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Theorem basellem9 24615

Description: Lemma for basel 24616. Since by basellem8 24614 𝐹 is bounded by two expressions that tend to π↑2 / 6, 𝐹 must also go to π↑2 / 6 by the squeeze theorem climsqz 14219. But the series 𝐹 is exactly the partial sums of 𝑘↑-2, so it follows that this is also the value of the infinite sum Σ𝑘 ∈ ℕ(𝑘↑-2). (Contributed by Mario Carneiro, 28-Jul-2014.)

**Hypotheses**
Ref	Expression
basel.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (1 / ((2 · 𝑛) + 1)))
basel.f	⊢ 𝐹 = seq1( + , (𝑛 ∈ ℕ ↦ (𝑛↑-2)))
basel.h	⊢ 𝐻 = ((ℕ × {((π↑2) / 6)}) ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 − 𝐺))
basel.j	⊢ 𝐽 = (𝐻 ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺)))
basel.k	⊢ 𝐾 = (𝐻 ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 + 𝐺))

**Assertion**
Ref	Expression
basellem9	⊢ Σ𝑘 ∈ ℕ (𝑘↑-2) = ((π↑2) / 6)

Distinct variable groups: 𝑘,𝑛,𝐹 𝑘,𝐺 𝑘,𝐻 𝑘,𝐽,𝑛 𝑘,𝐾 Allowed substitution hints: 𝐺(𝑛) 𝐻(𝑛) 𝐾(𝑛)

Proof of Theorem basellem9 Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnuz 11599	. . 3 ⊢ ℕ = (ℤ_≥‘1)
2		1zzd 11285	. . 3 ⊢ (⊤ → 1 ∈ ℤ)
3		oveq1 6556	. . . . 5 ⊢ (𝑛 = 𝑘 → (𝑛↑-2) = (𝑘↑-2))
4		eqid 2610	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ (𝑛↑-2)) = (𝑛 ∈ ℕ ↦ (𝑛↑-2))
5		ovex 6577	. . . . 5 ⊢ (𝑘↑-2) ∈ V
6	3, 4, 5	fvmpt 6191	. . . 4 ⊢ (𝑘 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑛↑-2))‘𝑘) = (𝑘↑-2))
7	6	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑛↑-2))‘𝑘) = (𝑘↑-2))
8		nnre 10904	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
9		nnne0 10930	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
10		2z 11286	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
11		znegcl 11289	. . . . . . . . . . 11 ⊢ (2 ∈ ℤ → -2 ∈ ℤ)
12	10, 11	ax-mp 5	. . . . . . . . . 10 ⊢ -2 ∈ ℤ
13	12	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → -2 ∈ ℤ)
14	8, 9, 13	reexpclzd 12896	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (𝑛↑-2) ∈ ℝ)
15	14	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (𝑛↑-2) ∈ ℝ)
16	15, 4	fmptd 6292	. . . . . 6 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (𝑛↑-2)):ℕ⟶ℝ)
17	16	ffvelrnda 6267	. . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑛↑-2))‘𝑘) ∈ ℝ)
18	7, 17	eqeltrrd 2689	. . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘↑-2) ∈ ℝ)
19	18	recnd 9947	. . 3 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝑘↑-2) ∈ ℂ)
20	1, 2, 17	serfre 12692	. . . . . . . . . . 11 ⊢ (⊤ → seq1( + , (𝑛 ∈ ℕ ↦ (𝑛↑-2))):ℕ⟶ℝ)
21		basel.f	. . . . . . . . . . . 12 ⊢ 𝐹 = seq1( + , (𝑛 ∈ ℕ ↦ (𝑛↑-2)))
22	21	feq1i 5949	. . . . . . . . . . 11 ⊢ (𝐹:ℕ⟶ℝ ↔ seq1( + , (𝑛 ∈ ℕ ↦ (𝑛↑-2))):ℕ⟶ℝ)
23	20, 22	sylibr 223	. . . . . . . . . 10 ⊢ (⊤ → 𝐹:ℕ⟶ℝ)
24	23	ffvelrnda 6267	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ℝ)
25	24	recnd 9947	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ℂ)
26		remulcl 9900	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
27	26	adantl 481	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
28		ovex 6577	. . . . . . . . . . . . . . . 16 ⊢ ((π↑2) / 6) ∈ V
29	28	fconst 6004	. . . . . . . . . . . . . . 15 ⊢ (ℕ × {((π↑2) / 6)}):ℕ⟶{((π↑2) / 6)}
30		pire 24014	. . . . . . . . . . . . . . . . . . 19 ⊢ π ∈ ℝ
31	30	resqcli 12811	. . . . . . . . . . . . . . . . . 18 ⊢ (π↑2) ∈ ℝ
32		6re 10978	. . . . . . . . . . . . . . . . . 18 ⊢ 6 ∈ ℝ
33		6nn 11066	. . . . . . . . . . . . . . . . . . 19 ⊢ 6 ∈ ℕ
34	33	nnne0i 10932	. . . . . . . . . . . . . . . . . 18 ⊢ 6 ≠ 0
