fourierdlem31 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem31 46136

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 12197	. . . 4 ⊢ 1 ∈ ℕ
2		rzal 4472	. . . . 5 ⊢ (𝐴 = ∅ → ∀𝑖 ∈ 𝐴 𝜒)
3	2	ralrimivw 3129	. . . 4 ⊢ (𝐴 = ∅ → ∀𝑟 ∈ (1(,)+∞)∀𝑖 ∈ 𝐴 𝜒)
4		oveq1 7394	. . . . . 6 ⊢ (𝑛 = 1 → (𝑛(,)+∞) = (1(,)+∞))
5	4	raleqdv 3299	. . . . 5 ⊢ (𝑛 = 1 → (∀𝑟 ∈ (𝑛(,)+∞)∀𝑖 ∈ 𝐴 𝜒 ↔ ∀𝑟 ∈ (1(,)+∞)∀𝑖 ∈ 𝐴 𝜒))
6	5	rspcev 3588	. . . 4 ⊢ ((1 ∈ ℕ ∧ ∀𝑟 ∈ (1(,)+∞)∀𝑖 ∈ 𝐴 𝜒) → ∃𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)∀𝑖 ∈ 𝐴 𝜒)
7	1, 3, 6	sylancr 587	. . 3 ⊢ (𝐴 = ∅ → ∃𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)∀𝑖 ∈ 𝐴 𝜒)
8	7	adantl 481	. 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 9429	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → inf(𝑀, ℝ, < ) = inf({𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}, ℝ, < ))
15		ssrab2 4043	. . . . . . . . 9 ⊢ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ⊆ ℕ
16		nnuz 12836	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
17	15, 16	sseqtri 3995	. . . . . . . . . 10 ⊢ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ⊆ (ℤ_≥‘1)
18		fourierdlem31.exm	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑖 ∈ 𝐴 ∃𝑚 ∈ ℕ ∀𝑟 ∈ (𝑚(,)+∞)𝜒)
19	18	r19.21bi 3229	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ∃𝑚 ∈ ℕ ∀𝑟 ∈ (𝑚(,)+∞)𝜒)
20		rabn0 4352	. . . . . . . . . . 11 ⊢ ({𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ≠ ∅ ↔ ∃𝑚 ∈ ℕ ∀𝑟 ∈ (𝑚(,)+∞)𝜒)
21	19, 20	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ≠ ∅)
22		infssuzcl 12891	. . . . . . . . . 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 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → inf({𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}, ℝ, < ) ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒})
24	15, 23	sselid 3944	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → inf({𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}, ℝ, < ) ∈ ℕ)
25	14, 24	eqeltrd 2828	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → inf(𝑀, ℝ, < ) ∈ ℕ)
26	10, 11, 25	rnmptssd 45190	. . . . . 6 ⊢ (𝜑 → ran 𝑉 ⊆ ℕ)
27	26	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ran 𝑉 ⊆ ℕ)
28		ltso 11254	. . . . . . 7 ⊢ < Or ℝ
29	28	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → < Or ℝ)
30		fourierdlem31.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ Fin)
31		mptfi 9302	. . . . . . . . . 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 2832	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ Fin)
34		rnfi 9291	. . . . . . . 8 ⊢ (𝑉 ∈ Fin → ran 𝑉 ∈ Fin)
35	33, 34	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝑉 ∈ Fin)
36	35	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ran 𝑉 ∈ Fin)
37		neqne 2933	. . . . . . . . 9 ⊢ (¬ 𝐴 = ∅ → 𝐴 ≠ ∅)
38		n0 4316	. . . . . . . . 9 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑖 𝑖 ∈ 𝐴)
39	37, 38	sylib 218	. . . . . . . 8 ⊢ (¬ 𝐴 = ∅ → ∃𝑖 𝑖 ∈ 𝐴)
40	39	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ∃𝑖 𝑖 ∈ 𝐴)
41		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑖 ¬ 𝐴 = ∅
42	10, 41	nfan 1899	. . . . . . . 8 ⊢ Ⅎ𝑖(𝜑 ∧ ¬ 𝐴 = ∅)
43		fourierdlem31.iv	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝑉
44	43	nfrn 5916	. . . . . . . . 9 ⊢ Ⅎ𝑖ran 𝑉
45		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑖∅
46	44, 45	nfne 3026	. . . . . . . 8 ⊢ Ⅎ𝑖ran 𝑉 ≠ ∅
47		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝑖 ∈ 𝐴)
48	11	elrnmpt1 5924	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ 𝐴 ∧ inf(𝑀, ℝ, < ) ∈ ℕ) → inf(𝑀, ℝ, < ) ∈ ran 𝑉)
49	47, 25, 48	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → inf(𝑀, ℝ, < ) ∈ ran 𝑉)
