sge0rpcpnf - Mathbox for Glauco Siliprandi

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Theorem sge0rpcpnf 46342

Description: The sum of an infinite number of a positive constant, is +∞ (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypotheses**
Ref	Expression
sge0rpcpnf.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
sge0rpcpnf.nfi	⊢ (𝜑 → ¬ 𝐴 ∈ Fin)
sge0rpcpnf.b	⊢ (𝜑 → 𝐵 ∈ ℝ⁺)

**Assertion**
Ref	Expression
sge0rpcpnf	⊢ (𝜑 → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝜑,𝑥 Allowed substitution hint: 𝑉(𝑥)

Proof of Theorem sge0rpcpnf Dummy variables 𝑛 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sge0rpcpnf.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2	1	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → 𝐴 ∈ 𝑉)
3		0xr 11337	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ^*
4	3	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℝ^*)
5		pnfxr 11344	. . . . . . . . . . . . . . . 16 ⊢ +∞ ∈ ℝ^*
6	5	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → +∞ ∈ ℝ^*)
7		sge0rpcpnf.b	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
8	7	rpxrd 13100	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
9	7	rpge0d 13103	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ 𝐵)
10	7	rpred 13099	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℝ)
11		ltpnf 13183	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
12	10, 11	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 < +∞)
13	8, 6, 12	xrltled 13212	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≤ +∞)
14	4, 6, 8, 9, 13	eliccxrd 45445	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞))
15	14	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
16		eqid 2740	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
17	15, 16	fmptd 7148	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
18	17	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
19	2, 18	sge0xrcl 46306	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ^*)
20	5	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → +∞ ∈ ℝ^*)
21		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞)
22	19, 20, 21	xrgtned 45237	. . . . . . . . 9 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → +∞ ≠ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)))
23	22	necomd 3002	. . . . . . . 8 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ≠ +∞)
24	23	neneqd 2951	. . . . . . 7 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞)
25	2, 18	sge0repnf 46307	. . . . . . 7 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ ↔ ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞))
26	24, 25	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)
27	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → 𝐵 ∈ ℝ)
28	7	rpne0d 13104	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≠ 0)
29	28	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → 𝐵 ≠ 0)
30	26, 27, 29	redivcld 12122	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) ∈ ℝ)
31		arch 12550	. . . . 5 ⊢ (((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) ∈ ℝ → ∃𝑛 ∈ ℕ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛)
32	30, 31	syl 17	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ∃𝑛 ∈ ℕ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛)
33		sge0rpcpnf.nfi	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝐴 ∈ Fin)
34		ishashinf 14512	. . . . . . . . . . . . 13 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ℕ ∃𝑦 ∈ 𝒫 𝐴(♯‘𝑦) = 𝑛)
35	33, 34	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∃𝑦 ∈ 𝒫 𝐴(♯‘𝑦) = 𝑛)
36	35	r19.21bi 3257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑦 ∈ 𝒫 𝐴(♯‘𝑦) = 𝑛)
37		df-rex 3077	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝒫 𝐴(♯‘𝑦) = 𝑛 ↔ ∃𝑦(𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛))
38	36, 37	sylib 218	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑦(𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛))
39	38	adantlr 714	. . . . . . . . 9 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ) → ∃𝑦(𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛))
40	39	3adant3 1132	. . . . . . . 8 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → ∃𝑦(𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛))
41		nfv 1913	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛)
42		simprl 770	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → 𝑦 ∈ 𝒫 𝐴)
43		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ (♯‘𝑦) = 𝑛) → (♯‘𝑦) = 𝑛)
44		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ (♯‘𝑦) = 𝑛) → 𝑛 ∈ ℕ)
45	43, 44	eqeltrd 2844	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (♯‘𝑦) = 𝑛) → (♯‘𝑦) ∈ ℕ)
46		nnnn0 12560	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑦) ∈ ℕ → (♯‘𝑦) ∈ ℕ₀)
47		vex 3492	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
48	47	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑦) ∈ ℕ → 𝑦 ∈ V)
49		hashclb 14407	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ V → (𝑦 ∈ Fin ↔ (♯‘𝑦) ∈ ℕ₀))
50	48, 49	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑦) ∈ ℕ → (𝑦 ∈ Fin ↔ (♯‘𝑦) ∈ ℕ₀))
51	46, 50	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑦) ∈ ℕ → 𝑦 ∈ Fin)
52	45, 51	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ (♯‘𝑦) = 𝑛) → 𝑦 ∈ Fin)
53	52	adantrl 715	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → 𝑦 ∈ Fin)
54	53	3ad2antl2 1186	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → 𝑦 ∈ Fin)
55	42, 54	elind 4223	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
56		simp3 1138	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛)
57	26	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)
58		nnre 12300	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
59	58	3ad2ant2 1134	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → 𝑛 ∈ ℝ)
60	7	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → 𝐵 ∈ ℝ⁺)
61	60	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → 𝐵 ∈ ℝ⁺)
62	57, 59, 61	ltdivmul2d 13151	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → (((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛 ↔ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (𝑛 · 𝐵)))
63	56, 62	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (𝑛 · 𝐵))