35	31, 32, 34	redivcli 10671	. . . . . . . . . . . . . . . . 17 ⊢ ((π↑2) / 6) ∈ ℝ
36	35	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → ((π↑2) / 6) ∈ ℝ)
37	36	snssd 4281	. . . . . . . . . . . . . . 15 ⊢ (⊤ → {((π↑2) / 6)} ⊆ ℝ)
38		fss 5969	. . . . . . . . . . . . . . 15 ⊢ (((ℕ × {((π↑2) / 6)}):ℕ⟶{((π↑2) / 6)} ∧ {((π↑2) / 6)} ⊆ ℝ) → (ℕ × {((π↑2) / 6)}):ℕ⟶ℝ)
39	29, 37, 38	sylancr 694	. . . . . . . . . . . . . 14 ⊢ (⊤ → (ℕ × {((π↑2) / 6)}):ℕ⟶ℝ)
40		resubcl 10224	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 − 𝑦) ∈ ℝ)
41	40	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 − 𝑦) ∈ ℝ)
42		1ex 9914	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ V
43	42	fconst 6004	. . . . . . . . . . . . . . . 16 ⊢ (ℕ × {1}):ℕ⟶{1}
44		1red 9934	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → 1 ∈ ℝ)
45	44	snssd 4281	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → {1} ⊆ ℝ)
46		fss 5969	. . . . . . . . . . . . . . . 16 ⊢ (((ℕ × {1}):ℕ⟶{1} ∧ {1} ⊆ ℝ) → (ℕ × {1}):ℕ⟶ℝ)
47	43, 45, 46	sylancr 694	. . . . . . . . . . . . . . 15 ⊢ (⊤ → (ℕ × {1}):ℕ⟶ℝ)
48		2nn 11062	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℕ
49	48	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (⊤ → 2 ∈ ℕ)
50		nnmulcl 10920	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ) → (2 · 𝑛) ∈ ℕ)
51	49, 50	sylan 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (2 · 𝑛) ∈ ℕ)
52	51	peano2nnd 10914	. . . . . . . . . . . . . . . . 17 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → ((2 · 𝑛) + 1) ∈ ℕ)
53	52	nnrecred 10943	. . . . . . . . . . . . . . . 16 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (1 / ((2 · 𝑛) + 1)) ∈ ℝ)
54		basel.g	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (1 / ((2 · 𝑛) + 1)))
55	53, 54	fmptd 6292	. . . . . . . . . . . . . . 15 ⊢ (⊤ → 𝐺:ℕ⟶ℝ)
56		nnex 10903	. . . . . . . . . . . . . . . 16 ⊢ ℕ ∈ V
57	56	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (⊤ → ℕ ∈ V)
58		inidm 3784	. . . . . . . . . . . . . . 15 ⊢ (ℕ ∩ ℕ) = ℕ
59	41, 47, 55, 57, 57, 58	off 6810	. . . . . . . . . . . . . 14 ⊢ (⊤ → ((ℕ × {1}) ∘_𝑓 − 𝐺):ℕ⟶ℝ)
60	27, 39, 59, 57, 57, 58	off 6810	. . . . . . . . . . . . 13 ⊢ (⊤ → ((ℕ × {((π↑2) / 6)}) ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 − 𝐺)):ℕ⟶ℝ)
61		basel.h	. . . . . . . . . . . . . 14 ⊢ 𝐻 = ((ℕ × {((π↑2) / 6)}) ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 − 𝐺))
62	61	feq1i 5949	. . . . . . . . . . . . 13 ⊢ (𝐻:ℕ⟶ℝ ↔ ((ℕ × {((π↑2) / 6)}) ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 − 𝐺)):ℕ⟶ℝ)
63	60, 62	sylibr 223	. . . . . . . . . . . 12 ⊢ (⊤ → 𝐻:ℕ⟶ℝ)
64		readdcl 9898	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ)
65	64	adantl 481	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
66		negex 10158	. . . . . . . . . . . . . . . 16 ⊢ -2 ∈ V
67	66	fconst 6004	. . . . . . . . . . . . . . 15 ⊢ (ℕ × {-2}):ℕ⟶{-2}
68	12	zrei 11260	. . . . . . . . . . . . . . . . 17 ⊢ -2 ∈ ℝ
69	68	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → -2 ∈ ℝ)
70	69	snssd 4281	. . . . . . . . . . . . . . 15 ⊢ (⊤ → {-2} ⊆ ℝ)
71		fss 5969	. . . . . . . . . . . . . . 15 ⊢ (((ℕ × {-2}):ℕ⟶{-2} ∧ {-2} ⊆ ℝ) → (ℕ × {-2}):ℕ⟶ℝ)
72	67, 70, 71	sylancr 694	. . . . . . . . . . . . . 14 ⊢ (⊤ → (ℕ × {-2}):ℕ⟶ℝ)
73	27, 72, 55, 57, 57, 58	off 6810	. . . . . . . . . . . . 13 ⊢ (⊤ → ((ℕ × {-2}) ∘_𝑓 · 𝐺):ℕ⟶ℝ)
74	65, 47, 73, 57, 57, 58	off 6810	. . . . . . . . . . . 12 ⊢ (⊤ → ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺)):ℕ⟶ℝ)
75	27, 63, 74, 57, 57, 58	off 6810	. . . . . . . . . . 11 ⊢ (⊤ → (𝐻 ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))):ℕ⟶ℝ)
