fourierdlem31 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem31 43308

Description: If 𝐴 is finite and for any element in 𝐴 there is a number 𝑚 such that a property holds for all numbers larger than 𝑚, then there is a number 𝑛 such that the property holds for all numbers larger than 𝑛 and for all elements in 𝐴. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 29-Sep-2020.)

**Hypotheses**
Ref	Expression
fourierdlem31.i	⊢ Ⅎ𝑖𝜑
fourierdlem31.r	⊢ Ⅎ𝑟𝜑
fourierdlem31.iv	⊢ Ⅎ𝑖𝑉
fourierdlem31.a	⊢ (𝜑 → 𝐴 ∈ Fin)
fourierdlem31.exm	⊢ (𝜑 → ∀𝑖 ∈ 𝐴 ∃𝑚 ∈ ℕ ∀𝑟 ∈ (𝑚(,)+∞)𝜒)
fourierdlem31.m	⊢ 𝑀 = {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}
fourierdlem31.v	⊢ 𝑉 = (𝑖 ∈ 𝐴 ↦ inf(𝑀, ℝ, < ))
fourierdlem31.n	⊢ 𝑁 = sup(ran 𝑉, ℝ, < )

**Assertion**
Ref	Expression
fourierdlem31	⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)∀𝑖 ∈ 𝐴 𝜒)

Distinct variable groups: 𝐴,𝑖,𝑚,𝑟 𝐴,𝑛,𝑖,𝑟 𝑛,𝑁 𝜒,𝑚 𝜒,𝑛 Allowed substitution hints: 𝜑(𝑖,𝑚,𝑛,𝑟) 𝜒(𝑖,𝑟) 𝑀(𝑖,𝑚,𝑛,𝑟) 𝑁(𝑖,𝑚,𝑟) 𝑉(𝑖,𝑚,𝑛,𝑟)

Proof of Theorem fourierdlem31 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1nn 11824	. . . 4 ⊢ 1 ∈ ℕ
2		rzal 4410	. . . . 5 ⊢ (𝐴 = ∅ → ∀𝑖 ∈ 𝐴 𝜒)
3	2	ralrimivw 3099	. . . 4 ⊢ (𝐴 = ∅ → ∀𝑟 ∈ (1(,)+∞)∀𝑖 ∈ 𝐴 𝜒)
4		oveq1 7209	. . . . . 6 ⊢ (𝑛 = 1 → (𝑛(,)+∞) = (1(,)+∞))
5	4	raleqdv 3318	. . . . 5 ⊢ (𝑛 = 1 → (∀𝑟 ∈ (𝑛(,)+∞)∀𝑖 ∈ 𝐴 𝜒 ↔ ∀𝑟 ∈ (1(,)+∞)∀𝑖 ∈ 𝐴 𝜒))
6	5	rspcev 3530	. . . 4 ⊢ ((1 ∈ ℕ ∧ ∀𝑟 ∈ (1(,)+∞)∀𝑖 ∈ 𝐴 𝜒) → ∃𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)∀𝑖 ∈ 𝐴 𝜒)
7	1, 3, 6	sylancr 590	. . 3 ⊢ (𝐴 = ∅ → ∃𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)∀𝑖 ∈ 𝐴 𝜒)
8	7	adantl 485	. 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → ∃𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)∀𝑖 ∈ 𝐴 𝜒)
9		fourierdlem31.n	. . . 4 ⊢ 𝑁 = sup(ran 𝑉, ℝ, < )
10		fourierdlem31.i	. . . . . . 7 ⊢ Ⅎ𝑖𝜑
11		fourierdlem31.v	. . . . . . 7 ⊢ 𝑉 = (𝑖 ∈ 𝐴 ↦ inf(𝑀, ℝ, < ))
12		fourierdlem31.m	. . . . . . . . . 10 ⊢ 𝑀 = {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}
13	12	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝑀 = {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒})
14	13	infeq1d 9082	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → inf(𝑀, ℝ, < ) = inf({𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}, ℝ, < ))
15		ssrab2 3983	. . . . . . . . 9 ⊢ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ⊆ ℕ
16		nnuz 12460	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
17	15, 16	sseqtri 3927	. . . . . . . . . 10 ⊢ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ⊆ (ℤ_≥‘1)
18		fourierdlem31.exm	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑖 ∈ 𝐴 ∃𝑚 ∈ ℕ ∀𝑟 ∈ (𝑚(,)+∞)𝜒)
19	18	r19.21bi 3123	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ∃𝑚 ∈ ℕ ∀𝑟 ∈ (𝑚(,)+∞)𝜒)
20		rabn0 4290	. . . . . . . . . . 11 ⊢ ({𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ≠ ∅ ↔ ∃𝑚 ∈ ℕ ∀𝑟 ∈ (𝑚(,)+∞)𝜒)
21	19, 20	sylibr 237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ≠ ∅)
22		infssuzcl 12511	. . . . . . . . . 10 ⊢ (({𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ⊆ (ℤ_≥‘1) ∧ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ≠ ∅) → inf({𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}, ℝ, < ) ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒})
23	17, 21, 22	sylancr 590	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → inf({𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}, ℝ, < ) ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒})
