sge0rpcpnf - Mathbox for Glauco Siliprandi

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Theorem sge0rpcpnf 47062

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 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → 𝐴 ∈ 𝑉)
3		0xr 11256	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ^*
4	3	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℝ^*)
5		pnfxr 11263	. . . . . . . . . . . . . . . 16 ⊢ +∞ ∈ ℝ^*
6	5	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → +∞ ∈ ℝ^*)
7		sge0rpcpnf.b	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
8	7	rpxrd 13061	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
9	7	rpge0d 13064	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ 𝐵)
10	7	rpred 13060	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℝ)
11		ltpnf 13145	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
12	10, 11	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 < +∞)
13	8, 6, 12	xrltled 13175	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≤ +∞)
14	4, 6, 8, 9, 13	eliccxrd 46170	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞))
15	14	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
16		eqid 2769	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
17	15, 16	fmptd 7110	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
18	17	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
19	2, 18	sge0xrcl 47026	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ^*)
20	5	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → +∞ ∈ ℝ^*)
21		simpr 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞)
22	19, 20, 21	xrgtned 13189	. . . . . . . . 9 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → +∞ ≠ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)))
23	22	necomd 3019	. . . . . . . 8 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ≠ +∞)
24	23	neneqd 2969	. . . . . . 7 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞)
25	2, 18	sge0repnf 47027	. . . . . . 7 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ ↔ ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞))
26	24, 25	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)
27	10	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → 𝐵 ∈ ℝ)
28	7	rpne0d 13065	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≠ 0)
29	28	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → 𝐵 ≠ 0)
30	26, 27, 29	redivcld 12043	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) ∈ ℝ)
31		arch 12501	. . . . 5 ⊢ (((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) ∈ ℝ → ∃𝑛 ∈ ℕ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛)
32	30, 31	syl 18	. . . 4 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ∃𝑛 ∈ ℕ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛)
33		sge0rpcpnf.nfi	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝐴 ∈ Fin)
34		ishashinf 14500	. . . . . . . . . . . . 13 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ℕ ∃𝑦 ∈ 𝒫 𝐴(♯‘𝑦) = 𝑛)
35	33, 34	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∃𝑦 ∈ 𝒫 𝐴(♯‘𝑦) = 𝑛)
36	35	r19.21bi 3263	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑦 ∈ 𝒫 𝐴(♯‘𝑦) = 𝑛)
37		df-rex 3096	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝒫 𝐴(♯‘𝑦) = 𝑛 ↔ ∃𝑦(𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛))
38	36, 37	sylib 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑦(𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛))
39	38	adantlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ) → ∃𝑦(𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛))
40	39	3adant3 1148	. . . . . . . 8 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → ∃𝑦(𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛))
41		nfv 1941	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛)
42		simprl 782	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → 𝑦 ∈ 𝒫 𝐴)
43		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ (♯‘𝑦) = 𝑛) → (♯‘𝑦) = 𝑛)
44		simpl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ (♯‘𝑦) = 𝑛) → 𝑛 ∈ ℕ)
45	43, 44	eqeltrd 2869	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (♯‘𝑦) = 𝑛) → (♯‘𝑦) ∈ ℕ)
46		nnnn0 12511	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑦) ∈ ℕ → (♯‘𝑦) ∈ ℕ₀)
47		vex 3465	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
48	47	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑦) ∈ ℕ → 𝑦 ∈ V)
49		hashclb 14394	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ V → (𝑦 ∈ Fin ↔ (♯‘𝑦) ∈ ℕ₀))
50	48, 49	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑦) ∈ ℕ → (𝑦 ∈ Fin ↔ (♯‘𝑦) ∈ ℕ₀))
51	46, 50	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑦) ∈ ℕ → 𝑦 ∈ Fin)
52	45, 51	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ (♯‘𝑦) = 𝑛) → 𝑦 ∈ Fin)
53	52	adantrl 728	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → 𝑦 ∈ Fin)
54	53	3ad2antl2 1203	. . . . . . . . . . . 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 4159	. . . . . . . . . . 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 1154	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛)
57	26	3ad2ant1 1149	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)
58		nnre 12240	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
59	58	3ad2ant2 1150	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → 𝑛 ∈ ℝ)
60	7	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → 𝐵 ∈ ℝ⁺)
61	60	3ad2ant1 1149	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → 𝐵 ∈ ℝ⁺)