76		basel.j	. . . . . . . . . . . 12 ⊢ 𝐽 = (𝐻 ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺)))
77	76	feq1i 5949	. . . . . . . . . . 11 ⊢ (𝐽:ℕ⟶ℝ ↔ (𝐻 ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))):ℕ⟶ℝ)
78	75, 77	sylibr 223	. . . . . . . . . 10 ⊢ (⊤ → 𝐽:ℕ⟶ℝ)
79	78	ffvelrnda 6267	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (𝐽‘𝑛) ∈ ℝ)
80	79	recnd 9947	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (𝐽‘𝑛) ∈ ℂ)
81	25, 80	npcand 10275	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → (((𝐹‘𝑛) − (𝐽‘𝑛)) + (𝐽‘𝑛)) = (𝐹‘𝑛))
82	81	mpteq2dva 4672	. . . . . 6 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ (((𝐹‘𝑛) − (𝐽‘𝑛)) + (𝐽‘𝑛))) = (𝑛 ∈ ℕ ↦ (𝐹‘𝑛)))
83		ovex 6577	. . . . . . . 8 ⊢ ((𝐹‘𝑛) − (𝐽‘𝑛)) ∈ V
84	83	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) − (𝐽‘𝑛)) ∈ V)
85	23	feqmptd 6159	. . . . . . . 8 ⊢ (⊤ → 𝐹 = (𝑛 ∈ ℕ ↦ (𝐹‘𝑛)))
86	78	feqmptd 6159	. . . . . . . 8 ⊢ (⊤ → 𝐽 = (𝑛 ∈ ℕ ↦ (𝐽‘𝑛)))
87	57, 24, 79, 85, 86	offval2 6812	. . . . . . 7 ⊢ (⊤ → (𝐹 ∘_𝑓 − 𝐽) = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛) − (𝐽‘𝑛))))
88	57, 84, 79, 87, 86	offval2 6812	. . . . . 6 ⊢ (⊤ → ((𝐹 ∘_𝑓 − 𝐽) ∘_𝑓 + 𝐽) = (𝑛 ∈ ℕ ↦ (((𝐹‘𝑛) − (𝐽‘𝑛)) + (𝐽‘𝑛))))
89	82, 88, 85	3eqtr4d 2654	. . . . 5 ⊢ (⊤ → ((𝐹 ∘_𝑓 − 𝐽) ∘_𝑓 + 𝐽) = 𝐹)
90	65, 47, 55, 57, 57, 58	off 6810	. . . . . . . . . 10 ⊢ (⊤ → ((ℕ × {1}) ∘_𝑓 + 𝐺):ℕ⟶ℝ)
91		recn 9905	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
92		recn 9905	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
93		recn 9905	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℂ)
94		subdi 10342	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 · (𝑦 − 𝑧)) = ((𝑥 · 𝑦) − (𝑥 · 𝑧)))
95	91, 92, 93, 94	syl3an 1360	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 · (𝑦 − 𝑧)) = ((𝑥 · 𝑦) − (𝑥 · 𝑧)))
96	95	adantl 481	. . . . . . . . . 10 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)) → (𝑥 · (𝑦 − 𝑧)) = ((𝑥 · 𝑦) − (𝑥 · 𝑧)))
97	57, 63, 90, 74, 96	caofdi 6831	. . . . . . . . 9 ⊢ (⊤ → (𝐻 ∘_𝑓 · (((ℕ × {1}) ∘_𝑓 + 𝐺) ∘_𝑓 − ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺)))) = ((𝐻 ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 + 𝐺)) ∘_𝑓 − (𝐻 ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺)))))
98		basel.k	. . . . . . . . . 10 ⊢ 𝐾 = (𝐻 ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 + 𝐺))
99	98, 76	oveq12i 6561	. . . . . . . . 9 ⊢ (𝐾 ∘_𝑓 − 𝐽) = ((𝐻 ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 + 𝐺)) ∘_𝑓 − (𝐻 ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))))
100	97, 99	syl6eqr 2662	. . . . . . . 8 ⊢ (⊤ → (𝐻 ∘_𝑓 · (((ℕ × {1}) ∘_𝑓 + 𝐺) ∘_𝑓 − ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺)))) = (𝐾 ∘_𝑓 − 𝐽))
101	35	recni 9931	. . . . . . . . . . . . . 14 ⊢ ((π↑2) / 6) ∈ ℂ
102	1	eqimss2i 3623	. . . . . . . . . . . . . . 15 ⊢ (ℤ_≥‘1) ⊆ ℕ
103	102, 56	climconst2 14127	. . . . . . . . . . . . . 14 ⊢ ((((π↑2) / 6) ∈ ℂ ∧ 1 ∈ ℤ) → (ℕ × {((π↑2) / 6)}) ⇝ ((π↑2) / 6))
104	101, 2, 103	sylancr 694	. . . . . . . . . . . . 13 ⊢ (⊤ → (ℕ × {((π↑2) / 6)}) ⇝ ((π↑2) / 6))
105		ovex 6577	. . . . . . . . . . . . . 14 ⊢ ((ℕ × {((π↑2) / 6)}) ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 − 𝐺)) ∈ V
106	105	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → ((ℕ × {((π↑2) / 6)}) ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 − 𝐺)) ∈ V)
107		ax-resscn 9872	. . . . . . . . . . . . . . . 16 ⊢ ℝ ⊆ ℂ
108		fss 5969	. . . . . . . . . . . . . . . 16 ⊢ (((ℕ × {1}):ℕ⟶ℝ ∧ ℝ ⊆ ℂ) → (ℕ × {1}):ℕ⟶ℂ)