24	15, 23	sseldi 3889	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → inf({𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}, ℝ, < ) ∈ ℕ)
25	14, 24	eqeltrd 2834	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → inf(𝑀, ℝ, < ) ∈ ℕ)
26	10, 11, 25	rnmptssd 42360	. . . . . 6 ⊢ (𝜑 → ran 𝑉 ⊆ ℕ)
27	26	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ran 𝑉 ⊆ ℕ)
28		ltso 10896	. . . . . . 7 ⊢ < Or ℝ
29	28	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → < Or ℝ)
30		fourierdlem31.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ Fin)
31		mptfi 8964	. . . . . . . . . 10 ⊢ (𝐴 ∈ Fin → (𝑖 ∈ 𝐴 ↦ inf(𝑀, ℝ, < )) ∈ Fin)
32	30, 31	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ 𝐴 ↦ inf(𝑀, ℝ, < )) ∈ Fin)
33	11, 32	eqeltrid 2838	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ Fin)
34		rnfi 8948	. . . . . . . 8 ⊢ (𝑉 ∈ Fin → ran 𝑉 ∈ Fin)
35	33, 34	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝑉 ∈ Fin)
36	35	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ran 𝑉 ∈ Fin)
37		neqne 2943	. . . . . . . . 9 ⊢ (¬ 𝐴 = ∅ → 𝐴 ≠ ∅)
38		n0 4251	. . . . . . . . 9 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑖 𝑖 ∈ 𝐴)
39	37, 38	sylib 221	. . . . . . . 8 ⊢ (¬ 𝐴 = ∅ → ∃𝑖 𝑖 ∈ 𝐴)
40	39	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ∃𝑖 𝑖 ∈ 𝐴)
41		nfv 1922	. . . . . . . . 9 ⊢ Ⅎ𝑖 ¬ 𝐴 = ∅
42	10, 41	nfan 1907	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ ¬ 𝐴 = ∅)
43		fourierdlem31.iv	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝑉
44	43	nfrn 5810	. . . . . . . . 9 ⊢ Ⅎ𝑖ran 𝑉
45		nfcv 2900	. . . . . . . . 9 ⊢ Ⅎ𝑖∅
46	44, 45	nfne 3035	. . . . . . . 8 ⊢ Ⅎ𝑖ran 𝑉 ≠ ∅
47		simpr 488	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝑖 ∈ 𝐴)
48	11	elrnmpt1 5816	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ 𝐴 ∧ inf(𝑀, ℝ, < ) ∈ ℕ) → inf(𝑀, ℝ, < ) ∈ ran 𝑉)
49	47, 25, 48	syl2anc 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → inf(𝑀, ℝ, < ) ∈ ran 𝑉)
50	49	ne0d 4240	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ran 𝑉 ≠ ∅)
51	50	ex 416	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ 𝐴 → ran 𝑉 ≠ ∅))
52	51	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → (𝑖 ∈ 𝐴 → ran 𝑉 ≠ ∅))
53	42, 46, 52	exlimd 2216	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → (∃𝑖 𝑖 ∈ 𝐴 → ran 𝑉 ≠ ∅))
54	40, 53	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ran 𝑉 ≠ ∅)
55		nnssre 11817	. . . . . . 7 ⊢ ℕ ⊆ ℝ
56	27, 55	sstrdi 3903	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ran 𝑉 ⊆ ℝ)
57		fisupcl 9074	. . . . . 6 ⊢ (( < Or ℝ ∧ (ran 𝑉 ∈ Fin ∧ ran 𝑉 ≠ ∅ ∧ ran 𝑉 ⊆ ℝ)) → sup(ran 𝑉, ℝ, < ) ∈ ran 𝑉)
58	29, 36, 54, 56, 57	syl13anc 1374	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → sup(ran 𝑉, ℝ, < ) ∈ ran 𝑉)
59	27, 58	sseldd 3892	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → sup(ran 𝑉, ℝ, < ) ∈ ℕ)
60	9, 59	eqeltrid 2838	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝑁 ∈ ℕ)
61		fourierdlem31.r	. . . . 5 ⊢ Ⅎ𝑟𝜑
62		nfcv 2900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖ℝ
63		nfcv 2900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖 <
64	44, 62, 63	nfsup 9056	. . . . . . . . . . 11 ⊢ Ⅎ𝑖sup(ran 𝑉, ℝ, < )
65	9, 64	nfcxfr 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝑁
66		nfcv 2900	. . . . . . . . . 10 ⊢ Ⅎ𝑖(,)
67		nfcv 2900	. . . . . . . . . 10 ⊢ Ⅎ𝑖+∞
68	65, 66, 67	nfov 7232	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝑁(,)+∞)
69	68	nfcri 2887	. . . . . . . 8 ⊢ Ⅎ𝑖 𝑟 ∈ (𝑁(,)+∞)
70	10, 69	nfan 1907	. . . . . . 7 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑟 ∈ (𝑁(,)+∞))
