Theorem fprodabs2 45121

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 15889	. . . 4 ⊢ (𝑥 = ∅ → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ ∅ 𝐵)
2	1	fveq2d 6900	. . 3 ⊢ (𝑥 = ∅ → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ ∅ 𝐵))
3		prodeq1 15889	. . 3 ⊢ (𝑥 = ∅ → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
4	2, 3	eqeq12d 2741	. 2 ⊢ (𝑥 = ∅ → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)))
5		prodeq1 15889	. . . 4 ⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ 𝑦 𝐵)
6	5	fveq2d 6900	. . 3 ⊢ (𝑥 = 𝑦 → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ 𝑦 𝐵))
7		prodeq1 15889	. . 3 ⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵))
8	6, 7	eqeq12d 2741	. 2 ⊢ (𝑥 = 𝑦 → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ 𝑦 𝐵) = ∏𝑘 ∈ 𝑦 (abs‘𝐵)))
9		prodeq1 15889	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
10	9	fveq2d 6900	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
11		prodeq1 15889	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵))
12	10, 11	eqeq12d 2741	. 2 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵)))
13		prodeq1 15889	. . . 4 ⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ 𝑥 𝐵 = ∏𝑘 ∈ 𝐴 𝐵)
14	13	fveq2d 6900	. . 3 ⊢ (𝑥 = 𝐴 → (abs‘∏𝑘 ∈ 𝑥 𝐵) = (abs‘∏𝑘 ∈ 𝐴 𝐵))
15		prodeq1 15889	. . 3 ⊢ (𝑥 = 𝐴 → ∏𝑘 ∈ 𝑥 (abs‘𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵))
16	14, 15	eqeq12d 2741	. 2 ⊢ (𝑥 = 𝐴 → ((abs‘∏𝑘 ∈ 𝑥 𝐵) = ∏𝑘 ∈ 𝑥 (abs‘𝐵) ↔ (abs‘∏𝑘 ∈ 𝐴 𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵)))
17		abs1 15280	. . . 4 ⊢ (abs‘1) = 1
18		prod0 15923	. . . . 5 ⊢ ∏𝑘 ∈ ∅ 𝐵 = 1
19	18	fveq2i 6899	. . . 4 ⊢ (abs‘∏𝑘 ∈ ∅ 𝐵) = (abs‘1)
20		prod0 15923	. . . 4 ⊢ ∏𝑘 ∈ ∅ (abs‘𝐵) = 1
21	17, 19, 20	3eqtr4i 2763	. . 3 ⊢ (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵)
22	21	a1i 11	. 2 ⊢ (𝜑 → (abs‘∏𝑘 ∈ ∅ 𝐵) = ∏𝑘 ∈ ∅ (abs‘𝐵))
23		eqidd 2726	. . . 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 1909	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦)))
25		nfcsb1v 3914	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
26		fprodabs2.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ Fin)
27	26	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝐴 ∈ Fin)
28		simpr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ 𝐴)
29		ssfi 9198	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin)
30	27, 28, 29	syl2anc 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝑦 ∈ Fin)
31	30	adantrr 715	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑦 ∈ Fin)
32		simprr 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
33	32	eldifbd 3957	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ¬ 𝑧 ∈ 𝑦)
34		simpll 765	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝜑)
35	28	sselda 3976	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ⊆ 𝐴) ∧ 𝑘 ∈ 𝑦) → 𝑘 ∈ 𝐴)
36	35	adantlrr 719	. . . . . . . . 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 582	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → 𝐵 ∈ ℂ)
39		csbeq1a 3903	. . . . . . . 8 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
40		simpl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝜑)
41	32	eldifad 3956	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ 𝐴)
42		nfv 1909	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑧 ∈ 𝐴)
43	25	nfel1 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ
44	42, 43	nfim 1891	. . . . . . . . . 10 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
45		eleq1w 2808	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑧 → (𝑘 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
46	45	anbi2d 628	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑧 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑧 ∈ 𝐴)))
47	39	eleq1d 2810	. . . . . . . . . . 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 2228	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
50	40, 41, 49	syl2anc 582	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ⦋𝑧 / 𝑘⦌𝐵 ∈ ℂ)
51	24, 25, 31, 32, 33, 38, 39, 50	fprodsplitsn 15969	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = (∏𝑘 ∈ 𝑦 𝐵 · ⦋𝑧 / 𝑘⦌𝐵))
52	51	adantr 479	. . . . . 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 6900	. . . . 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 15972	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ 𝑦 𝐵 ∈ ℂ)
55	54, 50	absmuld 15437	. . . . . 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 479	. . . . 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 7426	. . . . . 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 480	. . . . 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 2769	. . . 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 2891	. . . . . . 7 ⊢ Ⅎ𝑘abs
61	60, 25	nffv 6906	. . . . . 6 ⊢ Ⅎ𝑘(abs‘⦋𝑧 / 𝑘⦌𝐵)
62	38	abscld 15419	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → (abs‘𝐵) ∈ ℝ)
63	62	recnd 11274	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ 𝑘 ∈ 𝑦) → (abs‘𝐵) ∈ ℂ)
64	39	fveq2d 6900	. . . . . 6 ⊢ (𝑘 = 𝑧 → (abs‘𝐵) = (abs‘⦋𝑧 / 𝑘⦌𝐵))
65	50	abscld 15419	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (abs‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)
66	65	recnd 11274	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (abs‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ)
67	24, 61, 31, 32, 33, 63, 64, 66	fprodsplitsn 15969	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ∏𝑘 ∈ (𝑦 ∪ {𝑧})(abs‘𝐵) = (∏𝑘 ∈ 𝑦 (abs‘𝐵) · (abs‘⦋𝑧 / 𝑘⦌𝐵)))
68	67	adantr 479	. . . 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 2775	. . 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 411	. 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 9191	1 ⊢ (𝜑 → (abs‘∏𝑘 ∈ 𝐴 𝐵) = ∏𝑘 ∈ 𝐴 (abs‘𝐵))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ⦋csb 3889   ∖ cdif 3941   ∪ cun 3942   ⊆ wss 3944  ∅c0 4322  {csn 4630  ‘cfv 6549  (class class class)co 7419  Fincfn 8964  ℂcc 11138  1c1 11141   · cmul 11145  abscabs 15217  ∏cprod 15885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-oi 9535  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-n0 12506  df-z 12592  df-uz 12856  df-rp 13010  df-fz 13520  df-fzo 13663  df-seq 14003  df-exp 14063  df-hash 14326  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-clim 15468  df-prod 15886

This theorem is referenced by:  etransclem41  45801