109	47, 107, 108	sylancl 693	. . . . . . . . . . . . . . 15 ⊢ (⊤ → (ℕ × {1}):ℕ⟶ℂ)
110		fss 5969	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺:ℕ⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:ℕ⟶ℂ)
111	55, 107, 110	sylancl 693	. . . . . . . . . . . . . . 15 ⊢ (⊤ → 𝐺:ℕ⟶ℂ)
112		ofnegsub 10895	. . . . . . . . . . . . . . 15 ⊢ ((ℕ ∈ V ∧ (ℕ × {1}):ℕ⟶ℂ ∧ 𝐺:ℕ⟶ℂ) → ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-1}) ∘_𝑓 · 𝐺)) = ((ℕ × {1}) ∘_𝑓 − 𝐺))
113	57, 109, 111, 112	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (⊤ → ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-1}) ∘_𝑓 · 𝐺)) = ((ℕ × {1}) ∘_𝑓 − 𝐺))
114		neg1cn 11001	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℂ
115	54, 114	basellem7 24613	. . . . . . . . . . . . . 14 ⊢ ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-1}) ∘_𝑓 · 𝐺)) ⇝ 1
116	113, 115	syl6eqbrr 4623	. . . . . . . . . . . . 13 ⊢ (⊤ → ((ℕ × {1}) ∘_𝑓 − 𝐺) ⇝ 1)
117	39	ffvelrnda 6267	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((ℕ × {((π↑2) / 6)})‘𝑘) ∈ ℝ)
118	117	recnd 9947	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((ℕ × {((π↑2) / 6)})‘𝑘) ∈ ℂ)
119	59	ffvelrnda 6267	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℕ × {1}) ∘_𝑓 − 𝐺)‘𝑘) ∈ ℝ)
120	119	recnd 9947	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℕ × {1}) ∘_𝑓 − 𝐺)‘𝑘) ∈ ℂ)
121		ffn 5958	. . . . . . . . . . . . . . 15 ⊢ ((ℕ × {((π↑2) / 6)}):ℕ⟶ℝ → (ℕ × {((π↑2) / 6)}) Fn ℕ)
122	39, 121	syl 17	. . . . . . . . . . . . . 14 ⊢ (⊤ → (ℕ × {((π↑2) / 6)}) Fn ℕ)
123		fnconstg 6006	. . . . . . . . . . . . . . . 16 ⊢ (1 ∈ ℤ → (ℕ × {1}) Fn ℕ)
124	2, 123	syl 17	. . . . . . . . . . . . . . 15 ⊢ (⊤ → (ℕ × {1}) Fn ℕ)
125		ffn 5958	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:ℕ⟶ℝ → 𝐺 Fn ℕ)
126	55, 125	syl 17	. . . . . . . . . . . . . . 15 ⊢ (⊤ → 𝐺 Fn ℕ)
127	124, 126, 57, 57, 58	offn 6806	. . . . . . . . . . . . . 14 ⊢ (⊤ → ((ℕ × {1}) ∘_𝑓 − 𝐺) Fn ℕ)
128		eqidd 2611	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((ℕ × {((π↑2) / 6)})‘𝑘) = ((ℕ × {((π↑2) / 6)})‘𝑘))
129		eqidd 2611	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℕ × {1}) ∘_𝑓 − 𝐺)‘𝑘) = (((ℕ × {1}) ∘_𝑓 − 𝐺)‘𝑘))
130	122, 127, 57, 57, 58, 128, 129	ofval 6804	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℕ × {((π↑2) / 6)}) ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 − 𝐺))‘𝑘) = (((ℕ × {((π↑2) / 6)})‘𝑘) · (((ℕ × {1}) ∘_𝑓 − 𝐺)‘𝑘)))
131	1, 2, 104, 106, 116, 118, 120, 130	climmul 14211	. . . . . . . . . . . 12 ⊢ (⊤ → ((ℕ × {((π↑2) / 6)}) ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 − 𝐺)) ⇝ (((π↑2) / 6) · 1))
132	101	mulid1i 9921	. . . . . . . . . . . 12 ⊢ (((π↑2) / 6) · 1) = ((π↑2) / 6)
133	131, 132	syl6breq 4624	. . . . . . . . . . 11 ⊢ (⊤ → ((ℕ × {((π↑2) / 6)}) ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 − 𝐺)) ⇝ ((π↑2) / 6))
134	61, 133	syl5eqbr 4618	. . . . . . . . . 10 ⊢ (⊤ → 𝐻 ⇝ ((π↑2) / 6))
135		ovex 6577	. . . . . . . . . . 11 ⊢ (𝐻 ∘_𝑓 · (((ℕ × {1}) ∘_𝑓 + 𝐺) ∘_𝑓 − ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺)))) ∈ V
136	135	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (𝐻 ∘_𝑓 · (((ℕ × {1}) ∘_𝑓 + 𝐺) ∘_𝑓 − ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺)))) ∈ V)
137		3cn 10972	. . . . . . . . . . . . 13 ⊢ 3 ∈ ℂ
138	102, 56	climconst2 14127	. . . . . . . . . . . . 13 ⊢ ((3 ∈ ℂ ∧ 1 ∈ ℤ) → (ℕ × {3}) ⇝ 3)
139	137, 2, 138	sylancr 694	. . . . . . . . . . . 12 ⊢ (⊤ → (ℕ × {3}) ⇝ 3)
140		ovex 6577	. . . . . . . . . . . . 13 ⊢ ((ℕ × {3}) ∘_𝑓 · 𝐺) ∈ V
141	140	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → ((ℕ × {3}) ∘_𝑓 · 𝐺) ∈ V)
