sge0rpcpnf - Mathbox for Glauco Siliprandi

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Theorem sge0rpcpnf 47000

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 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → 𝐴 ∈ 𝑉)
3		0xr 11231	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ^*
4	3	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℝ^*)
5		pnfxr 11238	. . . . . . . . . . . . . . . 16 ⊢ +∞ ∈ ℝ^*
6	5	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → +∞ ∈ ℝ^*)
7		sge0rpcpnf.b	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ ℝ⁺)
8	7	rpxrd 13040	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
9	7	rpge0d 13043	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ 𝐵)
10	7	rpred 13039	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℝ)
11		ltpnf 13124	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
12	10, 11	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 < +∞)
13	8, 6, 12	xrltled 13154	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≤ +∞)
14	4, 6, 8, 9, 13	eliccxrd 46108	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞))
15	14	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
16		eqid 2764	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
17	15, 16	fmptd 7097	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
18	17	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞))
19	2, 18	sge0xrcl 46964	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ^*)
20	5	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → +∞ ∈ ℝ^*)
21		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞)
22	19, 20, 21	xrgtned 13168	. . . . . . . . 9 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → +∞ ≠ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)))
23	22	necomd 3014	. . . . . . . 8 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ≠ +∞)
24	23	neneqd 2964	. . . . . . 7 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞)
25	2, 18	sge0repnf 46965	. . . . . . 7 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ ↔ ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞))
26	24, 25	mpbird 259	. . . . . 6 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)
27	10	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → 𝐵 ∈ ℝ)
28	7	rpne0d 13044	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≠ 0)
29	28	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → 𝐵 ≠ 0)
30	26, 27, 29	redivcld 12021	. . . . 5 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) ∈ ℝ)
31		arch 12480	. . . . 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 14478	. . . . . . . . . . . . 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 3256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑦 ∈ 𝒫 𝐴(♯‘𝑦) = 𝑛)
37		df-rex 3089	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝒫 𝐴(♯‘𝑦) = 𝑛 ↔ ∃𝑦(𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛))
38	36, 37	sylib 220	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑦(𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛))
39	38	adantlr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ) → ∃𝑦(𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛))
40	39	3adant3 1146	. . . . . . . 8 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → ∃𝑦(𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛))
41		nfv 1936	. . . . . . . . 9 ⊢ Ⅎ𝑦((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛)
42		simprl 780	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → 𝑦 ∈ 𝒫 𝐴)
43		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ (♯‘𝑦) = 𝑛) → (♯‘𝑦) = 𝑛)
44		simpl 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ ∧ (♯‘𝑦) = 𝑛) → 𝑛 ∈ ℕ)
45	43, 44	eqeltrd 2864	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ ∧ (♯‘𝑦) = 𝑛) → (♯‘𝑦) ∈ ℕ)
46		nnnn0 12490	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑦) ∈ ℕ → (♯‘𝑦) ∈ ℕ₀)
47		vex 3460	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
48	47	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑦) ∈ ℕ → 𝑦 ∈ V)
49		hashclb 14373	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ V → (𝑦 ∈ Fin ↔ (♯‘𝑦) ∈ ℕ₀))
50	48, 49	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑦) ∈ ℕ → (𝑦 ∈ Fin ↔ (♯‘𝑦) ∈ ℕ₀))
51	46, 50	mpbird 259	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑦) ∈ ℕ → 𝑦 ∈ Fin)
52	45, 51	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ ∧ (♯‘𝑦) = 𝑛) → 𝑦 ∈ Fin)
53	52	adantrl 726	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → 𝑦 ∈ Fin)
54	53	3ad2antl2 1201	. . . . . . . . . . . 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 4154	. . . . . . . . . . 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 1152	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛)
57	26	3ad2ant1 1147	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ)
58		nnre 12219	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
59	58	3ad2ant2 1148	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → 𝑛 ∈ ℝ)
60	7	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → 𝐵 ∈ ℝ⁺)
61	60	3ad2ant1 1147	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → 𝐵 ∈ ℝ⁺)
62	57, 59, 61	ltdivmul2d 13091	. . . . . . . . . . . . . 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 234	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ ∧ ((Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) / 𝐵) < 𝑛) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (𝑛 · 𝐵))
