Theorem fprodabs2 44611

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 15858	. . . 4 ⊢ (𝑥 = ∅ → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
2	1	fveq2d 6896	. . 3 ⊢ (𝑥 = ∅ → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ ∅ 𝐵))
3		prodeq1 15858	. . 3 ⊢ (𝑥 = ∅ → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
4	2, 3	eqeq12d 2747	. 2 ⊢ (𝑥 = ∅ → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)))
5		prodeq1 15858	. . . 4 ⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ 𝑦 𝐵)
6	5	fveq2d 6896	. . 3 ⊢ (𝑥 = 𝑦 → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ 𝑦 𝐵))
7		prodeq1 15858	. . 3 ⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵))
8	6, 7	eqeq12d 2747	. 2 ⊢ (𝑥 = 𝑦 → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)))
9		prodeq1 15858	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
10	9	fveq2d 6896	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
11		prodeq1 15858	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵))
12	10, 11	eqeq12d 2747	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵)))
13		prodeq1 15858	. . . 4 ⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ 𝐴 𝐵)
14	13	fveq2d 6896	. . 3 ⊢ (𝑥 = 𝐴 → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ 𝐴 𝐵))
15		prodeq1 15858	. . 3 ⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵))
16	14, 15	eqeq12d 2747	. 2 ⊢ (𝑥 = 𝐴 → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ 𝐴 𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵)))
17		abs1 15249	. . . 4 ⊢ (abs‘1) = 1
18		prod0 15892	. . . . 5 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
19	18	fveq2i 6895	. . . 4 ⊢ (abs‘∏𝑘 ∈ ∅ 𝐵) = (abs‘1)
20		prod0 15892	. . . 4 ⊢ ∏𝑘 ∈ ∅ (abs‘𝐵) = 1
21	17, 19, 20	3eqtr4i 2769	. . 3 ⊢ (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)
22	21	a1i 11	. 2 ⊢ (𝜑 → (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
23		eqidd 2732	. . . 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 1916	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)))
25		nfcsb1v 3919	. . . . . . . 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 9176	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin)
30	27, 28, 29	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin)
31	30	adantrr 714	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
32		simprr 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
33	32	eldifbd 3962	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
34		simpll 764	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝜑)
35	28	sselda 3983	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝐴)
36	35	adantlrr 718	. . . . . . . . 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 583	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℂ)
39		csbeq1a 3908	. . . . . . . 8 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
40		simpl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝜑)
41	32	eldifad 3961	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
42		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑧 ∈ 𝐴)
43	25	nfel1 2918	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ
44	42, 43	nfim 1898	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
45		eleq1w 2815	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑧 → (𝑘 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
46	45	anbi2d 628	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑧 ∈ 𝐴)))
47	39	eleq1d 2817	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ ℂ ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ))
48	46, 47	imbi12d 343	. . . . . . . . . 10 ⊢ (𝑘 = 𝑧 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)))
49	44, 48, 37	chvarfv 2232	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
50	40, 41, 49	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
51	24, 25, 31, 32, 33, 38, 39, 50	fprodsplitsn 15938	. . . . . . 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 6896	. . . . 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 15941	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ 𝑦 𝐵 ∈ ℂ)
55	54, 50	absmuld 15406	. . . . . 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 7419	. . . . . 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 2775	. . . 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 2902	. . . . . . 7 ⊢ Ⅎ𝑘abs
61	60, 25	nffv 6902	. . . . . 6 ⊢ Ⅎ𝑘(abs‘⦋𝑧 / 𝑘⦌𝐵)
62	38	abscld 15388	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → (abs‘𝐵) ∈ ℝ)
63	62	recnd 11247	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → (abs‘𝐵) ∈ ℂ)
64	39	fveq2d 6896	. . . . . 6 ⊢ (𝑘 = 𝑧 → (abs‘𝐵) = (abs‘⦋𝑧 / 𝑘⦌𝐵))
65	50	abscld 15388	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (abs‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)
66	65	recnd 11247	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (abs‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ)
67	24, 61, 31, 32, 33, 63, 64, 66	fprodsplitsn 15938	. . . . 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 2781	. . 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 9169	1 ⊢ (𝜑 → (abs‘∏𝑘 ∈ 𝐴 𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ⦋csb 3894   ∖ cdif 3946   ∪ cun 3947   ⊆ wss 3949  ∅c0 4323  {csn 4629  ‘cfv 6544  (class class class)co 7412  Fincfn 8942  ℂcc 11111  1c1 11114   · cmul 11118  abscabs 15186  ∏cprod 15854

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-inf2 9639  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190  ax-pre-sup 11191

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-er 8706  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-sup 9440  df-oi 9508  df-card 9937  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-3 12281  df-n0 12478  df-z 12564  df-uz 12828  df-rp 12980  df-fz 13490  df-fzo 13633  df-seq 13972  df-exp 14033  df-hash 14296  df-cj 15051  df-re 15052  df-im 15053  df-sqrt 15187  df-abs 15188  df-clim 15437  df-prod 15855

This theorem is referenced by:  etransclem41  45291