142	54	basellem6 24612	. . . . . . . . . . . . 13 ⊢ 𝐺 ⇝ 0
143	142	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → 𝐺 ⇝ 0)
144		3ex 10973	. . . . . . . . . . . . . . . 16 ⊢ 3 ∈ V
145	144	fconst 6004	. . . . . . . . . . . . . . 15 ⊢ (ℕ × {3}):ℕ⟶{3}
146		3re 10971	. . . . . . . . . . . . . . . . 17 ⊢ 3 ∈ ℝ
147	146	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → 3 ∈ ℝ)
148	147	snssd 4281	. . . . . . . . . . . . . . 15 ⊢ (⊤ → {3} ⊆ ℝ)
149		fss 5969	. . . . . . . . . . . . . . 15 ⊢ (((ℕ × {3}):ℕ⟶{3} ∧ {3} ⊆ ℝ) → (ℕ × {3}):ℕ⟶ℝ)
150	145, 148, 149	sylancr 694	. . . . . . . . . . . . . 14 ⊢ (⊤ → (ℕ × {3}):ℕ⟶ℝ)
151	150	ffvelrnda 6267	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((ℕ × {3})‘𝑘) ∈ ℝ)
152	151	recnd 9947	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((ℕ × {3})‘𝑘) ∈ ℂ)
153	55	ffvelrnda 6267	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
154	153	recnd 9947	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℂ)
155		ffn 5958	. . . . . . . . . . . . . 14 ⊢ ((ℕ × {3}):ℕ⟶ℝ → (ℕ × {3}) Fn ℕ)
156	150, 155	syl 17	. . . . . . . . . . . . 13 ⊢ (⊤ → (ℕ × {3}) Fn ℕ)
157		eqidd 2611	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((ℕ × {3})‘𝑘) = ((ℕ × {3})‘𝑘))
158		eqidd 2611	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) = (𝐺‘𝑘))
159	156, 126, 57, 57, 58, 157, 158	ofval 6804	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℕ × {3}) ∘_𝑓 · 𝐺)‘𝑘) = (((ℕ × {3})‘𝑘) · (𝐺‘𝑘)))
160	1, 2, 139, 141, 143, 152, 154, 159	climmul 14211	. . . . . . . . . . 11 ⊢ (⊤ → ((ℕ × {3}) ∘_𝑓 · 𝐺) ⇝ (3 · 0))
161	137	mul01i 10105	. . . . . . . . . . 11 ⊢ (3 · 0) = 0
162	160, 161	syl6breq 4624	. . . . . . . . . 10 ⊢ (⊤ → ((ℕ × {3}) ∘_𝑓 · 𝐺) ⇝ 0)
163	63	ffvelrnda 6267	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) ∈ ℝ)
164	163	recnd 9947	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) ∈ ℂ)
165	27, 150, 55, 57, 57, 58	off 6810	. . . . . . . . . . . 12 ⊢ (⊤ → ((ℕ × {3}) ∘_𝑓 · 𝐺):ℕ⟶ℝ)
166	165	ffvelrnda 6267	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℕ × {3}) ∘_𝑓 · 𝐺)‘𝑘) ∈ ℝ)
167	166	recnd 9947	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℕ × {3}) ∘_𝑓 · 𝐺)‘𝑘) ∈ ℂ)
168		ffn 5958	. . . . . . . . . . . 12 ⊢ (𝐻:ℕ⟶ℝ → 𝐻 Fn ℕ)
169	63, 168	syl 17	. . . . . . . . . . 11 ⊢ (⊤ → 𝐻 Fn ℕ)
170	41, 90, 74, 57, 57, 58	off 6810	. . . . . . . . . . . 12 ⊢ (⊤ → (((ℕ × {1}) ∘_𝑓 + 𝐺) ∘_𝑓 − ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))):ℕ⟶ℝ)
171		ffn 5958	. . . . . . . . . . . 12 ⊢ ((((ℕ × {1}) ∘_𝑓 + 𝐺) ∘_𝑓 − ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))):ℕ⟶ℝ → (((ℕ × {1}) ∘_𝑓 + 𝐺) ∘_𝑓 − ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))) Fn ℕ)
172	170, 171	syl 17	. . . . . . . . . . 11 ⊢ (⊤ → (((ℕ × {1}) ∘_𝑓 + 𝐺) ∘_𝑓 − ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))) Fn ℕ)
173		eqidd 2611	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) = (𝐻‘𝑘))
174	154	mulid2d 9937	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (1 · (𝐺‘𝑘)) = (𝐺‘𝑘))
175		2cn 10968	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℂ
176		mulneg1 10345	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℂ ∧ (𝐺‘𝑘) ∈ ℂ) → (-2 · (𝐺‘𝑘)) = -(2 · (𝐺‘𝑘)))
177	175, 154, 176	sylancr 694	. . . . . . . . . . . . . . . . 17 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (-2 · (𝐺‘𝑘)) = -(2 · (𝐺‘𝑘)))
178	177	negeqd 10154	. . . . . . . . . . . . . . . 16 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → -(-2 · (𝐺‘𝑘)) = --(2 · (𝐺‘𝑘)))
179		mulcl 9899	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 ∈ ℂ ∧ (𝐺‘𝑘) ∈ ℂ) → (2 · (𝐺‘𝑘)) ∈ ℂ)
