Theorem sge0le 43835

Description: If all of the terms of sums compare, so do the sums. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0le.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
sge0le.F	⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))
sge0le.g	⊢ (𝜑 → 𝐺:𝑋⟶(0[,]+∞))
sge0le.le	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ≤ (𝐺‘𝑥))

Assertion

Ref	Expression
sge0le	⊢ (𝜑 → (Σ^‘𝐹) ≤ (Σ^‘𝐺))

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

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem sge0le

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sge0le.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
2		sge0le.F	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))
3	1, 2	sge0xrcl 43813	. . . . 5 ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ*)
4		pnfge 12795	. . . . 5 ⊢ ((Σ^‘𝐹) ∈ ℝ* → (Σ^‘𝐹) ≤ +∞)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → (Σ^‘𝐹) ≤ +∞)
6	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ (Σ^‘𝐺) = +∞) → (Σ^‘𝐹) ≤ +∞)
7		id 22	. . . . 5 ⊢ ((Σ^‘𝐺) = +∞ → (Σ^‘𝐺) = +∞)
8	7	eqcomd 2744	. . . 4 ⊢ ((Σ^‘𝐺) = +∞ → +∞ = (Σ^‘𝐺))
9	8	adantl 481	. . 3 ⊢ ((𝜑 ∧ (Σ^‘𝐺) = +∞) → +∞ = (Σ^‘𝐺))
10	6, 9	breqtrd 5096	. 2 ⊢ ((𝜑 ∧ (Σ^‘𝐺) = +∞) → (Σ^‘𝐹) ≤ (Σ^‘𝐺))
11		elinel2 4126	. . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 𝑋 ∩ Fin) → 𝑦 ∈ Fin)
12	11	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑦 ∈ Fin)
13	2	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → 𝐹:𝑋⟶(0[,]+∞))
14	1	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
15		sge0le.g	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:𝑋⟶(0[,]+∞))
16	15	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐺:𝑋⟶(0[,]+∞))
17		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
18	2	ffnd 6585	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 Fn 𝑋)
19		fvelrnb 6812	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn 𝑋 → (+∞ ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = +∞))
20	18, 19	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (+∞ ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = +∞))
21	20	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (+∞ ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = +∞))
22	17, 21	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = +∞)
23		iccssxr 13091	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0[,]+∞) ⊆ ℝ*
24	15	ffvelrnda 6943	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ (0[,]+∞))
25	23, 24	sselid 3915	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ ℝ*)
26	25	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → (𝐺‘𝑥) ∈ ℝ*)
27		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹‘𝑥) = +∞ → (𝐹‘𝑥) = +∞)
28	27	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑥) = +∞ → +∞ = (𝐹‘𝑥))
29	28	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ = (𝐹‘𝑥))
30		sge0le.le	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ≤ (𝐺‘𝑥))
31	30	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → (𝐹‘𝑥) ≤ (𝐺‘𝑥))
32	29, 31	eqbrtrd 5092	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ ≤ (𝐺‘𝑥))
33	26, 32	xrgepnfd 42760	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → (𝐺‘𝑥) = +∞)
34	33	eqcomd 2744	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ = (𝐺‘𝑥))
35	15	ffnd 6585	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐺 Fn 𝑋)
36	35	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐺 Fn 𝑋)
37		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
38		fnfvelrn 6940	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 Fn 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ ran 𝐺)
39	36, 37, 38	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ ran 𝐺)
40	39	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → (𝐺‘𝑥) ∈ ran 𝐺)
41	34, 40	eqeltrd 2839	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ ∈ ran 𝐺)
42	41	ex 412	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = +∞ → +∞ ∈ ran 𝐺))
43	42	adantlr 711	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = +∞ → +∞ ∈ ran 𝐺))
44	43	rexlimdva 3212	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = +∞ → +∞ ∈ ran 𝐺))
45	22, 44	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐺)
46	14, 16, 45	sge0pnfval 43801	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐺) = +∞)
47	46	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐺) = +∞)
48		simplr 765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ +∞ ∈ ran 𝐹) → ¬ (Σ^‘𝐺) = +∞)
49	47, 48	pm2.65da 813	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ¬ +∞ ∈ ran 𝐹)
50	13, 49	fge0iccico 43798	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → 𝐹:𝑋⟶(0[,)+∞))
51	50	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,)+∞))
52		elpwinss 42486	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 𝑋 ∩ Fin) → 𝑦 ⊆ 𝑋)
