Theorem sge0le 47154

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 47132	. . . . 5 ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ*)
4		pnfge 13166	. . . . 5 ⊢ ((Σ^‘𝐹) ∈ ℝ* → (Σ^‘𝐹) ≤ +∞)
5	3, 4	syl 18	. . . 4 ⊢ (𝜑 → (Σ^‘𝐹) ≤ +∞)
6	5	adantr 486	. . 3 ⊢ ((𝜑 ∧ (Σ^‘𝐺) = +∞) → (Σ^‘𝐹) ≤ +∞)
7		id 23	. . . . 5 ⊢ ((Σ^‘𝐺) = +∞ → (Σ^‘𝐺) = +∞)
8	7	eqcomd 2771	. . . 4 ⊢ ((Σ^‘𝐺) = +∞ → +∞ = (Σ^‘𝐺))
9	8	adantl 487	. . 3 ⊢ ((𝜑 ∧ (Σ^‘𝐺) = +∞) → +∞ = (Σ^‘𝐺))
10	6, 9	breqtrd 5139	. 2 ⊢ ((𝜑 ∧ (Σ^‘𝐺) = +∞) → (Σ^‘𝐹) ≤ (Σ^‘𝐺))
11		elinel2 4155	. . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 𝑋 ∩ Fin) → 𝑦 ∈ Fin)
12	11	adantl 487	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑦 ∈ Fin)
13	2	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → 𝐹:𝑋⟶(0[,]+∞))
14	1	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
15		sge0le.g	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:𝑋⟶(0[,]+∞))
16	15	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐺:𝑋⟶(0[,]+∞))
17		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
18	2	ffnd 6710	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 Fn 𝑋)
19		fvelrnb 6945	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 Fn 𝑋 → (+∞ ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = +∞))
20	18, 19	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (+∞ ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = +∞))
21	20	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (+∞ ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = +∞))
22	17, 21	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = +∞)
23		iccssxr 13468	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0[,]+∞) ⊆ ℝ*
24	15	ffvelcdmda 7083	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ (0[,]+∞))
25	23, 24	sselid 3936	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ ℝ*)
26	25	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → (𝐺‘𝑥) ∈ ℝ*)
27		id 23	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹‘𝑥) = +∞ → (𝐹‘𝑥) = +∞)
28	27	eqcomd 2771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑥) = +∞ → +∞ = (𝐹‘𝑥))
29	28	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ = (𝐹‘𝑥))
30		sge0le.le	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ≤ (𝐺‘𝑥))
31	30	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → (𝐹‘𝑥) ≤ (𝐺‘𝑥))
32	29, 31	eqbrtrd 5135	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ ≤ (𝐺‘𝑥))
33	26, 32	xrgepnfd 46080	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → (𝐺‘𝑥) = +∞)
34	33	eqcomd 2771	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ = (𝐺‘𝑥))
35	15	ffnd 6710	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐺 Fn 𝑋)
36	35	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐺 Fn 𝑋)
37		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
38		fnfvelrn 7079	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 Fn 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ ran 𝐺)
39	36, 37, 38	syl2anc 596	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ ran 𝐺)
40	39	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → (𝐺‘𝑥) ∈ ran 𝐺)
41	34, 40	eqeltrd 2865	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ ∈ ran 𝐺)
42	41	ex 418	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = +∞ → +∞ ∈ ran 𝐺))
43	42	adantlr 728	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = +∞ → +∞ ∈ ran 𝐺))
44	43	rexlimdva 3168	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (∃𝑥 ∈ 𝑋 (𝐹‘𝑥) = +∞ → +∞ ∈ ran 𝐺))
45	22, 44	mpd 16	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐺)
46	14, 16, 45	sge0pnfval 47120	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐺) = +∞)
47	46	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐺) = +∞)
48		simplr 781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ +∞ ∈ ran 𝐹) → ¬ (Σ^‘𝐺) = +∞)
49	47, 48	pm2.65da 829	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ¬ +∞ ∈ ran 𝐹)
50	13, 49	fge0iccico 47117	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → 𝐹:𝑋⟶(0[,)+∞))
51	50	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,)+∞))
52		elpwinss 45802	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 𝑋 ∩ Fin) → 𝑦 ⊆ 𝑋)
53	52	adantl 487	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑦 ⊆ 𝑋)
54	51, 53	fssresd 6749	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑦):𝑦⟶(0[,)+∞))
55	12, 54	sge0fsum 47134	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑦)) = Σ𝑥 ∈ 𝑦 ((𝐹 ↾ 𝑦)‘𝑥))
56		rge0ssre 13494	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
57	54	ffvelcdmda 7083	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐹 ↾ 𝑦)‘𝑥) ∈ (0[,)+∞))
58	56, 57	sselid 3936	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐹 ↾ 𝑦)‘𝑥) ∈ ℝ)