180	175, 154, 179	sylancr 694	. . . . . . . . . . . . . . . . 17 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · (𝐺‘𝑘)) ∈ ℂ)
181	180	negnegd 10262	. . . . . . . . . . . . . . . 16 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → --(2 · (𝐺‘𝑘)) = (2 · (𝐺‘𝑘)))
182	178, 181	eqtr2d 2645	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (2 · (𝐺‘𝑘)) = -(-2 · (𝐺‘𝑘)))
183	174, 182	oveq12d 6567	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((1 · (𝐺‘𝑘)) + (2 · (𝐺‘𝑘))) = ((𝐺‘𝑘) + -(-2 · (𝐺‘𝑘))))
184		remulcl 9900	. . . . . . . . . . . . . . . . 17 ⊢ ((-2 ∈ ℝ ∧ (𝐺‘𝑘) ∈ ℝ) → (-2 · (𝐺‘𝑘)) ∈ ℝ)
185	68, 153, 184	sylancr 694	. . . . . . . . . . . . . . . 16 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (-2 · (𝐺‘𝑘)) ∈ ℝ)
186	185	recnd 9947	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (-2 · (𝐺‘𝑘)) ∈ ℂ)
187	154, 186	negsubd 10277	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘) + -(-2 · (𝐺‘𝑘))) = ((𝐺‘𝑘) − (-2 · (𝐺‘𝑘))))
188	183, 187	eqtrd 2644	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((1 · (𝐺‘𝑘)) + (2 · (𝐺‘𝑘))) = ((𝐺‘𝑘) − (-2 · (𝐺‘𝑘))))
189		df-3 10957	. . . . . . . . . . . . . . . 16 ⊢ 3 = (2 + 1)
190		ax-1cn 9873	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
191	175, 190	addcomi 10106	. . . . . . . . . . . . . . . 16 ⊢ (2 + 1) = (1 + 2)
192	189, 191	eqtri 2632	. . . . . . . . . . . . . . 15 ⊢ 3 = (1 + 2)
193	192	oveq1i 6559	. . . . . . . . . . . . . 14 ⊢ (3 · (𝐺‘𝑘)) = ((1 + 2) · (𝐺‘𝑘))
194		1cnd 9935	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 1 ∈ ℂ)
195	175	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 2 ∈ ℂ)
196	194, 195, 154	adddird 9944	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((1 + 2) · (𝐺‘𝑘)) = ((1 · (𝐺‘𝑘)) + (2 · (𝐺‘𝑘))))
197	193, 196	syl5eq 2656	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (3 · (𝐺‘𝑘)) = ((1 · (𝐺‘𝑘)) + (2 · (𝐺‘𝑘))))
198	194, 154, 186	pnpcand 10308	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((1 + (𝐺‘𝑘)) − (1 + (-2 · (𝐺‘𝑘)))) = ((𝐺‘𝑘) − (-2 · (𝐺‘𝑘))))
199	188, 197, 198	3eqtr4rd 2655	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((1 + (𝐺‘𝑘)) − (1 + (-2 · (𝐺‘𝑘)))) = (3 · (𝐺‘𝑘)))
200	124, 126, 57, 57, 58	offn 6806	. . . . . . . . . . . . 13 ⊢ (⊤ → ((ℕ × {1}) ∘_𝑓 + 𝐺) Fn ℕ)
201	12	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → -2 ∈ ℤ)
202		fnconstg 6006	. . . . . . . . . . . . . . . 16 ⊢ (-2 ∈ ℤ → (ℕ × {-2}) Fn ℕ)
203	201, 202	syl 17	. . . . . . . . . . . . . . 15 ⊢ (⊤ → (ℕ × {-2}) Fn ℕ)
204	203, 126, 57, 57, 58	offn 6806	. . . . . . . . . . . . . 14 ⊢ (⊤ → ((ℕ × {-2}) ∘_𝑓 · 𝐺) Fn ℕ)
205	124, 204, 57, 57, 58	offn 6806	. . . . . . . . . . . . 13 ⊢ (⊤ → ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺)) Fn ℕ)
206	57, 44, 126, 158	ofc1 6818	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℕ × {1}) ∘_𝑓 + 𝐺)‘𝑘) = (1 + (𝐺‘𝑘)))
207	57, 69, 126, 158	ofc1 6818	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℕ × {-2}) ∘_𝑓 · 𝐺)‘𝑘) = (-2 · (𝐺‘𝑘)))
208	57, 44, 204, 207	ofc1 6818	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))‘𝑘) = (1 + (-2 · (𝐺‘𝑘))))
209	200, 205, 57, 57, 58, 206, 208	ofval 6804	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((((ℕ × {1}) ∘_𝑓 + 𝐺) ∘_𝑓 − ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺)))‘𝑘) = ((1 + (𝐺‘𝑘)) − (1 + (-2 · (𝐺‘𝑘)))))
210	57, 147, 126, 158	ofc1 6818	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℕ × {3}) ∘_𝑓 · 𝐺)‘𝑘) = (3 · (𝐺‘𝑘)))