64	63	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (𝑛 · 𝐵))
65	53	adantll 713	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → 𝑦 ∈ Fin)
66	3	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 0 ∈ ℝ^*)
67	5	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → +∞ ∈ ℝ^*)
68	8	ad3antrrr 729	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 𝐵 ∈ ℝ^*)
69	9	ad3antrrr 729	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 0 ≤ 𝐵)
70	12	ad3antrrr 729	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 𝐵 < +∞)
71	66, 67, 68, 69, 70	elicod 13457	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 𝐵 ∈ (0[,)+∞))
72	65, 71	sge0fsummpt 46311	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)) = Σ𝑥 ∈ 𝑦 𝐵)
73	10	recnd 11318	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℂ)
74	73	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → 𝐵 ∈ ℂ)
75		fsumconst 15838	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ ℂ) → Σ𝑥 ∈ 𝑦 𝐵 = ((♯‘𝑦) · 𝐵))
76	65, 74, 75	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → Σ𝑥 ∈ 𝑦 𝐵 = ((♯‘𝑦) · 𝐵))
77		oveq1 7455	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑦) = 𝑛 → ((♯‘𝑦) · 𝐵) = (𝑛 · 𝐵))
78	77	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛) → ((♯‘𝑦) · 𝐵) = (𝑛 · 𝐵))
79	78	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → ((♯‘𝑦) · 𝐵) = (𝑛 · 𝐵))
80	72, 76, 79	3eqtrrd 2785	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → (𝑛 · 𝐵) = (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
81	80	adantllr 718	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → (𝑛 · 𝐵) = (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
82	81	3adantl3 1168	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → (𝑛 · 𝐵) = (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
83	64, 82	breqtrd 5192	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
84	55, 83	jca 511	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵))))
85	84	ex 412	. . . . . . . . 9 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → ((𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛) → (𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))))
86	41, 85	eximd 2217	. . . . . . . 8 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → (∃𝑦(𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛) → ∃𝑦(𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))))
87	40, 86	mpd 15	. . . . . . 7 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → ∃𝑦(𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵))))
88		df-rex 3077	. . . . . . 7 ⊢ (∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)) ↔ ∃𝑦(𝑦 ∈ (𝒫 𝐴 ∩ Fin) ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵))))
89	87, 88	sylibr 234	. . . . . 6 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
90	89	3exp 1119	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (𝑛 ∈ ℕ → (((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))))
91	90	rexlimdv 3159	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (∃𝑛 ∈ ℕ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛 → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵))))
92	32, 91	mpd 15	. . 3 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
93	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐴 ∈ 𝑉)
94	15	adantlr 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
95		elpwinss 44951	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ⊆ 𝐴)
96	95	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ⊆ 𝐴)
97	93, 94, 96	sge0lessmpt 46320	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)) ≤ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)))
98		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
99	14	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑦) → 𝐵 ∈ (0[,]+∞))
100		eqid 2740	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑦 ↦ 𝐵) = (𝑥 ∈ 𝑦 ↦ 𝐵)
101	99, 100	fmptd 7148	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑦 ↦ 𝐵):𝑦⟶(0[,]+∞))
102	101	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑥 ∈ 𝑦 ↦ 𝐵):𝑦⟶(0[,]+∞))
103	98, 102	sge0xrcl 46306	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)) ∈ ℝ^*)
104	1, 17	sge0xrcl 46306	. . . . . . . . 9 ⊢ (𝜑 → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ^*)
105	104	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ^*)
106	103, 105	xrlenltd 11356	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ((Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)) ≤ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ↔ ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵))))
107	97, 106	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
108	107	ralrimiva 3152	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ (𝒫 𝐴 ∩ Fin) ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
109		ralnex 3078	. . . . 5 ⊢ (∀𝑦 ∈ (𝒫 𝐴 ∩ Fin) ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)) ↔ ¬ ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
110	108, 109	sylib 218	. . . 4 ⊢ (𝜑 → ¬ ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
111	110	adantr 480	. . 3 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ¬ ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
112	92, 111	pm2.65da 816	. 2 ⊢ (𝜑 → ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞)
113		nltpnft 13226	. . 3 ⊢ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ^* → ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞ ↔ ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞))
114	104, 113	syl 17	. 2 ⊢ (𝜑 → ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞ ↔ ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞))
115	112, 114	mpbird 257	1 ⊢ (𝜑 → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∃wex 1777 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 Vcvv 3488 ∩ cin 3975 ⊆ wss 3976 𝒫 cpw 4622 class class class wbr 5166 ↦ cmpt 5249 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 Fincfn 9003 ℂcc 11182 ℝcr 11183 0cc0 11184 · cmul 11189 +∞cpnf 11321 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 / cdiv 11947 ℕcn 12293 ℕ₀cn0 12553 ℝ⁺crp 13057 [,]cicc 13410 ♯chash 14379 Σcsu 15734 Σ^{^}csumge0 46283

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-sum 15735 df-sumge0 46284

This theorem is referenced by: hoicvrrex 46477

W3C validator