sge0rpcpnf - Mathbox for Glauco Siliprandi

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Theorem sge0rpcpnf 46426

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 11228	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ^*
4	3	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℝ^*)
5		pnfxr 11235	. . . . . . . . . . . . . . . 16 ⊢ +∞ ∈ ℝ^*
6	5	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → +∞ ∈ ℝ^*)
7		sge0rpcpnf.b	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
8	7	rpxrd 13003	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
9	7	rpge0d 13006	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ 𝐵)
10	7	rpred 13002	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℝ)
11		ltpnf 13087	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
12	10, 11	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 < +∞)
13	8, 6, 12	xrltled 13117	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≤ +∞)
14	4, 6, 8, 9, 13	eliccxrd 45532	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞))
15	14	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
16		eqid 2730	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
17	15, 16	fmptd 7089	. . . . . . . . . . . 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 46390	. . . . . . . . . 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 45325	. . . . . . . . 9 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → +∞ ≠ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)))
23	22	necomd 2981	. . . . . . . 8 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ≠ +∞)
24	23	neneqd 2931	. . . . . . 7 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞)
25	2, 18	sge0repnf 46391	. . . . . . 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 13007	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≠ 0)
29	28	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → 𝐵 ≠ 0)
30	26, 27, 29	redivcld 12017	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) ∈ ℝ)
31		arch 12446	. . . . 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 14435	. . . . . . . . . . . . 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 3230	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑦 ∈ 𝒫 𝐴(♯‘𝑦) = 𝑛)
37		df-rex 3055	. . . . . . . . . . 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 2829	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (♯‘𝑦) = 𝑛) → (♯‘𝑦) ∈ ℕ)
46		nnnn0 12456	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑦) ∈ ℕ → (♯‘𝑦) ∈ ℕ₀)
47		vex 3454	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
48	47	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑦) ∈ ℕ → 𝑦 ∈ V)
49		hashclb 14330	. . . . . . . . . . . . . . . . 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 4166	. . . . . . . . . . 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 12200	. . . . . . . . . . . . . . . 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 13054	. . . . . . . . . . . . . 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 13363	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 𝐵 ∈ (0[,)+∞))
72	65, 71	sge0fsummpt 46395	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)) = Σ𝑥 ∈ 𝑦 𝐵)
73	10	recnd 11209	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℂ)
74	73	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → 𝐵 ∈ ℂ)
75		fsumconst 15763	. . . . . . . . . . . . . . . 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 7397	. . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . 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 5136	. . . . . . . . . . 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 3055	. . . . . . 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 3133	. . . 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 45050	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ⊆ 𝐴)
96	95	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ⊆ 𝐴)
97	93, 94, 96	sge0lessmpt 46404	. . . . . . 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 2730	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑦 ↦ 𝐵) = (𝑥 ∈ 𝑦 ↦ 𝐵)
101	99, 100	fmptd 7089	. . . . . . . . . 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 46390	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)) ∈ ℝ^*)
104	1, 17	sge0xrcl 46390	. . . . . . . . 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 11247	. . . . . . 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 3126	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ (𝒫 𝐴 ∩ Fin) ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
109		ralnex 3056	. . . . 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 13131	. . 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 2109 ≠ wne 2926 ∀wral 3045 ∃wrex 3054 Vcvv 3450 ∩ cin 3916 ⊆ wss 3917 𝒫 cpw 4566 class class class wbr 5110 ↦ cmpt 5191 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 Fincfn 8921 ℂcc 11073 ℝcr 11074 0cc0 11075 · cmul 11080 +∞cpnf 11212 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 / cdiv 11842 ℕcn 12193 ℕ₀cn0 12449 ℝ⁺crp 12958 [,]cicc 13316 ♯chash 14302 Σcsu 15659 Σ^{^}csumge0 46367

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

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 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-rp 12959 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-sum 15660 df-sumge0 46368

This theorem is referenced by: hoicvrrex 46561

W3C validator