Theorem sge0sup 44622

Description: The arbitrary sum of nonnegative extended reals is the supremum of finite subsums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0sup.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
sge0sup.f	⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))

Assertion

Ref	Expression
sge0sup	⊢ (𝜑 → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))), ℝ*, < ))

Distinct variable groups:   𝑥,𝐹   𝑥,𝑋   𝜑,𝑥

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem sge0sup

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2737	. . 3 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ = +∞)
2		sge0sup.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
3	2	adantr 481	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
4		sge0sup.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))
5	4	adantr 481	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
6		simpr 485	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
7	3, 5, 6	sge0pnfval 44604	. . 3 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = +∞)
8		vex 3449	. . . . . . . . 9 ⊢ 𝑥 ∈ V
9	8	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ V)
10	4	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,]+∞))
11		elinel1 4155	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
12		elpwi 4567	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ⊆ 𝑋)
14	13	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ⊆ 𝑋)
15	10, 14	fssresd 6709	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥):𝑥⟶(0[,]+∞))
16	9, 15	sge0xrcl 44616	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑥)) ∈ ℝ*)
17	16	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑥)) ∈ ℝ*)
18	17	ralrimiva 3143	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹 ↾ 𝑥)) ∈ ℝ*)
19		eqid 2736	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥)))
20	19	rnmptss 7070	. . . . 5 ⊢ (∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹 ↾ 𝑥)) ∈ ℝ* → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))) ⊆ ℝ*)
21	18, 20	syl 17	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))) ⊆ ℝ*)
22	4	ffnd 6669	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑋)
23		fvelrnb 6903	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞))
24	22, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞))
25	24	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (+∞ ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞))
26	6, 25	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞)
27		snelpwi 5400	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ∈ 𝒫 𝑋)
28		snfi 8988	. . . . . . . . . . . . 13 ⊢ {𝑦} ∈ Fin
29	28	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ∈ Fin)
30	27, 29	elind 4154	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
31	30	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
32		simp2 1137	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → 𝑦 ∈ 𝑋)
33	4	3ad2ant1 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → 𝐹:𝑋⟶(0[,]+∞))
34	32	snssd 4769	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → {𝑦} ⊆ 𝑋)
35	33, 34	fssresd 6709	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐹 ↾ {𝑦}):{𝑦}⟶(0[,]+∞))
36	32, 35	sge0sn 44610	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (Σ^‘(𝐹 ↾ {𝑦})) = ((𝐹 ↾ {𝑦})‘𝑦))
37		vsnid 4623	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ {𝑦}
38		fvres 6861	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑦} → ((𝐹 ↾ {𝑦})‘𝑦) = (𝐹‘𝑦))
39	37, 38	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝐹 ↾ {𝑦})‘𝑦) = (𝐹‘𝑦)
40	39	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → ((𝐹 ↾ {𝑦})‘𝑦) = (𝐹‘𝑦))
41		simp3 1138	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐹‘𝑦) = +∞)
42	36, 40, 41	3eqtrrd 2781	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → +∞ = (Σ^‘(𝐹 ↾ {𝑦})))
43		reseq2 5932	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑦} → (𝐹 ↾ 𝑥) = (𝐹 ↾ {𝑦}))
44	43	fveq2d 6846	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑦} → (Σ^‘(𝐹 ↾ 𝑥)) = (Σ^‘(𝐹 ↾ {𝑦})))
45	44	rspceeqv 3595	. . . . . . . . . 10 ⊢ (({𝑦} ∈ (𝒫 𝑋 ∩ Fin) ∧ +∞ = (Σ^‘(𝐹 ↾ {𝑦}))) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (Σ^‘(𝐹 ↾ 𝑥)))
46	31, 42, 45	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (Σ^‘(𝐹 ↾ 𝑥)))
47		pnfex 11208	. . . . . . . . . 10 ⊢ +∞ ∈ V
48	47	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ V)
49	19, 46, 48	elrnmptd 5916	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))))
50	49	3exp 1119	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝑋 → ((𝐹‘𝑦) = +∞ → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))))))
51	50	rexlimdv 3150	. . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞ → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥)))))
52	51	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞ → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥)))))
53	26, 52	mpd 15	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))))
54		supxrpnf 13237	. . . 4 ⊢ ((ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))) ⊆ ℝ* ∧ +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥)))) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)
55	21, 53, 54	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)
56	1, 7, 55	3eqtr4d 2786	. 2 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))), ℝ*, < ))
57	2	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
58	4	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
59		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
60	58, 59	fge0iccico 44601	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
61	57, 60	sge0reval 44603	. . 3 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
62		elinel2 4156	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
63	62	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
64	15	adantlr 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥):𝑥⟶(0[,]+∞))
65		nelrnres 43396	. . . . . . . . . 10 ⊢ (¬ +∞ ∈ ran 𝐹 → ¬ +∞ ∈ ran (𝐹 ↾ 𝑥))
66	65	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ¬ +∞ ∈ ran (𝐹 ↾ 𝑥))
67	64, 66	fge0iccico 44601	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥):𝑥⟶(0[,)+∞))
68	63, 67	sge0fsum 44618	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑥)) = Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝑥)‘𝑦))
69		simpr 485	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
70		fvres 6861	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑥 → ((𝐹 ↾ 𝑥)‘𝑦) = (𝐹‘𝑦))
71	69, 70	syl 17	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → ((𝐹 ↾ 𝑥)‘𝑦) = (𝐹‘𝑦))
72	71	sumeq2dv 15588	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝑥)‘𝑦) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
73	72	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝑥)‘𝑦) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
74	68, 73	eqtrd 2776	. . . . . 6 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑥)) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
75	74	mpteq2dva 5205	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
76	75	rneqd 5893	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))) = ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
77	76	supeq1d 9382	. . 3 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
78	61, 77	eqtr4d 2779	. 2 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))), ℝ*, < ))
79	56, 78	pm2.61dan 811	1 ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))), ℝ*, < ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  Vcvv 3445   ∩ cin 3909   ⊆ wss 3910  𝒫 cpw 4560  {csn 4586   ↦ cmpt 5188  ran crn 5634   ↾ cres 5635   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  Fincfn 8883  supcsup 9376  0cc0 11051  +∞cpnf 11186  ℝ^*cxr 11188   < clt 11189  [,]cicc 13267  Σcsu 15570  Σ^{^}csumge0 44593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-sumge0 44594

This theorem is referenced by:  sge0gerp  44626  sge0pnffigt  44627  sge0lefi  44629