Theorem sge0le 46028

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 46006	. . . . 5 ⊢ (𝜑 → (Σ^‘𝐹) ∈ ℝ*)
4		pnfge 13164	. . . . 5 ⊢ ((Σ^‘𝐹) ∈ ℝ* → (Σ^‘𝐹) ≤ +∞)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → (Σ^‘𝐹) ≤ +∞)
6	5	adantr 479	. . 3 ⊢ ((𝜑 ∧ (Σ^‘𝐺) = +∞) → (Σ^‘𝐹) ≤ +∞)
7		id 22	. . . . 5 ⊢ ((Σ^‘𝐺) = +∞ → (Σ^‘𝐺) = +∞)
8	7	eqcomd 2732	. . . 4 ⊢ ((Σ^‘𝐺) = +∞ → +∞ = (Σ^‘𝐺))
9	8	adantl 480	. . 3 ⊢ ((𝜑 ∧ (Σ^‘𝐺) = +∞) → +∞ = (Σ^‘𝐺))
10	6, 9	breqtrd 5179	. 2 ⊢ ((𝜑 ∧ (Σ^‘𝐺) = +∞) → (Σ^‘𝐹) ≤ (Σ^‘𝐺))
11		elinel2 4197	. . . . . . . 8 ⊢ (𝑦 ∈ (𝒫 𝑋 ∩ Fin) → 𝑦 ∈ Fin)
12	11	adantl 480	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑦 ∈ Fin)
13	2	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → 𝐹:𝑋⟶(0[,]+∞))
14	1	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
15		sge0le.g	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺:𝑋⟶(0[,]+∞))
16	15	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐺:𝑋⟶(0[,]+∞))
17		simpr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
18	2	ffnd 6729	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 Fn 𝑋)
19		fvelrnb 6963	. . . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . 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 13461	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0[,]+∞) ⊆ ℝ*
24	15	ffvelcdmda 7098	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ (0[,]+∞))
25	23, 24	sselid 3977	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ ℝ*)
26	25	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → (𝐺‘𝑥) ∈ ℝ*)
27		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐹‘𝑥) = +∞ → (𝐹‘𝑥) = +∞)
28	27	eqcomd 2732	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑥) = +∞ → +∞ = (𝐹‘𝑥))
29	28	adantl 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ = (𝐹‘𝑥))
30		sge0le.le	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ≤ (𝐺‘𝑥))
31	30	adantr 479	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → (𝐹‘𝑥) ≤ (𝐺‘𝑥))
32	29, 31	eqbrtrd 5175	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ ≤ (𝐺‘𝑥))
33	26, 32	xrgepnfd 44946	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → (𝐺‘𝑥) = +∞)
34	33	eqcomd 2732	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ = (𝐺‘𝑥))
35	15	ffnd 6729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐺 Fn 𝑋)
36	35	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐺 Fn 𝑋)
37		simpr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
38		fnfvelrn 7094	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 Fn 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ ran 𝐺)
39	36, 37, 38	syl2anc 582	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐺‘𝑥) ∈ ran 𝐺)
40	39	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → (𝐺‘𝑥) ∈ ran 𝐺)
41	34, 40	eqeltrd 2826	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ (𝐹‘𝑥) = +∞) → +∞ ∈ ran 𝐺)
42	41	ex 411	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = +∞ → +∞ ∈ ran 𝐺))
43	42	adantlr 713	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = +∞ → +∞ ∈ ran 𝐺))
44	43	rexlimdva 3145	. . . . . . . . . . . . . 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 45994	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐺) = +∞)
47	46	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐺) = +∞)
48		simplr 767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ +∞ ∈ ran 𝐹) → ¬ (Σ^‘𝐺) = +∞)
49	47, 48	pm2.65da 815	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ¬ +∞ ∈ ran 𝐹)
50	13, 49	fge0iccico 45991	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → 𝐹:𝑋⟶(0[,)+∞))
51	50	adantr 479	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,)+∞))
52		elpwinss 44650	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 𝑋 ∩ Fin) → 𝑦 ⊆ 𝑋)
53	52	adantl 480	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑦 ⊆ 𝑋)
54	51, 53	fssresd 6769	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑦):𝑦⟶(0[,)+∞))
55	12, 54	sge0fsum 46008	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑦)) = Σ𝑥 ∈ 𝑦 ((𝐹 ↾ 𝑦)‘𝑥))
56		rge0ssre 13487	. . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ
57	54	ffvelcdmda 7098	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐹 ↾ 𝑦)‘𝑥) ∈ (0[,)+∞))
58	56, 57	sselid 3977	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐹 ↾ 𝑦)‘𝑥) ∈ ℝ)
59	12, 58	fsumrecl 15738	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑥 ∈ 𝑦 ((𝐹 ↾ 𝑦)‘𝑥) ∈ ℝ)
60	55, 59	eqeltrd 2826	. . . . 5 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑦)) ∈ ℝ)
61	15	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → 𝐺:𝑋⟶(0[,]+∞))
62	1	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → 𝑋 ∈ 𝑉)
63		simpr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ¬ (Σ^‘𝐺) = +∞)
64	62, 61	sge0repnf 46007	. . . . . . . . . . . 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 46009	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ¬ +∞ ∈ ran 𝐺)
67	61, 66	fge0iccico 45991	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → 𝐺:𝑋⟶(0[,)+∞))
68	67	adantr 479	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐺:𝑋⟶(0[,)+∞))
69	68, 53	fssresd 6769	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 ↾ 𝑦):𝑦⟶(0[,)+∞))
70	12, 69	sge0fsum 46008	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) = Σ𝑥 ∈ 𝑦 ((𝐺 ↾ 𝑦)‘𝑥))
71	69	ffvelcdmda 7098	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐺 ↾ 𝑦)‘𝑥) ∈ (0[,)+∞))
72	56, 71	sselid 3977	. . . . . . 7 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐺 ↾ 𝑦)‘𝑥) ∈ ℝ)
73	12, 72	fsumrecl 15738	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑥 ∈ 𝑦 ((𝐺 ↾ 𝑦)‘𝑥) ∈ ℝ)
74	70, 73	eqeltrd 2826	. . . . 5 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) ∈ ℝ)
75	65	adantr 479	. . . . 5 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘𝐺) ∈ ℝ)
76		simplll 773	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → 𝜑)
77	53	sselda 3979	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝑋)
78	76, 77, 30	syl2anc 582	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → (𝐹‘𝑥) ≤ (𝐺‘𝑥))
79		fvres 6920	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑦 → ((𝐹 ↾ 𝑦)‘𝑥) = (𝐹‘𝑥))
80	79	adantl 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐹 ↾ 𝑦)‘𝑥) = (𝐹‘𝑥))
81		fvres 6920	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑦 → ((𝐺 ↾ 𝑦)‘𝑥) = (𝐺‘𝑥))
82	81	adantl 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑥 ∈ 𝑦) → ((𝐺 ↾ 𝑦)‘𝑥) = (𝐺‘𝑥))
83	80, 82	breq12d 5166	. . . . . . . 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 15803	. . . . . 6 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑥 ∈ 𝑦 ((𝐹 ↾ 𝑦)‘𝑥) ≤ Σ𝑥 ∈ 𝑦 ((𝐺 ↾ 𝑦)‘𝑥))
86	55, 70	breq12d 5166	. . . . . 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 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑋 ∈ 𝑉)
89	15	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐺:𝑋⟶(0[,]+∞))
90	88, 89	sge0less 46013	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) ≤ (Σ^‘𝐺))
91	90	adantlr 713	. . . . 5 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐺 ↾ 𝑦)) ≤ (Σ^‘𝐺))
92	60, 74, 75, 87, 91	letrd 11421	. . . 4 ⊢ (((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) ∧ 𝑦 ∈ (𝒫 𝑋 ∩ Fin)) → (Σ^‘(𝐹 ↾ 𝑦)) ≤ (Σ^‘𝐺))
93	92	ralrimiva 3136	. . 3 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → ∀𝑦 ∈ (𝒫 𝑋 ∩ Fin)(Σ^‘(𝐹 ↾ 𝑦)) ≤ (Σ^‘𝐺))
94	62, 61	sge0xrcl 46006	. . . 4 ⊢ ((𝜑 ∧ ¬ (Σ^‘𝐺) = +∞) → (Σ^‘𝐺) ∈ ℝ*)
95	62, 13, 94	sge0lefi 46019	. . 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 811	1 ⊢ (𝜑 → (Σ^‘𝐹) ≤ (Σ^‘𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ∀wral 3051  ∃wrex 3060   ∩ cin 3946   ⊆ wss 3947  𝒫 cpw 4607   class class class wbr 5153  ran crn 5683   ↾ cres 5684   Fn wfn 6549  ⟶wf 6550  ‘cfv 6554  (class class class)co 7424  Fincfn 8974  ℝcr 11157  0cc0 11158  +∞cpnf 11295  ℝ^*cxr 11297   ≤ cle 11299  [,)cico 13380  [,]cicc 13381  Σcsu 15690  Σ^{^}csumge0 45983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-sup 9485  df-oi 9553  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12611  df-uz 12875  df-rp 13029  df-ico 13384  df-icc 13385  df-fz 13539  df-fzo 13682  df-seq 14022  df-exp 14082  df-hash 14348  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241  df-clim 15490  df-sum 15691  df-sumge0 45984

This theorem is referenced by:  sge0lempt  46031