59	12, 58	fsumrecl 15803	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑥 ∈ 𝑦 ((𝐹 ↾ 𝑦)‘𝑥) ∈ ℝ)
60	55, 59	eqeltrd 2865	. . . . 5 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑦)) ∈ ℝ)
61	15	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → 𝐺:𝑋⟶(0[,]+∞))
62	1	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → 𝑋 ∈ 𝑉)
63		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ¬ (Σ^‘𝐺) = +∞)
64	62, 61	sge0repnf 47133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ((Σ^‘𝐺) ∈ ℝ ↔ ¬ (Σ^‘𝐺) = +∞))
65	63, 64	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → (Σ^‘𝐺) ∈ ℝ)
66	62, 61, 65	sge0rern 47135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ¬ +∞ ∈ ran 𝐺)
67	61, 66	fge0iccico 47117	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → 𝐺:𝑋⟶(0[,)+∞))
68	67	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐺:𝑋⟶(0[,)+∞))
69	68, 53	fssresd 6749	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 ↾ 𝑦):𝑦⟶(0[,)+∞))
70	12, 69	sge0fsum 47134	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) = Σ𝑥 ∈ 𝑦 ((𝐺 ↾ 𝑦)‘𝑥))
71	69	ffvelcdmda 7083	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐺 ↾ 𝑦)‘𝑥) ∈ (0[,)+∞))
72	56, 71	sselid 3936	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐺 ↾ 𝑦)‘𝑥) ∈ ℝ)
73	12, 72	fsumrecl 15803	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑥 ∈ 𝑦 ((𝐺 ↾ 𝑦)‘𝑥) ∈ ℝ)
74	70, 73	eqeltrd 2865	. . . . 5 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) ∈ ℝ)
75	65	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘𝐺) ∈ ℝ)
76		simplll 787	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → 𝜑)
77	53	sselda 3938	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝑋)
78	76, 77, 30	syl2anc 596	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → (𝐹‘𝑥) ≤ (𝐺‘𝑥))
79		fvres 6904	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑦 → ((𝐹 ↾ 𝑦)‘𝑥) = (𝐹‘𝑥))
80	79	adantl 487	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐹 ↾ 𝑦)‘𝑥) = (𝐹‘𝑥))
81		fvres 6904	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑦 → ((𝐺 ↾ 𝑦)‘𝑥) = (𝐺‘𝑥))
82	81	adantl 487	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐺 ↾ 𝑦)‘𝑥) = (𝐺‘𝑥))
83	80, 82	breq12d 5124	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → (((𝐹 ↾ 𝑦)‘𝑥) ≤ ((𝐺 ↾ 𝑦)‘𝑥) ↔ (𝐹‘𝑥) ≤ (𝐺‘𝑥)))
84	78, 83	mpbird 260	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐹 ↾ 𝑦)‘𝑥) ≤ ((𝐺 ↾ 𝑦)‘𝑥))
85	12, 58, 72, 84	fsumle 15869	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑥 ∈ 𝑦 ((𝐹 ↾ 𝑦)‘𝑥) ≤ Σ𝑥 ∈ 𝑦 ((𝐺 ↾ 𝑦)‘𝑥))
86	55, 70	breq12d 5124	. . . . . 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 260	. . . . 5 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑦)) ≤ (Σ^‘(𝐺 ↾ 𝑦)))
88	1	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑋 ∈ 𝑉)
89	15	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐺:𝑋⟶(0[,]+∞))
90	88, 89	sge0less 47139	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) ≤ (Σ^‘𝐺))
91	90	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) ≤ (Σ^‘𝐺))
92	60, 74, 75, 87, 91	letrd 11378	. . . 4 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑦)) ≤ (Σ^‘𝐺))
93	92	ralrimiva 3159	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ∀𝑦 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹 ↾ 𝑦)) ≤ (Σ^‘𝐺))
94	62, 61	sge0xrcl 47132	. . . 4 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → (Σ^‘𝐺) ∈ ℝ*)
95	62, 13, 94	sge0lefi 47145	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ((Σ^‘𝐹) ≤ (Σ^‘𝐺) ↔ ∀𝑦 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹 ↾ 𝑦)) ≤ (Σ^‘𝐺)))
96	93, 95	mpbird 260	. 2 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → (Σ^‘𝐹) ≤ (Σ^‘𝐺))
97	10, 96	pm2.61dan 825	1 ⊢ (𝜑 → (Σ^‘𝐹) ≤ (Σ^‘𝐺))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091   ∩ cin 3905   ⊆ wss 3906  𝒫 cpw 4564   class class class wbr 5111  ran crn 5664   ↾ cres 5665   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7416  Fincfn 8945  ℝcr 11110  0cc0 11111  +∞cpnf 11251  ℝ^*cxr 11253   ≤ cle 11255  [,)cico 13385  [,]cicc 13386  Σcsu 15756  Σ^{^}csumge0 47109

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-inf2 9613  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188  ax-pre-sup 11189

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9405  df-oi 9475  df-card 9937  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-div 11883  df-nn 12245  df-2 12314  df-3 12315  df-n0 12516  df-z 12603  df-uz 12874  df-rp 13028  df-ico 13389  df-icc 13390  df-fz 13547  df-fzo 13695  df-seq 14051  df-exp 14111  df-hash 14380  df-cj 15169  df-re 15170  df-im 15171  df-sqrt 15305  df-abs 15306  df-clim 15558  df-sum 15757  df-sumge0 47110

This theorem is used by:  sge0lempt  47157