71	11	fvmpt2 6818	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ 𝐴 ∧ inf(𝑀, ℝ, < ) ∈ ℕ) → (𝑉‘𝑖) = inf(𝑀, ℝ, < ))
72	47, 25, 71	syl2anc 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → (𝑉‘𝑖) = inf(𝑀, ℝ, < ))
73	25	nnxrd 42210	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → inf(𝑀, ℝ, < ) ∈ ℝ^*)
74	72, 73	eqeltrd 2834	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → (𝑉‘𝑖) ∈ ℝ^*)
75	74	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → (𝑉‘𝑖) ∈ ℝ^*)
76		pnfxr 10870	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
77	76	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → +∞ ∈ ℝ^*)
78		elioore 12948	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ (𝑁(,)+∞) → 𝑟 ∈ ℝ)
79	78	adantl 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → 𝑟 ∈ ℝ)
80	72, 25	eqeltrd 2834	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → (𝑉‘𝑖) ∈ ℕ)
81	80	nnred 11828	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → (𝑉‘𝑖) ∈ ℝ)
82	81	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → (𝑉‘𝑖) ∈ ℝ)
83		ne0i 4239	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ 𝐴 → 𝐴 ≠ ∅)
84	83	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝐴 ≠ ∅)
85	84	neneqd 2940	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ¬ 𝐴 = ∅)
86	85, 60	syldan 594	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝑁 ∈ ℕ)
87	86	nnred 11828	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝑁 ∈ ℝ)
88	87	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → 𝑁 ∈ ℝ)
89	85, 56	syldan 594	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ran 𝑉 ⊆ ℝ)
90	26, 55	sstrdi 3903	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran 𝑉 ⊆ ℝ)
91		fimaxre2 11760	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝑉 ⊆ ℝ ∧ ran 𝑉 ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑉 𝑦 ≤ 𝑥)
92	90, 35, 91	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑉 𝑦 ≤ 𝑥)
93	92	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑉 𝑦 ≤ 𝑥)
94	72, 49	eqeltrd 2834	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → (𝑉‘𝑖) ∈ ran 𝑉)
95		suprub 11776	. . . . . . . . . . . . . . 15 ⊢ (((ran 𝑉 ⊆ ℝ ∧ ran 𝑉 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑉 𝑦 ≤ 𝑥) ∧ (𝑉‘𝑖) ∈ ran 𝑉) → (𝑉‘𝑖) ≤ sup(ran 𝑉, ℝ, < ))
96	89, 50, 93, 94, 95	syl31anc 1375	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → (𝑉‘𝑖) ≤ sup(ran 𝑉, ℝ, < ))
97	96, 9	breqtrrdi 5085	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → (𝑉‘𝑖) ≤ 𝑁)
98	97	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → (𝑉‘𝑖) ≤ 𝑁)
99	88	rexrd 10866	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → 𝑁 ∈ ℝ^*)
100		simpr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → 𝑟 ∈ (𝑁(,)+∞))
101		ioogtlb 42660	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑟 ∈ (𝑁(,)+∞)) → 𝑁 < 𝑟)
102	99, 77, 100, 101	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → 𝑁 < 𝑟)
103	82, 88, 79, 98, 102	lelttrd 10973	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → (𝑉‘𝑖) < 𝑟)
104	79	ltpnfd 12696	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → 𝑟 < +∞)
105	75, 77, 79, 103, 104	eliood 42663	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → 𝑟 ∈ ((𝑉‘𝑖)(,)+∞))
106	14, 23	eqeltrd 2834	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → inf(𝑀, ℝ, < ) ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒})
107	72, 106	eqeltrd 2834	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → (𝑉‘𝑖) ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒})
108		nfcv 2900	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚𝐴
109		nfrab1 3289	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑚{𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}
110	12, 109	nfcxfr 2898	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚𝑀
111		nfcv 2900	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚ℝ
112		nfcv 2900	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚 <
113	110, 111, 112	nfinf 9087	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚inf(𝑀, ℝ, < )
114	108, 113	nfmpt 5141	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚(𝑖 ∈ 𝐴 ↦ inf(𝑀, ℝ, < ))
115	11, 114	nfcxfr 2898	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚𝑉
116		nfcv 2900	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚𝑖
117	115, 116	nffv 6716	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚(𝑉‘𝑖)
118	117, 109	nfel 2914	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚(𝑉‘𝑖) ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}
119	117	nfel1 2916	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚(𝑉‘𝑖) ∈ ℕ
120		nfcv 2900	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚(,)
121		nfcv 2900	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚+∞
122	117, 120, 121	nfov 7232	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚((𝑉‘𝑖)(,)+∞)
123		nfv 1922	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚𝜒
124	122, 123	nfralw 3140	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚∀𝑟 ∈ ((𝑉‘𝑖)(,)+∞)𝜒
125	119, 124	nfan 1907	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚((𝑉‘𝑖) ∈ ℕ ∧ ∀𝑟 ∈ ((𝑉‘𝑖)(,)+∞)𝜒)
126	118, 125	nfbi 1911	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚((𝑉‘𝑖) ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ↔ ((𝑉‘𝑖) ∈ ℕ ∧ ∀𝑟 ∈ ((𝑉‘𝑖)(,)+∞)𝜒))
127		eleq1 2821	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑉‘𝑖) → (𝑚 ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ↔ (𝑉‘𝑖) ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}))
128		eleq1 2821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (𝑉‘𝑖) → (𝑚 ∈ ℕ ↔ (𝑉‘𝑖) ∈ ℕ))
129		oveq1 7209	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = (𝑉‘𝑖) → (𝑚(,)+∞) = ((𝑉‘𝑖)(,)+∞))
130		nfcv 2900	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑟(𝑚(,)+∞)
131		nfcv 2900	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑟𝐴
132		nfra1 3133	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑟∀𝑟 ∈ (𝑚(,)+∞)𝜒
133		nfcv 2900	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑟ℕ
134	132, 133	nfrabw 3290	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑟{𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}
135	12, 134	nfcxfr 2898	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑟𝑀
136		nfcv 2900	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑟ℝ
137		nfcv 2900	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑟 <
138	135, 136, 137	nfinf 9087	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑟inf(𝑀, ℝ, < )
139	131, 138	nfmpt 5141	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑟(𝑖 ∈ 𝐴 ↦ inf(𝑀, ℝ, < ))
140	11, 139	nfcxfr 2898	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑟𝑉
141		nfcv 2900	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑟𝑖
142	140, 141	nffv 6716	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑟(𝑉‘𝑖)
143		nfcv 2900	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑟(,)
144		nfcv 2900	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑟+∞
145	142, 143, 144	nfov 7232	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑟((𝑉‘𝑖)(,)+∞)
