Theorem sge0sup 46312

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 2741	. . 3 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ = +∞)
2		sge0sup.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
3	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
4		sge0sup.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))
5	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
6		simpr 484	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
7	3, 5, 6	sge0pnfval 46294	. . 3 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = +∞)
8		vex 3492	. . . . . . . . 9 ⊢ 𝑥 ∈ V
9	8	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ V)
10	4	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,]+∞))
11		elinel1 4224	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
12		elpwi 4629	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ⊆ 𝑋)
14	13	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ⊆ 𝑋)
15	10, 14	fssresd 6788	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥):𝑥⟶(0[,]+∞))
16	9, 15	sge0xrcl 46306	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑥)) ∈ ℝ*)
17	16	adantlr 714	. . . . . 6 ⊢ (((𝜑 ∧ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑥)) ∈ ℝ*)
18	17	ralrimiva 3152	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹 ↾ 𝑥)) ∈ ℝ*)
19		eqid 2740	. . . . . 6 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥)))
20	19	rnmptss 7157	. . . . 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 6748	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑋)
23		fvelrnb 6982	. . . . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (+∞ ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞))
26	6, 25	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞)
27		snelpwi 5463	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ∈ 𝒫 𝑋)
28		snfi 9109	. . . . . . . . . . . . 13 ⊢ {𝑦} ∈ Fin
29	28	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ∈ Fin)
30	27, 29	elind 4223	. . . . . . . . . . 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 4834	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → {𝑦} ⊆ 𝑋)
35	33, 34	fssresd 6788	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐹 ↾ {𝑦}):{𝑦}⟶(0[,]+∞))
36	32, 35	sge0sn 46300	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (Σ^‘(𝐹 ↾ {𝑦})) = ((𝐹 ↾ {𝑦})‘𝑦))
37		vsnid 4685	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ {𝑦}
38		fvres 6939	. . . . . . . . . . . . 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 2785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → +∞ = (Σ^‘(𝐹 ↾ {𝑦})))
43		reseq2 6004	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑦} → (𝐹 ↾ 𝑥) = (𝐹 ↾ {𝑦}))
44	43	fveq2d 6924	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑦} → (Σ^‘(𝐹 ↾ 𝑥)) = (Σ^‘(𝐹 ↾ {𝑦})))
45	44	rspceeqv 3658	. . . . . . . . . 10 ⊢ (({𝑦} ∈ (𝒫 𝑋 ∩ Fin) ∧ +∞ = (Σ^‘(𝐹 ↾ {𝑦}))) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (Σ^‘(𝐹 ↾ 𝑥)))
46	31, 42, 45	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (Σ^‘(𝐹 ↾ 𝑥)))
47		pnfex 11343	. . . . . . . . . 10 ⊢ +∞ ∈ V
48	47	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ V)
49	19, 46, 48	elrnmptd 5986	. . . . . . . 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 3159	. . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞ → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥)))))
52	51	adantr 480	. . . . 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 13380	. . . 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 583	. . 3 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)
56	1, 7, 55	3eqtr4d 2790	. 2 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))), ℝ*, < ))
57	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
58	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
59		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
60	58, 59	fge0iccico 46291	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
61	57, 60	sge0reval 46293	. . 3 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
62		elinel2 4225	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
63	62	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
64	15	adantlr 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥):𝑥⟶(0[,]+∞))
65		nelrnres 45094	. . . . . . . . . 10 ⊢ (¬ +∞ ∈ ran 𝐹 → ¬ +∞ ∈ ran (𝐹 ↾ 𝑥))
66	65	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ¬ +∞ ∈ ran (𝐹 ↾ 𝑥))
67	64, 66	fge0iccico 46291	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥):𝑥⟶(0[,)+∞))
68	63, 67	sge0fsum 46308	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑥)) = Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝑥)‘𝑦))
69		simpr 484	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
70		fvres 6939	. . . . . . . . . 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 15750	. . . . . . . 8 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝑥)‘𝑦) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
73	72	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦 ∈ 𝑥 ((𝐹 ↾ 𝑥)‘𝑦) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
74	68, 73	eqtrd 2780	. . . . . 6 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑥)) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
75	74	mpteq2dva 5266	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
76	75	rneqd 5963	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))) = ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)))
77	76	supeq1d 9515	. . 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 2783	. 2 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))), ℝ*, < ))
79	56, 78	pm2.61dan 812	1 ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (Σ^‘(𝐹 ↾ 𝑥))), ℝ*, < ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  {csn 4648   ↦ cmpt 5249  ran crn 5701   ↾ cres 5702   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Fincfn 9003  supcsup 9509  0cc0 11184  +∞cpnf 11321  ℝ^*cxr 11323   < clt 11324  [,]cicc 13410  Σcsu 15734  Σ^{^}csumge0 46283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735  df-sumge0 46284

This theorem is referenced by:  sge0gerp  46316  sge0pnffigt  46317  sge0lefi  46319