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Theorem mbfinf 25415

Description: The infimum of a sequence of measurable, real-valued functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 13-Sep-2020.)

**Hypotheses**
Ref	Expression
mbfinf.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
mbfinf.2	⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))
mbfinf.3	⊢ (𝜑 → 𝑀 ∈ ℤ)
mbfinf.4	⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
mbfinf.5	⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)
mbfinf.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)

**Assertion**
Ref	Expression
mbfinf	⊢ (𝜑 → 𝐺 ∈ MblFn)

Distinct variable groups: 𝑥,𝑛,𝑦,𝐴 𝜑,𝑛,𝑥,𝑦 𝑛,𝑍,𝑥,𝑦 𝑦,𝐵 Allowed substitution hints: 𝐵(𝑥,𝑛) 𝐺(𝑥,𝑦,𝑛) 𝑀(𝑥,𝑦,𝑛)

Proof of Theorem mbfinf Dummy variables 𝑚 𝑟 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfinf.2	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ))
2		mbfinf.5	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℝ)
3	2	anass1rs 652	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ)
4	3	fmpttd 7116	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ)
5	4	frnd 6725	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) ⊆ ℝ)
6		mbfinf.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		uzid 12842	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
8	6, 7	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
9		mbfinf.1	. . . . . . . . . . 11 ⊢ 𝑍 = (ℤ_≥‘𝑀)
10	8, 9	eleqtrrdi 2843	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
11	10	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ 𝑍)
12		eqid 2731	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 ↦ 𝐵) = (𝑛 ∈ 𝑍 ↦ 𝐵)
13	12, 3	dmmptd 6695	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) = 𝑍)
14	11, 13	eleqtrrd 2835	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ dom (𝑛 ∈ 𝑍 ↦ 𝐵))
15	14	ne0d 4335	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
16		dm0rn0 5924	. . . . . . . 8 ⊢ (dom (𝑛 ∈ 𝑍 ↦ 𝐵) = ∅ ↔ ran (𝑛 ∈ 𝑍 ↦ 𝐵) = ∅)
17	16	necon3bii 2992	. . . . . . 7 ⊢ (dom (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅ ↔ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
18	15, 17	sylib 217	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
19		mbfinf.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)
20	4	ffnd 6718	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍)
21		breq2 5152	. . . . . . . . . . 11 ⊢ (𝑧 = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)))
22	21	ralrn 7089	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍 → (∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧 ↔ ∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)))
23	20, 22	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧 ↔ ∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)))
24		nfcv 2902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛𝑦
25		nfcv 2902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛 ≤
26		nffvmpt1 6902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)
27	24, 25, 26	nfbr 5195	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)
28		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑚 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)
29		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))
30	29	breq2d 5160	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)))
31	27, 28, 30	cbvralw 3302	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))
32		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
33	12	fvmpt2 7009	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝐵 ∈ ℝ) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵)
34	32, 3, 33	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵)
35	34	breq2d 5160	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ 𝑦 ≤ 𝐵))
36	35	ralbidva 3174	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵))
37	31, 36	bitrid 283	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵))
38	23, 37	bitrd 279	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧 ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵))
39	38	rexbidv 3177	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵))
40	19, 39	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧)
41		infrenegsup 12202	. . . . . 6 ⊢ ((ran (𝑛 ∈ 𝑍 ↦ 𝐵) ⊆ ℝ ∧ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧) → inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) = -sup({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}, ℝ, < ))
42	5, 18, 40, 41	syl3anc 1370	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) = -sup({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}, ℝ, < ))
43		rabid 3451	. . . . . . . . . 10 ⊢ (𝑟 ∈ {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} ↔ (𝑟 ∈ ℝ ∧ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)))
44	3	recnd 11247	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℂ)
45	44	adantlr 712	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℂ)
46		simplr 766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝑟 ∈ ℝ)
47	46	recnd 11247	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝑟 ∈ ℂ)
48		negcon2 11518	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℂ ∧ 𝑟 ∈ ℂ) → (𝐵 = -𝑟 ↔ 𝑟 = -𝐵))
49	45, 47, 48	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝐵 = -𝑟 ↔ 𝑟 = -𝐵))
50		eqcom 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 = -𝐵 ↔ -𝐵 = 𝑟)
51	49, 50	bitrdi 287	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝐵 = -𝑟 ↔ -𝐵 = 𝑟))
52	34	adantlr 712	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵)
53	52	eqeq1d 2733	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ 𝐵 = -𝑟))
54		negex 11463	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -𝐵 ∈ V
55		eqid 2731	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ 𝑍 ↦ -𝐵) = (𝑛 ∈ 𝑍 ↦ -𝐵)
56	55	fvmpt2 7009	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ 𝑍 ∧ -𝐵 ∈ V) → ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = -𝐵)
57	54, 56	mpan2 688	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = -𝐵)
58	57	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = -𝐵)
59	58	eqeq1d 2733	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟 ↔ -𝐵 = 𝑟))
60	51, 53, 59	3bitr4d 311	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟))
61	60	ralrimiva 3145	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟))
62	26	nfeq1 2917	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟
63		nffvmpt1 6902	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚)
64	63	nfeq1 2917	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟
65	62, 64	nfbi 1905	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟)
66		nfv 1916	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟)
67		fveqeq2 6900	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟))
68		fveqeq2 6900	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → (((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟))
69	67, 68	bibi12d 345	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑛 → ((((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟) ↔ (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟)))