50	49	ne0d 4305	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ran 𝑉 ≠ ∅)
51	50	ex 412	. . . . . . . . 9 ⊢ (𝜑 → (𝑖 ∈ 𝐴 → ran 𝑉 ≠ ∅))
52	51	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → (𝑖 ∈ 𝐴 → ran 𝑉 ≠ ∅))
53	42, 46, 52	exlimd 2219	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → (∃𝑖 𝑖 ∈ 𝐴 → ran 𝑉 ≠ ∅))
54	40, 53	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ran 𝑉 ≠ ∅)
55		nnssre 12190	. . . . . . 7 ⊢ ℕ ⊆ ℝ
56	27, 55	sstrdi 3959	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ran 𝑉 ⊆ ℝ)
57		fisupcl 9421	. . . . . 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 3947	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → sup(ran 𝑉, ℝ, < ) ∈ ℕ)
60	9, 59	eqeltrid 2832	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝑁 ∈ ℕ)
61		fourierdlem31.r	. . . . 5 ⊢ Ⅎ𝑟𝜑
62		nfcv 2891	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖ℝ
63		nfcv 2891	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖 <
64	44, 62, 63	nfsup 9402	. . . . . . . . . . 11 ⊢ Ⅎ𝑖sup(ran 𝑉, ℝ, < )
65	9, 64	nfcxfr 2889	. . . . . . . . . 10 ⊢ Ⅎ𝑖𝑁
66		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑖(,)
67		nfcv 2891	. . . . . . . . . 10 ⊢ Ⅎ𝑖+∞
68	65, 66, 67	nfov 7417	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝑁(,)+∞)
69	68	nfcri 2883	. . . . . . . 8 ⊢ Ⅎ𝑖 𝑟 ∈ (𝑁(,)+∞)
70	10, 69	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑟 ∈ (𝑁(,)+∞))
71	11	fvmpt2 6979	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ 𝐴 ∧ inf(𝑀, ℝ, < ) ∈ ℕ) → (𝑉‘𝑖) = inf(𝑀, ℝ, < ))
72	47, 25, 71	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → (𝑉‘𝑖) = inf(𝑀, ℝ, < ))
73	25	nnxrd 45272	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → inf(𝑀, ℝ, < ) ∈ ℝ^*)
74	72, 73	eqeltrd 2828	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → (𝑉‘𝑖) ∈ ℝ^*)
75	74	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → (𝑉‘𝑖) ∈ ℝ^*)
76		pnfxr 11228	. . . . . . . . . . . 12 ⊢ +∞ ∈ ℝ^*
77	76	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → +∞ ∈ ℝ^*)
78		elioore 13336	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ (𝑁(,)+∞) → 𝑟 ∈ ℝ)
79	78	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → 𝑟 ∈ ℝ)
80	72, 25	eqeltrd 2828	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → (𝑉‘𝑖) ∈ ℕ)
81	80	nnred 12201	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → (𝑉‘𝑖) ∈ ℝ)
82	81	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → (𝑉‘𝑖) ∈ ℝ)
83		ne0i 4304	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ 𝐴 → 𝐴 ≠ ∅)
84	83	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝐴 ≠ ∅)
85	84	neneqd 2930	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ¬ 𝐴 = ∅)
86	85, 60	syldan 591	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝑁 ∈ ℕ)
87	86	nnred 12201	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝑁 ∈ ℝ)
88	87	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → 𝑁 ∈ ℝ)
89	85, 56	syldan 591	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ran 𝑉 ⊆ ℝ)
90	26, 55	sstrdi 3959	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran 𝑉 ⊆ ℝ)
91		fimaxre2 12128	. . . . . . . . . . . . . . . . 17 ⊢ ((ran 𝑉 ⊆ ℝ ∧ ran 𝑉 ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑉 𝑦 ≤ 𝑥)
92	90, 35, 91	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑉 𝑦 ≤ 𝑥)
93	92	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran 𝑉 𝑦 ≤ 𝑥)
94	72, 49	eqeltrd 2828	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → (𝑉‘𝑖) ∈ ran 𝑉)
95		suprub 12144	. . . . . . . . . . . . . . 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 5149	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → (𝑉‘𝑖) ≤ 𝑁)