211	199, 209, 210	3eqtr4d 2654	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((((ℕ × {1}) ∘_𝑓 + 𝐺) ∘_𝑓 − ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺)))‘𝑘) = (((ℕ × {3}) ∘_𝑓 · 𝐺)‘𝑘))
212	169, 172, 57, 57, 58, 173, 211	ofval 6804	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝐻 ∘_𝑓 · (((ℕ × {1}) ∘_𝑓 + 𝐺) ∘_𝑓 − ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))))‘𝑘) = ((𝐻‘𝑘) · (((ℕ × {3}) ∘_𝑓 · 𝐺)‘𝑘)))
213	1, 2, 134, 136, 162, 164, 167, 212	climmul 14211	. . . . . . . . 9 ⊢ (⊤ → (𝐻 ∘_𝑓 · (((ℕ × {1}) ∘_𝑓 + 𝐺) ∘_𝑓 − ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺)))) ⇝ (((π↑2) / 6) · 0))
214	101	mul01i 10105	. . . . . . . . 9 ⊢ (((π↑2) / 6) · 0) = 0
215	213, 214	syl6breq 4624	. . . . . . . 8 ⊢ (⊤ → (𝐻 ∘_𝑓 · (((ℕ × {1}) ∘_𝑓 + 𝐺) ∘_𝑓 − ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺)))) ⇝ 0)
216	100, 215	eqbrtrrd 4607	. . . . . . 7 ⊢ (⊤ → (𝐾 ∘_𝑓 − 𝐽) ⇝ 0)
217		ovex 6577	. . . . . . . 8 ⊢ (𝐹 ∘_𝑓 − 𝐽) ∈ V
218	217	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝐹 ∘_𝑓 − 𝐽) ∈ V)
219	27, 63, 90, 57, 57, 58	off 6810	. . . . . . . . . 10 ⊢ (⊤ → (𝐻 ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 + 𝐺)):ℕ⟶ℝ)
220	98	feq1i 5949	. . . . . . . . . 10 ⊢ (𝐾:ℕ⟶ℝ ↔ (𝐻 ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 + 𝐺)):ℕ⟶ℝ)
221	219, 220	sylibr 223	. . . . . . . . 9 ⊢ (⊤ → 𝐾:ℕ⟶ℝ)
222	41, 221, 78, 57, 57, 58	off 6810	. . . . . . . 8 ⊢ (⊤ → (𝐾 ∘_𝑓 − 𝐽):ℕ⟶ℝ)
223	222	ffvelrnda 6267	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝐾 ∘_𝑓 − 𝐽)‘𝑘) ∈ ℝ)
224	41, 23, 78, 57, 57, 58	off 6810	. . . . . . . 8 ⊢ (⊤ → (𝐹 ∘_𝑓 − 𝐽):ℕ⟶ℝ)
225	224	ffvelrnda 6267	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝐹 ∘_𝑓 − 𝐽)‘𝑘) ∈ ℝ)
226	23	ffvelrnda 6267	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ)
227	221	ffvelrnda 6267	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐾‘𝑘) ∈ ℝ)
228	78	ffvelrnda 6267	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐽‘𝑘) ∈ ℝ)
229		eqid 2610	. . . . . . . . . . . 12 ⊢ ((2 · 𝑘) + 1) = ((2 · 𝑘) + 1)
230	54, 21, 61, 76, 98, 229	basellem8 24614	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → ((𝐽‘𝑘) ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) ≤ (𝐾‘𝑘)))
231	230	adantl 481	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝐽‘𝑘) ≤ (𝐹‘𝑘) ∧ (𝐹‘𝑘) ≤ (𝐾‘𝑘)))
232	231	simprd 478	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ≤ (𝐾‘𝑘))
233	226, 227, 228, 232	lesub1dd 10522	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘) − (𝐽‘𝑘)) ≤ ((𝐾‘𝑘) − (𝐽‘𝑘)))
234		ffn 5958	. . . . . . . . . 10 ⊢ (𝐹:ℕ⟶ℝ → 𝐹 Fn ℕ)
235	23, 234	syl 17	. . . . . . . . 9 ⊢ (⊤ → 𝐹 Fn ℕ)
236		ffn 5958	. . . . . . . . . 10 ⊢ (𝐽:ℕ⟶ℝ → 𝐽 Fn ℕ)
237	78, 236	syl 17	. . . . . . . . 9 ⊢ (⊤ → 𝐽 Fn ℕ)
238		eqidd 2611	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = (𝐹‘𝑘))
239		eqidd 2611	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐽‘𝑘) = (𝐽‘𝑘))
240	235, 237, 57, 57, 58, 238, 239	ofval 6804	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝐹 ∘_𝑓 − 𝐽)‘𝑘) = ((𝐹‘𝑘) − (𝐽‘𝑘)))
241		ffn 5958	. . . . . . . . . 10 ⊢ (𝐾:ℕ⟶ℝ → 𝐾 Fn ℕ)
242	221, 241	syl 17	. . . . . . . . 9 ⊢ (⊤ → 𝐾 Fn ℕ)
243		eqidd 2611	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐾‘𝑘) = (𝐾‘𝑘))
244	242, 237, 57, 57, 58, 243, 239	ofval 6804	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝐾 ∘_𝑓 − 𝐽)‘𝑘) = ((𝐾‘𝑘) − (𝐽‘𝑘)))
245	233, 240, 244	3brtr4d 4615	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝐹 ∘_𝑓 − 𝐽)‘𝑘) ≤ ((𝐾 ∘_𝑓 − 𝐽)‘𝑘))
246	231	simpld 474	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐽‘𝑘) ≤ (𝐹‘𝑘))
