finfdm - Mathbox for Glauco Siliprandi

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Theorem finfdm 47295

Description: The domain of the inf function is defined in Proposition 121F (c) of [Fremlin1], p. 39. See smfinf 47267. Note that this definition of the inf function is quite general, as it does not require the original functions to be sigma-measurable, and it could be applied to uncountable sets of functions. The equality proved here is part of the proof of the fifth statement of Proposition 121H in [Fremlin1], p. 39. (Contributed by Glauco Siliprandi, 1-Feb-2025.)

**Hypotheses**
Ref	Expression
finfdm.1	⊢ Ⅎ𝑛𝜑
finfdm.2	⊢ Ⅎ𝑥𝜑
finfdm.3	⊢ Ⅎ𝑚𝜑
finfdm.4	⊢ Ⅎ𝑥𝐹
finfdm.5	⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ^*)
finfdm.6	⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)}
finfdm.7	⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}))

**Assertion**
Ref	Expression
finfdm	⊢ (𝜑 → 𝐷 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))

Distinct variable groups: 𝐷,𝑚 𝑚,𝐹,𝑦 𝑦,𝐻 𝑚,𝑍,𝑛,𝑥,𝑦 𝜑,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑚,𝑛) 𝐷(𝑥,𝑦,𝑛) 𝐹(𝑥,𝑛) 𝐻(𝑥,𝑚,𝑛)