62	57, 59, 61	ltdivmul2d 13112	. . . . . . . . . . . . . 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 235	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (𝑛 · 𝐵))
64	63	adantr 485	. . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 𝐵 ∈ ℝ^*)
69	9	ad3antrrr 742	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 0 ≤ 𝐵)
70	12	ad3antrrr 742	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 𝐵 < +∞)
71	66, 67, 68, 69, 70	elicod 13422	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 𝐵 ∈ (0[,)+∞))
72	65, 71	sge0fsummpt 47031	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)) = Σ𝑥 ∈ 𝑦 𝐵)
73	10	recnd 11237	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℂ)
74	73	ad2antrr 738	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → 𝐵 ∈ ℂ)
75		fsumconst 15841	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ ℂ) → Σ𝑥 ∈ 𝑦 𝐵 = ((♯‘𝑦) · 𝐵))
76	65, 74, 75	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → Σ𝑥 ∈ 𝑦 𝐵 = ((♯‘𝑦) · 𝐵))
77		oveq1 7418	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑦) = 𝑛 → ((♯‘𝑦) · 𝐵) = (𝑛 · 𝐵))
78	77	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛) → ((♯‘𝑦) · 𝐵) = (𝑛 · 𝐵))
79	78	adantl 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → ((♯‘𝑦) · 𝐵) = (𝑛 · 𝐵))
80	72, 76, 79	3eqtrrd 2809	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → (𝑛 · 𝐵) = (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
81	80	adantllr 731	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → (𝑛 · 𝐵) = (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
82	81	3adantl3 1185	. . . . . . . . . . . 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 5139	. . . . . . . . . . 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 520	. . . . . . . . . 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 417	. . . . . . . . 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 2258	. . . . . . . 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 16	. . . . . . 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 3096	. . . . . . 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 237	. . . . . 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 1135	. . . . 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 3170	. . . 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 16	. . 3 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
93	1	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐴 ∈ 𝑉)
94	15	adantlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
95		elpwinss 45696	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ⊆ 𝐴)
96	95	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ⊆ 𝐴)
97	93, 94, 96	sge0lessmpt 47040	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)) ≤ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)))
98		simpr 489	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
99	14	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑦) → 𝐵 ∈ (0[,]+∞))
100		eqid 2769	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑦 ↦ 𝐵) = (𝑥 ∈ 𝑦 ↦ 𝐵)
101	99, 100	fmptd 7110	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑦 ↦ 𝐵):𝑦⟶(0[,]+∞))
102	101	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑥 ∈ 𝑦 ↦ 𝐵):𝑦⟶(0[,]+∞))
103	98, 102	sge0xrcl 47026	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)) ∈ ℝ^*)
104	1, 17	sge0xrcl 47026	. . . . . . . . 9 ⊢ (𝜑 → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ^*)
105	104	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ^*)
106	103, 105	xrlenltd 11275	. . . . . . 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 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
108	107	ralrimiva 3163	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ (𝒫 𝐴 ∩ Fin) ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
109		ralnex 3097	. . . . 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 221	. . . 4 ⊢ (𝜑 → ¬ ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
111	110	adantr 485	. . 3 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ¬ ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
112	92, 111	pm2.65da 828	. 2 ⊢ (𝜑 → ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞)
113		nltpnft 13190	. . 3 ⊢ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ^* → ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞ ↔ ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞))
114	104, 113	syl 18	. 2 ⊢ (𝜑 → ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞ ↔ ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞))
115	112, 114	mpbird 260	1 ⊢ (𝜑 → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∃wex 1806 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 Vcvv 3461 ∩ cin 3910 ⊆ wss 3911 𝒫 cpw 4565 class class class wbr 5111 ↦ cmpt 5194 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 Fincfn 8943 ℂcc 11098 ℝcr 11099 0cc0 11100 · cmul 11105 +∞cpnf 11240 ℝ^*cxr 11242 < clt 11243 ≤ cle 11244 / cdiv 11871 ℕcn 12233 ℕ₀cn0 12504 ℝ⁺crp 13016 [,]cicc 13375 ♯chash 14366 Σcsu 15737 Σ^{^}csumge0 47003

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3375 df-reu 3376 df-rab 3423 df-v 3463 df-sbc 3752 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9402 df-oi 9472 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-n0 12505 df-z 12592 df-uz 12863 df-rp 13017 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-sum 15738 df-sumge0 47004

This theorem is referenced by: hoicvrrex 47197

W3C validator