sge0rpcpnf - Mathbox for Glauco Siliprandi

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Theorem sge0rpcpnf 46450

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 11282	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ^*
4	3	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℝ^*)
5		pnfxr 11289	. . . . . . . . . . . . . . . 16 ⊢ +∞ ∈ ℝ^*
6	5	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → +∞ ∈ ℝ^*)
7		sge0rpcpnf.b	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
8	7	rpxrd 13052	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
9	7	rpge0d 13055	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ 𝐵)
10	7	rpred 13051	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℝ)
11		ltpnf 13136	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
12	10, 11	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 < +∞)
13	8, 6, 12	xrltled 13166	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≤ +∞)
14	4, 6, 8, 9, 13	eliccxrd 45556	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞))
15	14	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
16		eqid 2735	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
17	15, 16	fmptd 7104	. . . . . . . . . . . 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 46414	. . . . . . . . . 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 45349	. . . . . . . . 9 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → +∞ ≠ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)))
23	22	necomd 2987	. . . . . . . 8 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ≠ +∞)
24	23	neneqd 2937	. . . . . . 7 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞)
25	2, 18	sge0repnf 46415	. . . . . . 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 13056	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≠ 0)
29	28	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → 𝐵 ≠ 0)
30	26, 27, 29	redivcld 12069	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) ∈ ℝ)
31		arch 12498	. . . . 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 14481	. . . . . . . . . . . . 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 3234	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑦 ∈ 𝒫 𝐴(♯‘𝑦) = 𝑛)
37		df-rex 3061	. . . . . . . . . . 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 715	. . . . . . . . 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 1914	. . . . . . . . 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 2834	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (♯‘𝑦) = 𝑛) → (♯‘𝑦) ∈ ℕ)
46		nnnn0 12508	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑦) ∈ ℕ → (♯‘𝑦) ∈ ℕ₀)
47		vex 3463	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
48	47	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑦) ∈ ℕ → 𝑦 ∈ V)
49		hashclb 14376	. . . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → 𝑦 ∈ Fin)
54	53	3ad2antl2 1187	. . . . . . . . . . . 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 4175	. . . . . . . . . . 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 12247	. . . . . . . . . . . . . . . 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 13103	. . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 𝐵 ∈ ℝ^*)
69	9	ad3antrrr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 0 ≤ 𝐵)
70	12	ad3antrrr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 𝐵 < +∞)
71	66, 67, 68, 69, 70	elicod 13412	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 𝐵 ∈ (0[,)+∞))
72	65, 71	sge0fsummpt 46419	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)) = Σ𝑥 ∈ 𝑦 𝐵)
73	10	recnd 11263	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℂ)
74	73	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → 𝐵 ∈ ℂ)
75		fsumconst 15806	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ ℂ) → Σ𝑥 ∈ 𝑦 𝐵 = ((♯‘𝑦) · 𝐵))
76	65, 74, 75	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → Σ𝑥 ∈ 𝑦 𝐵 = ((♯‘𝑦) · 𝐵))
77		oveq1 7412	. . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → (𝑛 · 𝐵) = (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
81	80	adantllr 719	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → (𝑛 · 𝐵) = (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
82	81	3adantl3 1169	. . . . . . . . . . . 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 5145	. . . . . . . . . . 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 2216	. . . . . . . 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 3061	. . . . . . 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 3139	. . . 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 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
95		elpwinss 45073	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ⊆ 𝐴)
96	95	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ⊆ 𝐴)
97	93, 94, 96	sge0lessmpt 46428	. . . . . . 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 2735	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑦 ↦ 𝐵) = (𝑥 ∈ 𝑦 ↦ 𝐵)
101	99, 100	fmptd 7104	. . . . . . . . . 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 46414	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)) ∈ ℝ^*)
104	1, 17	sge0xrcl 46414	. . . . . . . . 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 11301	. . . . . . 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 3132	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ (𝒫 𝐴 ∩ Fin) ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
109		ralnex 3062	. . . . 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 13180	. . 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 1086 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 Vcvv 3459 ∩ cin 3925 ⊆ wss 3926 𝒫 cpw 4575 class class class wbr 5119 ↦ cmpt 5201 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 Fincfn 8959 ℂcc 11127 ℝcr 11128 0cc0 11129 · cmul 11134 +∞cpnf 11266 ℝ^*cxr 11268 < clt 11269 ≤ cle 11270 / cdiv 11894 ℕcn 12240 ℕ₀cn0 12501 ℝ⁺crp 13008 [,]cicc 13365 ♯chash 14348 Σcsu 15702 Σ^{^}csumge0 46391

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9454 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-n0 12502 df-z 12589 df-uz 12853 df-rp 13009 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-seq 14020 df-exp 14080 df-hash 14349 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-clim 15504 df-sum 15703 df-sumge0 46392

This theorem is referenced by: hoicvrrex 46585

W3C validator