finfdm - Mathbox for Glauco Siliprandi

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Theorem finfdm 46801

Description: The domain of the inf function is defined in Proposition 121F (c) of [Fremlin1], p. 39. See smfinf 46773. 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 2902	. . . . 5 ⊢ Ⅎ𝑥ℕ
4		nfcv 2902	. . . . . 6 ⊢ Ⅎ𝑥𝑍
5		finfdm.7	. . . . . . . . 9 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}))
6		nfrab1 3453	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}
7	3, 6	nfmpt 5254	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})
8	4, 7	nfmpt 5254	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}))
9	5, 8	nfcxfr 2900	. . . . . . . 8 ⊢ Ⅎ𝑥𝐻
10		nfcv 2902	. . . . . . . 8 ⊢ Ⅎ𝑥𝑛
11	9, 10	nffv 6916	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑛)
12		nfcv 2902	. . . . . . 7 ⊢ Ⅎ𝑥𝑚
13	11, 12	nffv 6916	. . . . . 6 ⊢ Ⅎ𝑥((𝐻‘𝑛)‘𝑚)
14	4, 13	nfiin 5028	. . . . 5 ⊢ Ⅎ𝑥∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)
15	3, 14	nfiun 5027	. . . 4 ⊢ Ⅎ𝑥∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)
16		finfdm.3	. . . . . . . . . . 11 ⊢ Ⅎ𝑚𝜑
17		nfv 1911	. . . . . . . . . . 11 ⊢ Ⅎ𝑚 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
18	16, 17	nfan 1896	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
19		nfv 1911	. . . . . . . . . 10 ⊢ Ⅎ𝑚 𝑦 ∈ ℝ
20	18, 19	nfan 1896	. . . . . . . . 9 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ)
21		nfv 1911	. . . . . . . . 9 ⊢ Ⅎ𝑚∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)
22	20, 21	nfan 1896	. . . . . . . 8 ⊢ Ⅎ𝑚(((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
23		finfdm.1	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝜑
24		nfii1 5033	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
25	24	nfel2 2921	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
26	23, 25	nfan 1896	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
27		nfv 1911	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛 𝑦 ∈ ℝ
28	26, 27	nfan 1896	. . . . . . . . . . 11 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ)
29		nfra1 3281	. . . . . . . . . . 11 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)
30	28, 29	nfan 1896	. . . . . . . . . 10 ⊢ Ⅎ𝑛(((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
31		nfv 1911	. . . . . . . . . 10 ⊢ Ⅎ𝑛 𝑚 ∈ ℕ
32		nfv 1911	. . . . . . . . . 10 ⊢ Ⅎ𝑛-𝑦 < 𝑚
33	30, 31, 32	nf3an 1898	. . . . . . . . 9 ⊢ Ⅎ𝑛((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)) ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚)
34		vex 3481	. . . . . . . . . 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 1184	. . . . . . . . . . . 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 45050	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
40	37, 38, 39	syl2anc 584	. . . . . . . . . . 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 1191	. . . . . . . . . . . . 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 12270	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
43	42	renegcld 11687	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → -𝑚 ∈ ℝ)
44	43	rexrd 11308	. . . . . . . . . . . . 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 11304	. . . . . . . . . . . . . 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 1184	. . . . . . . . . . . 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 1184	. . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑛)) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ^*)
54		simp3 1137	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑛)) → 𝑥 ∈ dom (𝐹‘𝑛))
55	53, 54	ffvelcdmd 7104	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑛)) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ^*)
56	51, 38, 40, 55	syl3anc 1370	. . . . . . . . . . . 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 1184	. . . . . . . . . . . . 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 1192	. . . . . . . . . . . . 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 1135	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) → 𝑦 ∈ ℝ)
60	42	3ad2ant2 1133	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) → 𝑚 ∈ ℝ)
61		simp3 1137	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℝ ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) → -𝑦 < 𝑚)
62	59, 60, 61	ltnegcon1d 11840	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ 𝑚 ∈ ℕ ∧ -𝑦 < 𝑚) → -𝑚 < 𝑦)
63	57, 41, 58, 62	syl3anc 1370	. . . . . . . . . . . 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 1224	. . . . . . . . . . . . 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 3245	. . . . . . . . . . . . 13 ⊢ ((∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥) ∧ 𝑛 ∈ 𝑍) → 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
