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Theorem mbfinf 24829

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 6989	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ)
5	4	frnd 6608	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) ⊆ ℝ)
6		mbfinf.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		uzid 12597	. . . . . . . . . . . 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 2850	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
11	10	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ 𝑍)
12		eqid 2738	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 ↦ 𝐵) = (𝑛 ∈ 𝑍 ↦ 𝐵)
13	12, 3	dmmptd 6578	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) = 𝑍)
14	11, 13	eleqtrrd 2842	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ dom (𝑛 ∈ 𝑍 ↦ 𝐵))
15	14	ne0d 4269	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
16		dm0rn0 5834	. . . . . . . 8 ⊢ (dom (𝑛 ∈ 𝑍 ↦ 𝐵) = ∅ ↔ ran (𝑛 ∈ 𝑍 ↦ 𝐵) = ∅)
17	16	necon3bii 2996	. . . . . . 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 6601	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍)
21		breq2 5078	. . . . . . . . . . 11 ⊢ (𝑧 = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)))
22	21	ralrn 6964	. . . . . . . . . 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 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛𝑦
25		nfcv 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛 ≤
26		nffvmpt1 6785	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)
27	24, 25, 26	nfbr 5121	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)
28		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑚 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)
29		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))
30	29	breq2d 5086	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)))
31	27, 28, 30	cbvralw 3373	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))
32		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
33	12	fvmpt2 6886	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝐵 ∈ ℝ) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵)
34	32, 3, 33	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵)
35	34	breq2d 5086	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ 𝑦 ≤ 𝐵))
36	35	ralbidva 3111	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵))
37	31, 36	bitrid 282	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵))
38	23, 37	bitrd 278	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧 ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵))
39	38	rexbidv 3226	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵))
40	19, 39	mpbird 256	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧)
41		infrenegsup 11958	. . . . . 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 3310	. . . . . . . . . 10 ⊢ (𝑟 ∈ {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} ↔ (𝑟 ∈ ℝ ∧ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)))
44	3	recnd 11003	. . . . . . . . . . . . . . . . . . . . 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 11003	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝑟 ∈ ℂ)
48		negcon2 11274	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℂ ∧ 𝑟 ∈ ℂ) → (𝐵 = -𝑟 ↔ 𝑟 = -𝐵))
49	45, 47, 48	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝐵 = -𝑟 ↔ 𝑟 = -𝐵))
50		eqcom 2745	. . . . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ 𝐵 = -𝑟))
54		negex 11219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -𝐵 ∈ V
55		eqid 2738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ 𝑍 ↦ -𝐵) = (𝑛 ∈ 𝑍 ↦ -𝐵)
56	55	fvmpt2 6886	. . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = -𝐵)
59	58	eqeq1d 2740	. . . . . . . . . . . . . . . . . 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 3103	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟))
62	26	nfeq1 2922	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟
63		nffvmpt1 6785	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚)
64	63	nfeq1 2922	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟
65	62, 64	nfbi 1906	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟)
66		nfv 1917	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟)
67		fveqeq2 6783	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟))
68		fveqeq2 6783	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → (((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟))
69	67, 68	bibi12d 346	. . . . . . . . . . . . . . . . 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 3373	. . . . . . . . . . . . . . . 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 3134	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑚 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟))
73	72	rexbidva 3225	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (∃𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ∃𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟))
74	20	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍)
75		fvelrnb 6830	. . . . . . . . . . . . . 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 11402	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → -𝐵 ∈ ℝ)
78	77	fmpttd 6989	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ -𝐵):𝑍⟶ℝ)
79	78	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (𝑛 ∈ 𝑍 ↦ -𝐵):𝑍⟶ℝ)
80	79	ffnd 6601	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (𝑛 ∈ 𝑍 ↦ -𝐵) Fn 𝑍)
81		fvelrnb 6830	. . . . . . . . . . . . . 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 579	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑟 ∈ ℝ ∧ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)) ↔ (𝑟 ∈ ℝ ∧ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵))))
85	78	frnd 6608	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ -𝐵) ⊆ ℝ)
86	85	sseld 3920	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵) → 𝑟 ∈ ℝ))
87	86	pm4.71rd 563	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵) ↔ (𝑟 ∈ ℝ ∧ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵))))
88	84, 87	bitr4d 281	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑟 ∈ ℝ ∧ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)) ↔ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵)))
89	43, 88	bitrid 282	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑟 ∈ {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} ↔ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵)))
90	89	alrimiv 1930	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑟(𝑟 ∈ {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} ↔ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵)))
91		nfrab1 3317	. . . . . . . . 9 ⊢ Ⅎ𝑟{𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}
92		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑟ran (𝑛 ∈ 𝑍 ↦ -𝐵)
93	91, 92	cleqf 2938	. . . . . . . 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 9205	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}, ℝ, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ))
96	95	negeqd 11215	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -sup({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}, ℝ, < ) = -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ))
97	42, 96	eqtrd 2778	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) = -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ))
98	97	mpteq2dva 5174	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )) = (𝑥 ∈ 𝐴 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )))
99	1, 98	eqtrid 2790	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )))
100		ltso 11055	. . . . 5 ⊢ < Or ℝ
101	100	supex 9222	. . . 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 2738	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) = (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ))
104	2	anassrs 468	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
105		mbfinf.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
106	104, 105	mbfneg 24814	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ MblFn)
107	2	renegcld 11402	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → -𝐵 ∈ ℝ)
108		renegcl 11284	. . . . . . 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 11554	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑦 ≤ 𝐵 ↔ -𝐵 ≤ -𝑦))
113	112	ralbidva 3111	. . . . . . . 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 455	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)) → ∀𝑛 ∈ 𝑍 -𝐵 ≤ -𝑦)
116		brralrspcev 5134	. . . . . 6 ⊢ ((-𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 -𝐵 ≤ -𝑦) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -𝐵 ≤ 𝑧)
117	109, 115, 116	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -𝐵 ≤ 𝑧)
118	19, 117	rexlimddv 3220	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -𝐵 ≤ 𝑧)
119	9, 103, 6, 106, 107, 118	mbfsup 24828	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) ∈ MblFn)
120	102, 119	mbfneg 24814	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) ∈ MblFn)
121	99, 120	eqeltrd 2839	1 ⊢ (𝜑 → 𝐺 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1537 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 {crab 3068 Vcvv 3432 ⊆ wss 3887 ∅c0 4256 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 ran crn 5590 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 supcsup 9199 infcinf 9200 ℂcc 10869 ℝcr 10870 < clt 11009 ≤ cle 11010 -cneg 11206 ℤcz 12319 ℤ_≥cuz 12582 MblFncmbf 24778

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cc 10191 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-omul 8302 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-q 12689 df-rp 12731 df-xadd 12849 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-xmet 20590 df-met 20591 df-ovol 24628 df-vol 24629 df-mbf 24783

This theorem is referenced by: mbflimsup 24830

W3C validator