53	52	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑦 ⊆ 𝑋)
54	51, 53	fssresd 6625	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑦):𝑦⟶(0[,)+∞))
55	12, 54	sge0fsum 43815	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑦)) = Σ𝑥 ∈ 𝑦 ((𝐹 ↾ 𝑦)‘𝑥))
56		rge0ssre 13117	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
57	54	ffvelrnda 6943	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐹 ↾ 𝑦)‘𝑥) ∈ (0[,)+∞))
58	56, 57	sselid 3915	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐹 ↾ 𝑦)‘𝑥) ∈ ℝ)
59	12, 58	fsumrecl 15374	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑥 ∈ 𝑦 ((𝐹 ↾ 𝑦)‘𝑥) ∈ ℝ)
60	55, 59	eqeltrd 2839	. . . . 5 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑦)) ∈ ℝ)
61	15	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → 𝐺:𝑋⟶(0[,]+∞))
62	1	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → 𝑋 ∈ 𝑉)
63		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ¬ (Σ^‘𝐺) = +∞)
64	62, 61	sge0repnf 43814	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ((Σ^‘𝐺) ∈ ℝ ↔ ¬ (Σ^‘𝐺) = +∞))
65	63, 64	mpbird 256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → (Σ^‘𝐺) ∈ ℝ)
66	62, 61, 65	sge0rern 43816	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ¬ +∞ ∈ ran 𝐺)
67	61, 66	fge0iccico 43798	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → 𝐺:𝑋⟶(0[,)+∞))
68	67	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐺:𝑋⟶(0[,)+∞))
69	68, 53	fssresd 6625	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 ↾ 𝑦):𝑦⟶(0[,)+∞))
70	12, 69	sge0fsum 43815	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) = Σ𝑥 ∈ 𝑦 ((𝐺 ↾ 𝑦)‘𝑥))
71	69	ffvelrnda 6943	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐺 ↾ 𝑦)‘𝑥) ∈ (0[,)+∞))
72	56, 71	sselid 3915	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐺 ↾ 𝑦)‘𝑥) ∈ ℝ)
73	12, 72	fsumrecl 15374	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑥 ∈ 𝑦 ((𝐺 ↾ 𝑦)‘𝑥) ∈ ℝ)
74	70, 73	eqeltrd 2839	. . . . 5 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) ∈ ℝ)
75	65	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘𝐺) ∈ ℝ)
76		simplll 771	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → 𝜑)
77	53	sselda 3917	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝑋)
78	76, 77, 30	syl2anc 583	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → (𝐹‘𝑥) ≤ (𝐺‘𝑥))
79		fvres 6775	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑦 → ((𝐹 ↾ 𝑦)‘𝑥) = (𝐹‘𝑥))
80	79	adantl 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐹 ↾ 𝑦)‘𝑥) = (𝐹‘𝑥))
81		fvres 6775	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑦 → ((𝐺 ↾ 𝑦)‘𝑥) = (𝐺‘𝑥))
82	81	adantl 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐺 ↾ 𝑦)‘𝑥) = (𝐺‘𝑥))
83	80, 82	breq12d 5083	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → (((𝐹 ↾ 𝑦)‘𝑥) ≤ ((𝐺 ↾ 𝑦)‘𝑥) ↔ (𝐹‘𝑥) ≤ (𝐺‘𝑥)))
84	78, 83	mpbird 256	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐹 ↾ 𝑦)‘𝑥) ≤ ((𝐺 ↾ 𝑦)‘𝑥))
85	12, 58, 72, 84	fsumle 15439	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑥 ∈ 𝑦 ((𝐹 ↾ 𝑦)‘𝑥) ≤ Σ𝑥 ∈ 𝑦 ((𝐺 ↾ 𝑦)‘𝑥))
86	55, 70	breq12d 5083	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → ((Σ^‘(𝐹 ↾ 𝑦)) ≤ (Σ^‘(𝐺 ↾ 𝑦)) ↔ Σ𝑥 ∈ 𝑦 ((𝐹 ↾ 𝑦)‘𝑥) ≤ Σ𝑥 ∈ 𝑦 ((𝐺 ↾ 𝑦)‘𝑥)))
87	85, 86	mpbird 256	. . . . 5 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑦)) ≤ (Σ^‘(𝐺 ↾ 𝑦)))
88	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑋 ∈ 𝑉)
89	15	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐺:𝑋⟶(0[,]+∞))
90	88, 89	sge0less 43820	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) ≤ (Σ^‘𝐺))
91	90	adantlr 711	. . . . 5 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) ≤ (Σ^‘𝐺))
92	60, 74, 75, 87, 91	letrd 11062	. . . 4 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑦)) ≤ (Σ^‘𝐺))
93	92	ralrimiva 3107	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ∀𝑦 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹 ↾ 𝑦)) ≤ (Σ^‘𝐺))
94	62, 61	sge0xrcl 43813	. . . 4 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → (Σ^‘𝐺) ∈ ℝ*)
95	62, 13, 94	sge0lefi 43826	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ((Σ^‘𝐹) ≤ (Σ^‘𝐺) ↔ ∀𝑦 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹 ↾ 𝑦)) ≤ (Σ^‘𝐺)))
96	93, 95	mpbird 256	. 2 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → (Σ^‘𝐹) ≤ (Σ^‘𝐺))
97	10, 96	pm2.61dan 809	1 ⊢ (𝜑 → (Σ^‘𝐹) ≤ (Σ^‘𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530   class class class wbr 5070  ran crn 5581   ↾ cres 5582   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  Fincfn 8691  ℝcr 10801  0cc0 10802  +∞cpnf 10937  ℝ^*cxr 10939   ≤ cle 10941  [,)cico 13010  [,]cicc 13011  Σcsu 15325  Σ^{^}csumge0 43790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-rp 12660  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-sum 15326  df-sumge0 43791

This theorem is referenced by:  sge0lempt  43838