66	64, 38, 65	syl2anc 584	. . . . . . . . . . . 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 13199	. . . . . . . . . . 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 45097	. . . . . . . . . 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 12269	. . . . . . . . . . . . . . . 16 ⊢ ℕ ∈ V
71	70	mptex 7242	. . . . . . . . . . . . . . 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 7026	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 → (𝐻‘𝑛) = (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}))
75		finfdm.4	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥𝐹
76	75, 10	nffv 6916	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝐹‘𝑛)
77	76	nfdm 5964	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥dom (𝐹‘𝑛)
78		fvex 6919	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑛) ∈ V
79	78	dmex 7931	. . . . . . . . . . . . . . 15 ⊢ dom (𝐹‘𝑛) ∈ V
80	77, 79	rabexf 45073	. . . . . . . . . . . . . 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 7028	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ ℕ) → ((𝐻‘𝑛)‘𝑚) = {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})
83	82	eqcomd 2740	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ ℕ) → {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)} = ((𝐻‘𝑛)‘𝑚))
84	38, 41, 83	syl2anc 584	. . . . . . . . . 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 2840	. . . . . . . . 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 45069	. . . . . . . 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 11569	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
88	87	archd 45104	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → ∃𝑚 ∈ ℕ -𝑦 < 𝑚)
89	88	ad2antlr 727	. . . . . . . 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 45090	. . . . . . 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 3154	. . . . . 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 1116	. . . . 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 4999	. . . . 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 45081	. . 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 4043	. 2 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
97		nfcv 2902	. . 3 ⊢ Ⅎ𝑚𝐷
98		nfv 1911	. . . . 5 ⊢ Ⅎ𝑥 𝑚 ∈ ℕ
99	2, 98	nfan 1896	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑚 ∈ ℕ)
100		nfrab1 3453	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)}
101	1, 100	nfcxfr 2900	. . . 4 ⊢ Ⅎ𝑥𝐷
102	23, 31	nfan 1896	. . . . . . . 8 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑚 ∈ ℕ)
103		nfii1 5033	. . . . . . . . 9 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)
104	103	nfel2 2921	. . . . . . . 8 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)
105	102, 104	nfan 1896	. . . . . . 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 45050	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ((𝐻‘𝑛)‘𝑚))
108	107	adantll 714	. . . . . . . . 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 584	. . . . . . . . 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 2840	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})
113		rabidim1 3455	. . . . . . . 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 45069	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
116	43	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → -𝑚 ∈ ℝ)
117		breq1 5150	. . . . . . . . 9 ⊢ (𝑦 = -𝑚 → (𝑦 ≤ ((𝐹‘𝑛)‘𝑥) ↔ -𝑚 ≤ ((𝐹‘𝑛)‘𝑥)))
118	117	ralbidv 3175	. . . . . . . 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 1370	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ^*)
123		rabidim2 45041	. . . . . . . . . 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 13188	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → -𝑚 ≤ ((𝐹‘𝑛)‘𝑥))
126	105, 125	ralrimia 3255	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → ∀𝑛 ∈ 𝑍 -𝑚 ≤ ((𝐹‘𝑛)‘𝑥))
127	116, 119, 126	rspcedvd 3623	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥))
128	115, 127	rabidd 45097	. . . . 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 2849	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → 𝑥 ∈ 𝐷)
130	99, 14, 101, 129	ssdf2 45080	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ⊆ 𝐷)
131	16, 97, 130	iunssdf 45098	. 2 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ⊆ 𝐷)
132	96, 131	eqssd 4012	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 1086 = wceq 1536 Ⅎwnf 1779 ∈ wcel 2105 Ⅎwnfc 2887 ∀wral 3058 ∃wrex 3067 {crab 3432 Vcvv 3477 ∪ ciun 4995 ∩ ciin 4996 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5688 ⟶wf 6558 ‘cfv 6562 ℝcr 11151 ℝ^*cxr 11291 < clt 11292 ≤ cle 11293 -cneg 11490 ℕcn 12263

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264

This theorem is referenced by: finfdm2 46802

W3C validator