	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝜑)
35	28	sselda 3963	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝐴)
36	35	adantlrr 721	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℂ)
39		csbeq1a 3893	. . . . . . . 8 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
40		simpl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝜑)
41	32	eldifad 3943	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
42		nfv 1913	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑧 ∈ 𝐴)
43	25	nfel1 2914	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ
44	42, 43	nfim 1895	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
45		eleq1w 2816	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑧 → (𝑘 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
46	45	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑧 ∈ 𝐴)))
47	39	eleq1d 2818	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))
48	46, 47	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)))
49	44, 48, 37	chvarfv 2239	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
50	40, 41, 49	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
51	24, 25, 31, 32, 33, 38, 39, 50	fprodsplitsn 16007	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
52	51	adantr 480	. . . . . 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 6890	. . . . 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 16010	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ 𝑦 𝐵 ∈ ℂ)
55	54, 50	absmuld 15475	. . . . . 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 480	. . . . 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 7420	. . . . . 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 481	. . . . 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 2773	. . . 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 2897	. . . . . . 7 ⊢ Ⅎ𝑘abs
61	60, 25	nffv 6896	. . . . . 6 ⊢ Ⅎ𝑘(abs‘⦋𝑧 / 𝑘⦌𝐵)
62	38	abscld 15457	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → (abs‘𝐵) ∈ ℝ)
63	62	recnd 11271	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → (abs‘𝐵) ∈ ℂ)
64	39	fveq2d 6890	. . . . . 6 ⊢ (𝑘 = 𝑧 → (abs‘𝐵) = (abs‘⦋𝑧 / 𝑘⦌𝐵))
65	50	abscld 15457	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (abs‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)
66	65	recnd 11271	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (abs‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ)
67	24, 61, 31, 32, 33, 63, 64, 66	fprodsplitsn 16007	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵) = (∏𝑘 ∈ 𝑦 (abs‘𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵)))
68	67	adantr 480	. . . 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 2779	. . 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 412	. 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 9188	1 ⊢ (𝜑 → (abs‘∏𝑘 ∈ 𝐴 𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ⦋csb 3879   ∖ cdif 3928   ∪ cun 3929   ⊆ wss 3931  ∅c0 4313  {csn 4606  ‘cfv 6541  (class class class)co 7413  Fincfn 8967  ℂcc 11135  1c1 11138   · cmul 11142  abscabs 15255  ∏cprod 15921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-inf2 9663  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214  ax-pre-sup 11215

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-int 4927  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-se 5618  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  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 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-er 8727  df-en 8968  df-dom 8969  df-sdom 8970  df-fin 8971  df-sup 9464  df-oi 9532  df-card 9961  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-div 11903  df-nn 12249  df-2 12311  df-3 12312  df-n0 12510  df-z 12597  df-uz 12861  df-rp 13017  df-fz 13530  df-fzo 13677  df-seq 14025  df-exp 14085  df-hash 14352  df-cj 15120  df-re 15121  df-im 15122  df-sqrt 15256  df-abs 15257  df-clim 15506  df-prod 15922

This theorem is referenced by:  etransclem41  46247