70	65, 66, 69	cbvralw 3302	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑚 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟) ↔ ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟))
71	61, 70	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → ∀𝑚 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟))
72	71	r19.21bi 3247	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑚 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟))
73	72	rexbidva 3175	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (∃𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ∃𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟))
74	20	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍)
75		fvelrnb 6952	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍 → (-𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ↔ ∃𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟))
76	74, 75	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (-𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ↔ ∃𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟))
77	3	renegcld 11646	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → -𝐵 ∈ ℝ)
78	77	fmpttd 7116	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ -𝐵):𝑍⟶ℝ)
79	78	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (𝑛 ∈ 𝑍 ↦ -𝐵):𝑍⟶ℝ)
80	79	ffnd 6718	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (𝑛 ∈ 𝑍 ↦ -𝐵) Fn 𝑍)
81		fvelrnb 6952	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ 𝑍 ↦ -𝐵) Fn 𝑍 → (𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵) ↔ ∃𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟))
82	80, 81	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵) ↔ ∃𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟))
83	73, 76, 82	3bitr4d 311	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (-𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ↔ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵)))
84	83	pm5.32da 578	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑟 ∈ ℝ ∧ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)) ↔ (𝑟 ∈ ℝ ∧ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵))))
85	78	frnd 6725	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ -𝐵) ⊆ ℝ)
86	85	sseld 3981	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵) → 𝑟 ∈ ℝ))
87	86	pm4.71rd 562	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵) ↔ (𝑟 ∈ ℝ ∧ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵))))
88	84, 87	bitr4d 282	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑟 ∈ ℝ ∧ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)) ↔ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵)))
89	43, 88	bitrid 283	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑟 ∈ {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} ↔ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵)))
90	89	alrimiv 1929	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑟(𝑟 ∈ {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} ↔ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵)))
91		nfrab1 3450	. . . . . . . . 9 ⊢ Ⅎ𝑟{𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}
92		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑟ran (𝑛 ∈ 𝑍 ↦ -𝐵)
93	91, 92	cleqf 2933	. . . . . . . 8 ⊢ ({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} = ran (𝑛 ∈ 𝑍 ↦ -𝐵) ↔ ∀𝑟(𝑟 ∈ {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} ↔ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵)))
94	90, 93	sylibr 233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} = ran (𝑛 ∈ 𝑍 ↦ -𝐵))
95	94	supeq1d 9445	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}, ℝ, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ))
96	95	negeqd 11459	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -sup({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}, ℝ, < ) = -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ))
97	42, 96	eqtrd 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) = -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ))
98	97	mpteq2dva 5248	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )) = (𝑥 ∈ 𝐴 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )))
99	1, 98	eqtrid 2783	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )))
100		ltso 11299	. . . . 5 ⊢ < Or ℝ
101	100	supex 9462	. . . 4 ⊢ sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ) ∈ V
102	101	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ) ∈ V)
103		eqid 2731	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) = (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ))
104	2	anassrs 467	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
105		mbfinf.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
106	104, 105	mbfneg 25400	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ MblFn)
107	2	renegcld 11646	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → -𝐵 ∈ ℝ)
108		renegcl 11528	. . . . . . 7 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
109	108	ad2antrl 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)) → -𝑦 ∈ ℝ)
110		simplr 766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝑦 ∈ ℝ)
111	3	adantlr 712	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ)
112	110, 111	lenegd 11798	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑦 ≤ 𝐵 ↔ -𝐵 ≤ -𝑦))
113	112	ralbidva 3174	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) → (∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ↔ ∀𝑛 ∈ 𝑍 -𝐵 ≤ -𝑦))
114	113	biimpd 228	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) → (∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 → ∀𝑛 ∈ 𝑍 -𝐵 ≤ -𝑦))
115	114	impr 454	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)) → ∀𝑛 ∈ 𝑍 -𝐵 ≤ -𝑦)
116		brralrspcev 5208	. . . . . 6 ⊢ ((-𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 -𝐵 ≤ -𝑦) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -𝐵 ≤ 𝑧)
117	109, 115, 116	syl2anc 583	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -𝐵 ≤ 𝑧)
118	19, 117	rexlimddv 3160	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -𝐵 ≤ 𝑧)
119	9, 103, 6, 106, 107, 118	mbfsup 25414	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) ∈ MblFn)
120	102, 119	mbfneg 25400	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) ∈ MblFn)
121	99, 120	eqeltrd 2832	1 ⊢ (𝜑 → 𝐺 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 {crab 3431 Vcvv 3473 ⊆ wss 3948 ∅c0 4322 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ran crn 5677 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 supcsup 9439 infcinf 9440 ℂcc 11112 ℝcr 11113 < clt 11253 ≤ cle 11254 -cneg 11450 ℤcz 12563 ℤ_≥cuz 12827 MblFncmbf 25364

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cc 10434 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-oadd 8474 df-omul 8475 df-er 8707 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-inf 9442 df-oi 9509 df-dju 9900 df-card 9938 df-acn 9941 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-q 12938 df-rp 12980 df-xadd 13098 df-ioo 13333 df-ioc 13334 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-rlim 15438 df-sum 15638 df-xmet 21138 df-met 21139 df-ovol 25214 df-vol 25215 df-mbf 25369

This theorem is referenced by: mbflimsup 25416

W3C validator