**Proof of Theorem finfdm**
Step	Hyp	Ref	Expression
1		finfdm.6	. . 3 ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)}
2		finfdm.2	. . . 4 ⊢ Ⅎ𝑥𝜑
3		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑥ℕ
4		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥𝑍
5		finfdm.7	. . . . . . . . 9 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}))
6		nfrab1 3410	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}
7	3, 6	nfmpt 5184	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})
8	4, 7	nfmpt 5184	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}))
9	5, 8	nfcxfr 2897	. . . . . . . 8 ⊢ Ⅎ𝑥𝐻
10		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥𝑛
11	9, 10	nffv 6845	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑛)
12		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑥𝑚
13	11, 12	nffv 6845	. . . . . 6 ⊢ Ⅎ𝑥((𝐻‘𝑛)‘𝑚)
14	4, 13	nfiin 4967	. . . . 5 ⊢ Ⅎ𝑥∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)
15	3, 14	nfiun 4966	. . . 4 ⊢ Ⅎ𝑥∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)
16		finfdm.3	. . . . . . . . . . 11 ⊢ Ⅎ𝑚𝜑
17		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑚 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
18	16, 17	nfan 1901	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
19		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑚 𝑦 ∈ ℝ
20	18, 19	nfan 1901	. . . . . . . . 9 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ)
21		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑚∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)
22	20, 21	nfan 1901	. . . . . . . 8 ⊢ Ⅎ𝑚(((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
23		finfdm.1	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝜑
24		nfii1 4972	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
25	24	nfel2 2918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
26	23, 25	nfan 1901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
27		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛 𝑦 ∈ ℝ
28	26, 27	nfan 1901	. . . . . . . . . . 11 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ)
29		nfra1 3262	. . . . . . . . . . 11 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)
30	28, 29	nfan 1901	. . . . . . . . . 10 ⊢ Ⅎ𝑛(((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
31		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑛 𝑚 ∈ ℕ
32		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑛-𝑦 < 𝑚
33	30, 31, 32	nf3an 1903	. . . . . . . . 9 ⊢ Ⅎ𝑛((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚)
34		vex 3434	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
35	34	a1i 11	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) → 𝑥 ∈ V)
36		simp-4r 784	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
37	36	3ad2antl1 1187	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
38		simpr 484	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
39		eliinid 45562	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
40	37, 38, 39	syl2anc 585	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
41		simpl2 1194	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑚 ∈ ℕ)
42		nnre 12175	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
43	42	renegcld 11571	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → -𝑚 ∈ ℝ)
44	43	rexrd 11189	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → -𝑚 ∈ ℝ^*)
45	41, 44	syl 17	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → -𝑚 ∈ ℝ^*)
46		simpllr 776	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑛 ∈ 𝑍) → 𝑦 ∈ ℝ)
47		rexr 11185	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ^*)
48	46, 47	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑛 ∈ 𝑍) → 𝑦 ∈ ℝ^*)
49	48	3ad2antl1 1187	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑦 ∈ ℝ^*)
50		simp-4l 783	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑛 ∈ 𝑍) → 𝜑)
51	50	3ad2antl1 1187	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝜑)
52		finfdm.5	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ^*)
53	52	3adant3 1133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑛)) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ^*)
54		simp3 1139	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑛)) → 𝑥 ∈ dom (𝐹‘𝑛))
55	53, 54	ffvelcdmd 7032	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑛)) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ^*)
56	51, 38, 40, 55	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ^*)
57	46	3ad2antl1 1187	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑦 ∈ ℝ)
58		simpl3 1195	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → -𝑦 < 𝑚)
59		simp1 1137	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) → 𝑦 ∈ ℝ)
60	42	3ad2ant2 1135	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) → 𝑚 ∈ ℝ)
61		simp3 1139	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) → -𝑦 < 𝑚)
62	59, 60, 61	ltnegcon1d 11724	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) → -𝑚 < 𝑦)
63	57, 41, 58, 62	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → -𝑚 < 𝑦)
64		simpl1r 1227	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
65		rspa 3227	. . . . . . . . . . . . 13 ⊢ ((∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥) ∧ 𝑛 ∈ 𝑍) → 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
66	64, 38, 65	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
67	45, 49, 56, 63, 66	xrltletrd 13106	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → -𝑚 < ((𝐹‘𝑛)‘𝑥))
68	40, 67	rabidd 45606	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})
69		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ 𝑍)
70		nnex 12174	. . . . . . . . . . . . . . . 16 ⊢ ℕ ∈ V
71	70	mptex 7172	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}) ∈ V
72	71	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}) ∈ V)
73	5	fvmpt2 6954	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ 𝑍 ∧ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}) ∈ V) → (𝐻‘𝑛) = (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}))