146	130, 145	raleqf 3302	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚(,)+∞) = ((𝑉‘𝑖)(,)+∞) → (∀𝑟 ∈ (𝑚(,)+∞)𝜒 ↔ ∀𝑟 ∈ ((𝑉‘𝑖)(,)+∞)𝜒))
147	129, 146	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (𝑉‘𝑖) → (∀𝑟 ∈ (𝑚(,)+∞)𝜒 ↔ ∀𝑟 ∈ ((𝑉‘𝑖)(,)+∞)𝜒))
148	128, 147	anbi12d 634	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑉‘𝑖) → ((𝑚 ∈ ℕ ∧ ∀𝑟 ∈ (𝑚(,)+∞)𝜒) ↔ ((𝑉‘𝑖) ∈ ℕ ∧ ∀𝑟 ∈ ((𝑉‘𝑖)(,)+∞)𝜒)))
149	127, 148	bibi12d 349	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (𝑉‘𝑖) → ((𝑚 ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ↔ (𝑚 ∈ ℕ ∧ ∀𝑟 ∈ (𝑚(,)+∞)𝜒)) ↔ ((𝑉‘𝑖) ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ↔ ((𝑉‘𝑖) ∈ ℕ ∧ ∀𝑟 ∈ ((𝑉‘𝑖)(,)+∞)𝜒))))
150		rabid 3283	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ↔ (𝑚 ∈ ℕ ∧ ∀𝑟 ∈ (𝑚(,)+∞)𝜒))
151	117, 126, 149, 150	vtoclgf 3472	. . . . . . . . . . . . . 14 ⊢ ((𝑉‘𝑖) ∈ ℕ → ((𝑉‘𝑖) ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ↔ ((𝑉‘𝑖) ∈ ℕ ∧ ∀𝑟 ∈ ((𝑉‘𝑖)(,)+∞)𝜒)))
152	80, 151	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ((𝑉‘𝑖) ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ↔ ((𝑉‘𝑖) ∈ ℕ ∧ ∀𝑟 ∈ ((𝑉‘𝑖)(,)+∞)𝜒)))
153	107, 152	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ((𝑉‘𝑖) ∈ ℕ ∧ ∀𝑟 ∈ ((𝑉‘𝑖)(,)+∞)𝜒))
154	153	simprd 499	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ∀𝑟 ∈ ((𝑉‘𝑖)(,)+∞)𝜒)
155	154	r19.21bi 3123	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ ((𝑉‘𝑖)(,)+∞)) → 𝜒)
156	105, 155	syldan 594	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → 𝜒)
157	156	an32s 652	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ (𝑁(,)+∞)) ∧ 𝑖 ∈ 𝐴) → 𝜒)
158	157	ex 416	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝑁(,)+∞)) → (𝑖 ∈ 𝐴 → 𝜒))
159	70, 158	ralrimi 3130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝑁(,)+∞)) → ∀𝑖 ∈ 𝐴 𝜒)
160	159	ex 416	. . . . 5 ⊢ (𝜑 → (𝑟 ∈ (𝑁(,)+∞) → ∀𝑖 ∈ 𝐴 𝜒))
161	61, 160	ralrimi 3130	. . . 4 ⊢ (𝜑 → ∀𝑟 ∈ (𝑁(,)+∞)∀𝑖 ∈ 𝐴 𝜒)
162	161	adantr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ∀𝑟 ∈ (𝑁(,)+∞)∀𝑖 ∈ 𝐴 𝜒)
163		oveq1 7209	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛(,)+∞) = (𝑁(,)+∞))
164		nfcv 2900	. . . . . 6 ⊢ Ⅎ𝑟(𝑛(,)+∞)
165	140	nfrn 5810	. . . . . . . . 9 ⊢ Ⅎ𝑟ran 𝑉
166	165, 136, 137	nfsup 9056	. . . . . . . 8 ⊢ Ⅎ𝑟sup(ran 𝑉, ℝ, < )
167	9, 166	nfcxfr 2898	. . . . . . 7 ⊢ Ⅎ𝑟𝑁
168	167, 143, 144	nfov 7232	. . . . . 6 ⊢ Ⅎ𝑟(𝑁(,)+∞)
169	164, 168	raleqf 3302	. . . . 5 ⊢ ((𝑛(,)+∞) = (𝑁(,)+∞) → (∀𝑟 ∈ (𝑛(,)+∞)∀𝑖 ∈ 𝐴 𝜒 ↔ ∀𝑟 ∈ (𝑁(,)+∞)∀𝑖 ∈ 𝐴 𝜒))
170	163, 169	syl 17	. . . 4 ⊢ (𝑛 = 𝑁 → (∀𝑟 ∈ (𝑛(,)+∞)∀𝑖 ∈ 𝐴 𝜒 ↔ ∀𝑟 ∈ (𝑁(,)+∞)∀𝑖 ∈ 𝐴 𝜒))
171	170	rspcev 3530	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ ∀𝑟 ∈ (𝑁(,)+∞)∀𝑖 ∈ 𝐴 𝜒) → ∃𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)∀𝑖 ∈ 𝐴 𝜒)
172	60, 162, 171	syl2anc 587	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ∃𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)∀𝑖 ∈ 𝐴 𝜒)
173	8, 172	pm2.61dan 813	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)∀𝑖 ∈ 𝐴 𝜒)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 Ⅎwnf 1791 ∈ wcel 2110 Ⅎwnfc 2880 ≠ wne 2935 ∀wral 3054 ∃wrex 3055 {crab 3058 ⊆ wss 3857 ∅c0 4227 class class class wbr 5043 ↦ cmpt 5124 Or wor 5456 ran crn 5541 ‘cfv 6369 (class class class)co 7202 Fincfn 8615 supcsup 9045 infcinf 9046 ℝcr 10711 1c1 10713 +∞cpnf 10847 ℝ^*cxr 10849 < clt 10850 ≤ cle 10851 ℕcn 11813 ℤ_≥cuz 12421 (,)cioo 12918

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-inf 9048 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-n0 12074 df-z 12160 df-uz 12422 df-ioo 12922

This theorem is referenced by: fourierdlem73 43349

W3C validator