64	63	adantr 484	. . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . 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 740	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 𝐵 ∈ ℝ^*)
69	9	ad3antrrr 740	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 0 ≤ 𝐵)
70	12	ad3antrrr 740	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 𝐵 < +∞)
71	66, 67, 68, 69, 70	elicod 13401	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) ∧ 𝑥 ∈ 𝑦) → 𝐵 ∈ (0[,)+∞))
72	65, 71	sge0fsummpt 46969	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)) = Σ𝑥 ∈ 𝑦 𝐵)
73	10	recnd 11212	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ ℂ)
74	73	ad2antrr 736	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → 𝐵 ∈ ℂ)
75		fsumconst 15819	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Fin ∧ 𝐵 ∈ ℂ) → Σ𝑥 ∈ 𝑦 𝐵 = ((♯‘𝑦) · 𝐵))
76	65, 74, 75	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → Σ𝑥 ∈ 𝑦 𝐵 = ((♯‘𝑦) · 𝐵))
77		oveq1 7405	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑦) = 𝑛 → ((♯‘𝑦) · 𝐵) = (𝑛 · 𝐵))
78	77	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛) → ((♯‘𝑦) · 𝐵) = (𝑛 · 𝐵))
79	78	adantl 485	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → ((♯‘𝑦) · 𝐵) = (𝑛 · 𝐵))
80	72, 76, 79	3eqtrrd 2804	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → (𝑛 · 𝐵) = (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
81	80	adantllr 729	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) ∧ 𝑛 ∈ ℕ) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (♯‘𝑦) = 𝑛)) → (𝑛 · 𝐵) = (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
82	81	3adantl3 1183	. . . . . . . . . . . 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 5128	. . . . . . . . . . 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 519	. . . . . . . . . 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 416	. . . . . . . . 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 2253	. . . . . . . 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 3089	. . . . . . 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 236	. . . . . 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 1133	. . . . 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 3163	. . . 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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐴 ∈ 𝑉)
94	15	adantlr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
95		elpwinss 45634	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝒫 𝐴 ∩ Fin) → 𝑦 ⊆ 𝐴)
96	95	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ⊆ 𝐴)
97	93, 94, 96	sge0lessmpt 46978	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)) ≤ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)))
98		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑦 ∈ (𝒫 𝐴 ∩ Fin))
99	14	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑦) → 𝐵 ∈ (0[,]+∞))
100		eqid 2764	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑦 ↦ 𝐵) = (𝑥 ∈ 𝑦 ↦ 𝐵)
101	99, 100	fmptd 7097	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑦 ↦ 𝐵):𝑦⟶(0[,]+∞))
102	101	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑥 ∈ 𝑦 ↦ 𝐵):𝑦⟶(0[,]+∞))
103	98, 102	sge0xrcl 46964	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)) ∈ ℝ^*)
104	1, 17	sge0xrcl 46964	. . . . . . . . 9 ⊢ (𝜑 → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ^*)
105	104	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) ∈ ℝ^*)
106	103, 105	xrlenltd 11250	. . . . . . 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 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝒫 𝐴 ∩ Fin)) → ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
108	107	ralrimiva 3156	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ (𝒫 𝐴 ∩ Fin) ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
109		ralnex 3090	. . . . 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 220	. . . 4 ⊢ (𝜑 → ¬ ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
111	110	adantr 484	. . 3 ⊢ ((𝜑 ∧ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞) → ¬ ∃𝑦 ∈ (𝒫 𝐴 ∩ Fin)(Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < (Σ^{^}‘(𝑥 ∈ 𝑦 ↦ 𝐵)))
112	92, 111	pm2.65da 826	. 2 ⊢ (𝜑 → ¬ (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) < +∞)
113		nltpnft 13169	. . 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 259	1 ⊢ (𝜑 → (Σ^{^}‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = +∞)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1099 = wceq 1562 ∃wex 1801 ∈ wcel 2144 ≠ wne 2959 ∀wral 3078 ∃wrex 3088 Vcvv 3456 ∩ cin 3905 ⊆ wss 3906 𝒫 cpw 4557 class class class wbr 5102 ↦ cmpt 5183 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 Fincfn 8929 ℂcc 11073 ℝcr 11074 0cc0 11075 · cmul 11080 +∞cpnf 11215 ℝ^*cxr 11217 < clt 11218 ≤ cle 11219 / cdiv 11846 ℕcn 12212 ℕ₀cn0 12483 ℝ⁺crp 12995 [,]cicc 13354 ♯chash 14345 Σcsu 15715 Σ^{^}csumge0 46941

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-inf2 9598 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 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-om 7849 df-1st 7972 df-2nd 7973 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-er 8680 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-sup 9390 df-oi 9460 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-n0 12484 df-z 12571 df-uz 12842 df-rp 12996 df-ico 13357 df-icc 13358 df-fz 13515 df-fzo 13662 df-seq 14017 df-exp 14077 df-hash 14346 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-clim 15517 df-sum 15716 df-sumge0 46942

This theorem is referenced by: hoicvrrex 47135

W3C validator