247	226, 228	subge0d 10496	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (0 ≤ ((𝐹‘𝑘) − (𝐽‘𝑘)) ↔ (𝐽‘𝑘) ≤ (𝐹‘𝑘)))
248	246, 247	mpbird 246	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝐹‘𝑘) − (𝐽‘𝑘)))
249	248, 240	breqtrrd 4611	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 0 ≤ ((𝐹 ∘_𝑓 − 𝐽)‘𝑘))
250	1, 2, 216, 218, 223, 225, 245, 249	climsqz2 14220	. . . . . 6 ⊢ (⊤ → (𝐹 ∘_𝑓 − 𝐽) ⇝ 0)
251		ovex 6577	. . . . . . 7 ⊢ ((𝐹 ∘_𝑓 − 𝐽) ∘_𝑓 + 𝐽) ∈ V
252	251	a1i 11	. . . . . 6 ⊢ (⊤ → ((𝐹 ∘_𝑓 − 𝐽) ∘_𝑓 + 𝐽) ∈ V)
253		ovex 6577	. . . . . . . . . 10 ⊢ (𝐻 ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))) ∈ V
254	253	a1i 11	. . . . . . . . 9 ⊢ (⊤ → (𝐻 ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))) ∈ V)
255	68	recni 9931	. . . . . . . . . . 11 ⊢ -2 ∈ ℂ
256	54, 255	basellem7 24613	. . . . . . . . . 10 ⊢ ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺)) ⇝ 1
257	256	a1i 11	. . . . . . . . 9 ⊢ (⊤ → ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺)) ⇝ 1)
258	74	ffvelrnda 6267	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))‘𝑘) ∈ ℝ)
259	258	recnd 9947	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))‘𝑘) ∈ ℂ)
260		eqidd 2611	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))‘𝑘) = (((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))‘𝑘))
261	169, 205, 57, 57, 58, 173, 260	ofval 6804	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝐻 ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺)))‘𝑘) = ((𝐻‘𝑘) · (((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))‘𝑘)))
262	1, 2, 134, 254, 257, 164, 259, 261	climmul 14211	. . . . . . . 8 ⊢ (⊤ → (𝐻 ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))) ⇝ (((π↑2) / 6) · 1))
263	262, 132	syl6breq 4624	. . . . . . 7 ⊢ (⊤ → (𝐻 ∘_𝑓 · ((ℕ × {1}) ∘_𝑓 + ((ℕ × {-2}) ∘_𝑓 · 𝐺))) ⇝ ((π↑2) / 6))
264	76, 263	syl5eqbr 4618	. . . . . 6 ⊢ (⊤ → 𝐽 ⇝ ((π↑2) / 6))
265	225	recnd 9947	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝐹 ∘_𝑓 − 𝐽)‘𝑘) ∈ ℂ)
266	228	recnd 9947	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (𝐽‘𝑘) ∈ ℂ)
267		ffn 5958	. . . . . . . 8 ⊢ ((𝐹 ∘_𝑓 − 𝐽):ℕ⟶ℝ → (𝐹 ∘_𝑓 − 𝐽) Fn ℕ)
268	224, 267	syl 17	. . . . . . 7 ⊢ (⊤ → (𝐹 ∘_𝑓 − 𝐽) Fn ℕ)
269		eqidd 2611	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → ((𝐹 ∘_𝑓 − 𝐽)‘𝑘) = ((𝐹 ∘_𝑓 − 𝐽)‘𝑘))
270	268, 237, 57, 57, 58, 269, 239	ofval 6804	. . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((𝐹 ∘_𝑓 − 𝐽) ∘_𝑓 + 𝐽)‘𝑘) = (((𝐹 ∘_𝑓 − 𝐽)‘𝑘) + (𝐽‘𝑘)))
271	1, 2, 250, 252, 264, 265, 266, 270	climadd 14210	. . . . 5 ⊢ (⊤ → ((𝐹 ∘_𝑓 − 𝐽) ∘_𝑓 + 𝐽) ⇝ (0 + ((π↑2) / 6)))
272	89, 271	eqbrtrrd 4607	. . . 4 ⊢ (⊤ → 𝐹 ⇝ (0 + ((π↑2) / 6)))
273	101	addid2i 10103	. . . 4 ⊢ (0 + ((π↑2) / 6)) = ((π↑2) / 6)
274	272, 21, 273	3brtr3g 4616	. . 3 ⊢ (⊤ → seq1( + , (𝑛 ∈ ℕ ↦ (𝑛↑-2))) ⇝ ((π↑2) / 6))
275	1, 2, 7, 19, 274	isumclim 14330	. 2 ⊢ (⊤ → Σ𝑘 ∈ ℕ (𝑘↑-2) = ((π↑2) / 6))
276	275	trud 1484	1 ⊢ Σ𝑘 ∈ ℕ (𝑘↑-2) = ((π↑2) / 6)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 383 ∧ w3a 1031 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘_𝑓 cof 6793 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ≤ cle 9954 − cmin 10145 -cneg 10146 / cdiv 10563 ℕcn 10897 2c2 10947 3c3 10948 6c6 10951 ℤcz 11254 ℤ_≥cuz 11563 seqcseq 12663 ↑cexp 12722 ⇝ cli 14063 Σcsu 14264 πcpi 14636

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-tan 14641 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-0p 23243 df-limc 23436 df-dv 23437 df-ply 23748 df-idp 23749 df-coe 23750 df-dgr 23751 df-quot 23850

This theorem is referenced by: basel 24616

W3C validator