74	69, 72, 73	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 → (𝐻‘𝑛) = (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}))
75		finfdm.4	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥𝐹
76	75, 10	nffv 6845	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝐹‘𝑛)
77	76	nfdm 5901	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥dom (𝐹‘𝑛)
78		fvex 6848	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑛) ∈ V
79	78	dmex 7854	. . . . . . . . . . . . . . 15 ⊢ dom (𝐹‘𝑛) ∈ V
80	77, 79	rabexf 45585	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)} ∈ V
81	80	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ ℕ) → {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)} ∈ V)
82	74, 81	fvmpt2d 6956	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ ℕ) → ((𝐻‘𝑛)‘𝑚) = {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})
83	82	eqcomd 2743	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ ℕ) → {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)} = ((𝐻‘𝑛)‘𝑚))
84	38, 41, 83	syl2anc 585	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)} = ((𝐻‘𝑛)‘𝑚))
85	68, 84	eleqtrd 2839	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ((𝐻‘𝑛)‘𝑚))
86	33, 35, 85	eliind2 45581	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
87		renegcl 11451	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
88	87	archd 45613	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → ∃𝑚 ∈ ℕ -𝑦 < 𝑚)
89	88	ad2antlr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) → ∃𝑚 ∈ ℕ -𝑦 < 𝑚)
90	22, 86, 89	reximdd 45599	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) → ∃𝑚 ∈ ℕ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
91	90	rexlimdva2 3141	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) → (∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥) → ∃𝑚 ∈ ℕ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)))
92	91	3impia 1118	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) → ∃𝑚 ∈ ℕ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
93		eliun 4938	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
94	92, 93	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) → 𝑥 ∈ ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
95	2, 15, 94	rabssd 45593	. . 3 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} ⊆ ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
96	1, 95	eqsstrid 3961	. 2 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
97		nfcv 2899	. . 3 ⊢ Ⅎ𝑚𝐷
98		nfv 1916	. . . . 5 ⊢ Ⅎ𝑥 𝑚 ∈ ℕ
99	2, 98	nfan 1901	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑚 ∈ ℕ)
100		nfrab1 3410	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)}
101	1, 100	nfcxfr 2897	. . . 4 ⊢ Ⅎ𝑥𝐷
102	23, 31	nfan 1901	. . . . . . . 8 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑚 ∈ ℕ)
103		nfii1 4972	. . . . . . . . 9 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)
104	103	nfel2 2918	. . . . . . . 8 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)
105	102, 104	nfan 1901	. . . . . . 7 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
106		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
107		eliinid 45562	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ((𝐻‘𝑛)‘𝑚))
108	107	adantll 715	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ((𝐻‘𝑛)‘𝑚))
109	69	adantl 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
110		simpllr 776	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → 𝑚 ∈ ℕ)
111	109, 110, 82	syl2anc 585	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛)‘𝑚) = {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})
112	108, 111	eleqtrd 2839	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})
113		rabidim1 3412	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)} → 𝑥 ∈ dom (𝐹‘𝑛))
114	112, 113	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
115	105, 106, 114	eliind2 45581	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
116	43	ad2antlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → -𝑚 ∈ ℝ)
117		breq1 5089	. . . . . . . . 9 ⊢ (𝑦 = -𝑚 → (𝑦 ≤ ((𝐹‘𝑛)‘𝑥) ↔ -𝑚 ≤ ((𝐹‘𝑛)‘𝑥)))
118	117	ralbidv 3161	. . . . . . . 8 ⊢ (𝑦 = -𝑚 → (∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥) ↔ ∀𝑛 ∈ 𝑍 -𝑚 ≤ ((𝐹‘𝑛)‘𝑥)))
119	118	adantl 481	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑦 = -𝑚) → (∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥) ↔ ∀𝑛 ∈ 𝑍 -𝑚 ≤ ((𝐹‘𝑛)‘𝑥)))
120	110, 44	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → -𝑚 ∈ ℝ^*)
121		simplll 775	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → 𝜑)
122	121, 109, 114, 55	syl3anc 1374	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ^*)
123		rabidim2 45553	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)} → -𝑚 < ((𝐹‘𝑛)‘𝑥))
124	112, 123	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → -𝑚 < ((𝐹‘𝑛)‘𝑥))
125	120, 122, 124	xrltled 13095	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → -𝑚 ≤ ((𝐹‘𝑛)‘𝑥))
126	105, 125	ralrimia 3237	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → ∀𝑛 ∈ 𝑍 -𝑚 ≤ ((𝐹‘𝑛)‘𝑥))
127	116, 119, 126	rspcedvd 3567	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
128	115, 127	rabidd 45606	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → 𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)})
129	128, 1	eleqtrrdi 2848	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → 𝑥 ∈ 𝐷)
130	99, 14, 101, 129	ssdf2 45592	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ⊆ 𝐷)
131	16, 97, 130	iunssdf 45607	. 2 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ⊆ 𝐷)
132	96, 131	eqssd 3940	1 ⊢ (𝜑 → 𝐷 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 Ⅎwnf 1785 ∈ wcel 2114 Ⅎwnfc 2884 ∀wral 3052 ∃wrex 3062 {crab 3390 Vcvv 3430 ∪ ciun 4934 ∩ ciin 4935 class class class wbr 5086 ↦ cmpt 5167 dom cdm 5625 ⟶wf 6489 ‘cfv 6493 ℝcr 11031 ℝ^*cxr 11172 < clt 11173 ≤ cle 11174 -cneg 11372 ℕcn 12168

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169

This theorem is referenced by: finfdm2 47296

W3C validator