Theorem sge0le 46392

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 46370	. . . . 5 ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ*)
4		pnfge 13050	. . . . 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 2735	. . . 4 ⊢ ((Σ^‘𝐺) = +∞ → +∞ = (Σ^‘𝐺))
9	8	adantl 481	. . 3 ⊢ ((𝜑 ∧ (Σ^‘𝐺) = +∞) → +∞ = (Σ^‘𝐺))
10	6, 9	breqtrd 5121	. 2 ⊢ ((𝜑 ∧ (Σ^‘𝐺) = +∞) → (Σ^‘𝐹) ≤ (Σ^‘𝐺))
11		elinel2 4155	. . . . . . . 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 6657	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 Fn 𝑋)
19		fvelrnb 6887	. . . . . . . . . . . . . . . . 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 13351	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0[,]+∞) ⊆ ℝ*
24	15	ffvelcdmda 7022	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ (0[,]+∞))
25	23, 24	sselid 3935	. . . . . . . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . . . . . 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 5117	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ ≤ (𝐺‘𝑥))
33	26, 32	xrgepnfd 45314	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → (𝐺‘𝑥) = +∞)
34	33	eqcomd 2735	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ = (𝐺‘𝑥))
35	15	ffnd 6657	. . . . . . . . . . . . . . . . . . . . 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 7018	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 Fn 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ ran 𝐺)
39	36, 37, 38	syl2anc 584	. . . . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ ∈ ran 𝐺)
42	41	ex 412	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = +∞ → +∞ ∈ ran 𝐺))
43	42	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = +∞ → +∞ ∈ ran 𝐺))
44	43	rexlimdva 3130	. . . . . . . . . . . . . 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 46358	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐺) = +∞)
47	46	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐺) = +∞)
48		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ +∞ ∈ ran 𝐹) → ¬ (Σ^‘𝐺) = +∞)
49	47, 48	pm2.65da 816	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ¬ +∞ ∈ ran 𝐹)
50	13, 49	fge0iccico 46355	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → 𝐹:𝑋⟶(0[,)+∞))
51	50	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,)+∞))
52		elpwinss 45030	. . . . . . . . 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 6695	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑦):𝑦⟶(0[,)+∞))
55	12, 54	sge0fsum 46372	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑦)) = Σ𝑥 ∈ 𝑦 ((𝐹 ↾ 𝑦)‘𝑥))
56		rge0ssre 13377	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
57	54	ffvelcdmda 7022	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐹 ↾ 𝑦)‘𝑥) ∈ (0[,)+∞))
58	56, 57	sselid 3935	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐹 ↾ 𝑦)‘𝑥) ∈ ℝ)
59	12, 58	fsumrecl 15659	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑥 ∈ 𝑦 ((𝐹 ↾ 𝑦)‘𝑥) ∈ ℝ)
60	55, 59	eqeltrd 2828	. . . . 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 46371	. . . . . . . . . . . 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 46373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ¬ +∞ ∈ ran 𝐺)
67	61, 66	fge0iccico 46355	. . . . . . . . 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 6695	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 ↾ 𝑦):𝑦⟶(0[,)+∞))
70	12, 69	sge0fsum 46372	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) = Σ𝑥 ∈ 𝑦 ((𝐺 ↾ 𝑦)‘𝑥))
71	69	ffvelcdmda 7022	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐺 ↾ 𝑦)‘𝑥) ∈ (0[,)+∞))
72	56, 71	sselid 3935	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐺 ↾ 𝑦)‘𝑥) ∈ ℝ)
73	12, 72	fsumrecl 15659	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑥 ∈ 𝑦 ((𝐺 ↾ 𝑦)‘𝑥) ∈ ℝ)
74	70, 73	eqeltrd 2828	. . . . 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 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → 𝜑)
77	53	sselda 3937	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝑋)
78	76, 77, 30	syl2anc 584	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → (𝐹‘𝑥) ≤ (𝐺‘𝑥))
79		fvres 6845	. . . . . . . . . 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 6845	. . . . . . . . . 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 5108	. . . . . . . 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 15724	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑥 ∈ 𝑦 ((𝐹 ↾ 𝑦)‘𝑥) ≤ Σ𝑥 ∈ 𝑦 ((𝐺 ↾ 𝑦)‘𝑥))
86	55, 70	breq12d 5108	. . . . . 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 46377	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) ≤ (Σ^‘𝐺))
91	90	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) ≤ (Σ^‘𝐺))
92	60, 74, 75, 87, 91	letrd 11291	. . . 4 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑦)) ≤ (Σ^‘𝐺))
93	92	ralrimiva 3121	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ∀𝑦 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹 ↾ 𝑦)) ≤ (Σ^‘𝐺))
94	62, 61	sge0xrcl 46370	. . . 4 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → (Σ^‘𝐺) ∈ ℝ*)
95	62, 13, 94	sge0lefi 46383	. . 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 812	1 ⊢ (𝜑 → (Σ^‘𝐹) ≤ (Σ^‘𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4553   class class class wbr 5095  ran crn 5624   ↾ cres 5625   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  Fincfn 8879  ℝcr 11027  0cc0 11028  +∞cpnf 11165  ℝ^*cxr 11167   ≤ cle 11169  [,)cico 13268  [,]cicc 13269  Σcsu 15611  Σ^{^}csumge0 46347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-inf2 9556  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-sup 9351  df-oi 9421  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-div 11796  df-nn 12147  df-2 12209  df-3 12210  df-n0 12403  df-z 12490  df-uz 12754  df-rp 12912  df-ico 13272  df-icc 13273  df-fz 13429  df-fzo 13576  df-seq 13927  df-exp 13987  df-hash 14256  df-cj 15024  df-re 15025  df-im 15026  df-sqrt 15160  df-abs 15161  df-clim 15413  df-sum 15612  df-sumge0 46348

This theorem is referenced by:  sge0lempt  46395