Theorem sge0le 46762

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 46740	. . . . 5 ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ*)
4		pnfge 13056	. . . . 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 2743	. . . 4 ⊢ ((Σ^‘𝐺) = +∞ → +∞ = (Σ^‘𝐺))
9	8	adantl 481	. . 3 ⊢ ((𝜑 ∧ (Σ^‘𝐺) = +∞) → +∞ = (Σ^‘𝐺))
10	6, 9	breqtrd 5126	. 2 ⊢ ((𝜑 ∧ (Σ^‘𝐺) = +∞) → (Σ^‘𝐹) ≤ (Σ^‘𝐺))
11		elinel2 4156	. . . . . . . 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 6671	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 Fn 𝑋)
19		fvelrnb 6902	. . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = +∞)
23		iccssxr 13358	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0[,]+∞) ⊆ ℝ*
24	15	ffvelcdmda 7038	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ (0[,]+∞))
25	23, 24	sselid 3933	. . . . . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . . . . . . 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 5122	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ ≤ (𝐺‘𝑥))
33	26, 32	xrgepnfd 45687	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → (𝐺‘𝑥) = +∞)
34	33	eqcomd 2743	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ = (𝐺‘𝑥))
35	15	ffnd 6671	. . . . . . . . . . . . . . . . . . . . 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 7034	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 Fn 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ ran 𝐺)
39	36, 37, 38	syl2anc 585	. . . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ ∈ ran 𝐺)
42	41	ex 412	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = +∞ → +∞ ∈ ran 𝐺))
43	42	adantlr 716	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = +∞ → +∞ ∈ ran 𝐺))
44	43	rexlimdva 3139	. . . . . . . . . . . . . 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 46728	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐺) = +∞)
47	46	adantlr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐺) = +∞)
48		simplr 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ +∞ ∈ ran 𝐹) → ¬ (Σ^‘𝐺) = +∞)
49	47, 48	pm2.65da 817	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ¬ +∞ ∈ ran 𝐹)
50	13, 49	fge0iccico 46725	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → 𝐹:𝑋⟶(0[,)+∞))
51	50	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,)+∞))
52		elpwinss 45406	. . . . . . . . 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 6709	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑦):𝑦⟶(0[,)+∞))
55	12, 54	sge0fsum 46742	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑦)) = Σ𝑥 ∈ 𝑦 ((𝐹 ↾ 𝑦)‘𝑥))
56		rge0ssre 13384	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
57	54	ffvelcdmda 7038	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐹 ↾ 𝑦)‘𝑥) ∈ (0[,)+∞))
58	56, 57	sselid 3933	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐹 ↾ 𝑦)‘𝑥) ∈ ℝ)
59	12, 58	fsumrecl 15669	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑥 ∈ 𝑦 ((𝐹 ↾ 𝑦)‘𝑥) ∈ ℝ)
60	55, 59	eqeltrd 2837	. . . . 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 46741	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ((Σ^‘𝐺) ∈ ℝ ↔ ¬ (Σ^‘𝐺) = +∞))
65	63, 64	mpbird 257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → (Σ^‘𝐺) ∈ ℝ)
66	62, 61, 65	sge0rern 46743	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ¬ +∞ ∈ ran 𝐺)
67	61, 66	fge0iccico 46725	. . . . . . . . 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 6709	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 ↾ 𝑦):𝑦⟶(0[,)+∞))
70	12, 69	sge0fsum 46742	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) = Σ𝑥 ∈ 𝑦 ((𝐺 ↾ 𝑦)‘𝑥))
71	69	ffvelcdmda 7038	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐺 ↾ 𝑦)‘𝑥) ∈ (0[,)+∞))
72	56, 71	sselid 3933	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐺 ↾ 𝑦)‘𝑥) ∈ ℝ)
73	12, 72	fsumrecl 15669	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑥 ∈ 𝑦 ((𝐺 ↾ 𝑦)‘𝑥) ∈ ℝ)
74	70, 73	eqeltrd 2837	. . . . 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 775	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → 𝜑)
77	53	sselda 3935	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝑋)
78	76, 77, 30	syl2anc 585	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → (𝐹‘𝑥) ≤ (𝐺‘𝑥))
79		fvres 6861	. . . . . . . . . 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 6861	. . . . . . . . . 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 5113	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → (((𝐹 ↾ 𝑦)‘𝑥) ≤ ((𝐺 ↾ 𝑦)‘𝑥) ↔ (𝐹‘𝑥) ≤ (𝐺‘𝑥)))
84	78, 83	mpbird 257	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐹 ↾ 𝑦)‘𝑥) ≤ ((𝐺 ↾ 𝑦)‘𝑥))
85	12, 58, 72, 84	fsumle 15734	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑥 ∈ 𝑦 ((𝐹 ↾ 𝑦)‘𝑥) ≤ Σ𝑥 ∈ 𝑦 ((𝐺 ↾ 𝑦)‘𝑥))
86	55, 70	breq12d 5113	. . . . . 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 257	. . . . 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 46747	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) ≤ (Σ^‘𝐺))
91	90	adantlr 716	. . . . 5 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) ≤ (Σ^‘𝐺))
92	60, 74, 75, 87, 91	letrd 11302	. . . 4 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑦)) ≤ (Σ^‘𝐺))
93	92	ralrimiva 3130	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ∀𝑦 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹 ↾ 𝑦)) ≤ (Σ^‘𝐺))
94	62, 61	sge0xrcl 46740	. . . 4 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → (Σ^‘𝐺) ∈ ℝ*)
95	62, 13, 94	sge0lefi 46753	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ((Σ^‘𝐹) ≤ (Σ^‘𝐺) ↔ ∀𝑦 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹 ↾ 𝑦)) ≤ (Σ^‘𝐺)))
96	93, 95	mpbird 257	. 2 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → (Σ^‘𝐹) ≤ (Σ^‘𝐺))
97	10, 96	pm2.61dan 813	1 ⊢ (𝜑 → (Σ^‘𝐹) ≤ (Σ^‘𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556   class class class wbr 5100  ran crn 5633   ↾ cres 5634   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  Fincfn 8895  ℝcr 11037  0cc0 11038  +∞cpnf 11175  ℝ^*cxr 11177   ≤ cle 11179  [,)cico 13275  [,]cicc 13276  Σcsu 15621  Σ^{^}csumge0 46717

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9357  df-oi 9427  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-n0 12414  df-z 12501  df-uz 12764  df-rp 12918  df-ico 13279  df-icc 13280  df-fz 13436  df-fzo 13583  df-seq 13937  df-exp 13997  df-hash 14266  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-clim 15423  df-sum 15622  df-sumge0 46718

This theorem is referenced by:  sge0lempt  46765