98	97	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → (𝑉‘𝑖) ≤ 𝑁)
99	88	rexrd 11224	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → 𝑁 ∈ ℝ^*)
100		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → 𝑟 ∈ (𝑁(,)+∞))
101		ioogtlb 45493	. . . . . . . . . . . . 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 11332	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → (𝑉‘𝑖) < 𝑟)
104	79	ltpnfd 13081	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → 𝑟 < +∞)
105	75, 77, 79, 103, 104	eliood 45496	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → 𝑟 ∈ ((𝑉‘𝑖)(,)+∞))
106	14, 23	eqeltrd 2828	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → inf(𝑀, ℝ, < ) ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒})
107	72, 106	eqeltrd 2828	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → (𝑉‘𝑖) ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒})
108		nfcv 2891	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚𝐴
109		nfrab1 3426	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑚{𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}
110	12, 109	nfcxfr 2889	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚𝑀
111		nfcv 2891	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚ℝ
112		nfcv 2891	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚 <
113	110, 111, 112	nfinf 9434	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚inf(𝑀, ℝ, < )
114	108, 113	nfmpt 5205	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚(𝑖 ∈ 𝐴 ↦ inf(𝑀, ℝ, < ))
115	11, 114	nfcxfr 2889	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚𝑉
116		nfcv 2891	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚𝑖
117	115, 116	nffv 6868	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚(𝑉‘𝑖)
118	117, 109	nfel 2906	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚(𝑉‘𝑖) ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}
119	117	nfel1 2908	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚(𝑉‘𝑖) ∈ ℕ
120		nfcv 2891	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚(,)
121		nfcv 2891	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚+∞
122	117, 120, 121	nfov 7417	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚((𝑉‘𝑖)(,)+∞)
123		nfv 1914	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚𝜒
124	122, 123	nfralw 3285	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚∀𝑟 ∈ ((𝑉‘𝑖)(,)+∞)𝜒
125	119, 124	nfan 1899	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚((𝑉‘𝑖) ∈ ℕ ∧ ∀𝑟 ∈ ((𝑉‘𝑖)(,)+∞)𝜒)
126	118, 125	nfbi 1903	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚((𝑉‘𝑖) ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ↔ ((𝑉‘𝑖) ∈ ℕ ∧ ∀𝑟 ∈ ((𝑉‘𝑖)(,)+∞)𝜒))
127		eleq1 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑉‘𝑖) → (𝑚 ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ↔ (𝑉‘𝑖) ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}))
128		eleq1 2816	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (𝑉‘𝑖) → (𝑚 ∈ ℕ ↔ (𝑉‘𝑖) ∈ ℕ))
129		oveq1 7394	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = (𝑉‘𝑖) → (𝑚(,)+∞) = ((𝑉‘𝑖)(,)+∞))
130		nfcv 2891	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑟(𝑚(,)+∞)
131		nfcv 2891	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑟𝐴
132		nfra1 3261	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑟∀𝑟 ∈ (𝑚(,)+∞)𝜒
133		nfcv 2891	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑟ℕ
134	132, 133	nfrabw 3443	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑟{𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒}
135	12, 134	nfcxfr 2889	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑟𝑀
136		nfcv 2891	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑟ℝ
137		nfcv 2891	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑟 <
138	135, 136, 137	nfinf 9434	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑟inf(𝑀, ℝ, < )
139	131, 138	nfmpt 5205	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑟(𝑖 ∈ 𝐴 ↦ inf(𝑀, ℝ, < ))
140	11, 139	nfcxfr 2889	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑟𝑉
141		nfcv 2891	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑟𝑖
142	140, 141	nffv 6868	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑟(𝑉‘𝑖)
143		nfcv 2891	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑟(,)
144		nfcv 2891	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑟+∞
145	142, 143, 144	nfov 7417	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑟((𝑉‘𝑖)(,)+∞)
146	130, 145	raleqf 3329	. . . . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (𝑉‘𝑖) → ((𝑚 ∈ ℕ ∧ ∀𝑟 ∈ (𝑚(,)+∞)𝜒) ↔ ((𝑉‘𝑖) ∈ ℕ ∧ ∀𝑟 ∈ ((𝑉‘𝑖)(,)+∞)𝜒)))
149	127, 148	bibi12d 345	. . . . . . . . . . . . . . 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 3427	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ {𝑚 ∈ ℕ ∣ ∀𝑟 ∈ (𝑚(,)+∞)𝜒} ↔ (𝑚 ∈ ℕ ∧ ∀𝑟 ∈ (𝑚(,)+∞)𝜒))
151	117, 126, 149, 150	vtoclgf 3535	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ((𝑉‘𝑖) ∈ ℕ ∧ ∀𝑟 ∈ ((𝑉‘𝑖)(,)+∞)𝜒))
154	153	simprd 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → ∀𝑟 ∈ ((𝑉‘𝑖)(,)+∞)𝜒)
155	154	r19.21bi 3229	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ ((𝑉‘𝑖)(,)+∞)) → 𝜒)
156	105, 155	syldan 591	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝐴) ∧ 𝑟 ∈ (𝑁(,)+∞)) → 𝜒)
157	156	an32s 652	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ (𝑁(,)+∞)) ∧ 𝑖 ∈ 𝐴) → 𝜒)
158	157	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝑁(,)+∞)) → (𝑖 ∈ 𝐴 → 𝜒))
159	70, 158	ralrimi 3235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝑁(,)+∞)) → ∀𝑖 ∈ 𝐴 𝜒)
160	159	ex 412	. . . . 5 ⊢ (𝜑 → (𝑟 ∈ (𝑁(,)+∞) → ∀𝑖 ∈ 𝐴 𝜒))
161	61, 160	ralrimi 3235	. . . 4 ⊢ (𝜑 → ∀𝑟 ∈ (𝑁(,)+∞)∀𝑖 ∈ 𝐴 𝜒)
162	161	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ∀𝑟 ∈ (𝑁(,)+∞)∀𝑖 ∈ 𝐴 𝜒)
163		oveq1 7394	. . . . 5 ⊢ (𝑛 = 𝑁 → (𝑛(,)+∞) = (𝑁(,)+∞))
164		nfcv 2891	. . . . . 6 ⊢ Ⅎ𝑟(𝑛(,)+∞)
165	140	nfrn 5916	. . . . . . . . 9 ⊢ Ⅎ𝑟ran 𝑉
166	165, 136, 137	nfsup 9402	. . . . . . . 8 ⊢ Ⅎ𝑟sup(ran 𝑉, ℝ, < )
167	9, 166	nfcxfr 2889	. . . . . . 7 ⊢ Ⅎ𝑟𝑁
168	167, 143, 144	nfov 7417	. . . . . 6 ⊢ Ⅎ𝑟(𝑁(,)+∞)
169	164, 168	raleqf 3329	. . . . 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 3588	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ ∀𝑟 ∈ (𝑁(,)+∞)∀𝑖 ∈ 𝐴 𝜒) → ∃𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)∀𝑖 ∈ 𝐴 𝜒)
172	60, 162, 171	syl2anc 584	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ∃𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)∀𝑖 ∈ 𝐴 𝜒)
173	8, 172	pm2.61dan 812	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑟 ∈ (𝑛(,)+∞)∀𝑖 ∈ 𝐴 𝜒)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 Ⅎwnf 1783 ∈ wcel 2109 Ⅎwnfc 2876 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 {crab 3405 ⊆ wss 3914 ∅c0 4296 class class class wbr 5107 ↦ cmpt 5188 Or wor 5545 ran crn 5639 ‘cfv 6511 (class class class)co 7387 Fincfn 8918 supcsup 9391 infcinf 9392 ℝcr 11067 1c1 11069 +∞cpnf 11205 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 ℕcn 12186 ℤ_≥cuz 12793 (,)cioo 13306

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 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-n0 12443 df-z 12530 df-uz 12794 df-ioo 13310

This theorem is referenced by: fourierdlem73 46177

W3C validator