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Theorem mbfinf 25634

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 656	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ)
4	3	fmpttd 7069	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ)
5	4	frnd 6678	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) ⊆ ℝ)
6		mbfinf.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		uzid 12778	. . . . . . . . . . . 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 2848	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
11	10	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ 𝑍)
12		eqid 2737	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 ↦ 𝐵) = (𝑛 ∈ 𝑍 ↦ 𝐵)
13	12, 3	dmmptd 6645	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) = 𝑍)
14	11, 13	eleqtrrd 2840	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ dom (𝑛 ∈ 𝑍 ↦ 𝐵))
15	14	ne0d 4296	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
16		dm0rn0 5881	. . . . . . . 8 ⊢ (dom (𝑛 ∈ 𝑍 ↦ 𝐵) = ∅ ↔ ran (𝑛 ∈ 𝑍 ↦ 𝐵) = ∅)
17	16	necon3bii 2985	. . . . . . 7 ⊢ (dom (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅ ↔ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
18	15, 17	sylib 218	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
19		mbfinf.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)
20	4	ffnd 6671	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍)
21		breq2 5104	. . . . . . . . . . 11 ⊢ (𝑧 = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)))
22	21	ralrn 7042	. . . . . . . . . 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 2899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛𝑦
25		nfcv 2899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛 ≤
26		nffvmpt1 6853	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)
27	24, 25, 26	nfbr 5147	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)
28		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑚 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)
29		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))
30	29	breq2d 5112	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)))
31	27, 28, 30	cbvralw 3280	. . . . . . . . . 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 6961	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝐵 ∈ ℝ) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵)
34	32, 3, 33	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵)
35	34	breq2d 5112	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ 𝑦 ≤ 𝐵))
36	35	ralbidva 3159	. . . . . . . . . 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 3162	. . . . . . 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 12137	. . . . . 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 1374	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) = -sup({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}, ℝ, < ))
43		rabid 3422	. . . . . . . . . 10 ⊢ (𝑟 ∈ {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} ↔ (𝑟 ∈ ℝ ∧ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)))
44	3	recnd 11172	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℂ)
45	44	adantlr 716	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℂ)
46		simplr 769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝑟 ∈ ℝ)
47	46	recnd 11172	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝑟 ∈ ℂ)
48		negcon2 11446	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℂ ∧ 𝑟 ∈ ℂ) → (𝐵 = -𝑟 ↔ 𝑟 = -𝐵))
49	45, 47, 48	syl2anc 585	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝐵 = -𝑟 ↔ 𝑟 = -𝐵))
50		eqcom 2744	. . . . . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵)
53	52	eqeq1d 2739	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ 𝐵 = -𝑟))
54		negex 11390	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -𝐵 ∈ V
55		eqid 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ 𝑍 ↦ -𝐵) = (𝑛 ∈ 𝑍 ↦ -𝐵)
56	55	fvmpt2 6961	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ 𝑍 ∧ -𝐵 ∈ V) → ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = -𝐵)
57	54, 56	mpan2 692	. . . . . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . . 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 3130	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟))
62	26	nfeq1 2915	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟
63		nffvmpt1 6853	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚)
64	63	nfeq1 2915	. . . . . . . . . . . . . . . . . 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 6851	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟))
68		fveqeq2 6851	. . . . . . . . . . . . . . . . . 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 3280	. . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → ∀𝑚 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟))
72	71	r19.21bi 3230	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑚 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟))
73	72	rexbidva 3160	. . . . . . . . . . . . 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 6902	. . . . . . . . . . . . . 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 11576	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → -𝐵 ∈ ℝ)
78	77	fmpttd 7069	. . . . . . . . . . . . . . . 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 6671	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (𝑛 ∈ 𝑍 ↦ -𝐵) Fn 𝑍)
81		fvelrnb 6902	. . . . . . . . . . . . . 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 6678	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ -𝐵) ⊆ ℝ)
86	85	sseld 3934	. . . . . . . . . . . 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 3421	. . . . . . . . 9 ⊢ Ⅎ𝑟{𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}
92		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑟ran (𝑛 ∈ 𝑍 ↦ -𝐵)
93	91, 92	cleqf 2928	. . . . . . . 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 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} = ran (𝑛 ∈ 𝑍 ↦ -𝐵))
95	94	supeq1d 9361	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}, ℝ, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ))
96	95	negeqd 11386	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -sup({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}, ℝ, < ) = -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ))
97	42, 96	eqtrd 2772	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) = -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ))
98	97	mpteq2dva 5193	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )) = (𝑥 ∈ 𝐴 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )))
99	1, 98	eqtrid 2784	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )))
100		ltso 11225	. . . . 5 ⊢ < Or ℝ
101	100	supex 9379	. . . 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 2737	. . . 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 25619	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ MblFn)
107	2	renegcld 11576	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → -𝐵 ∈ ℝ)
108		renegcl 11456	. . . . . . 7 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
109	108	ad2antrl 729	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)) → -𝑦 ∈ ℝ)
110		simplr 769	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝑦 ∈ ℝ)
111	3	adantlr 716	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ)
112	110, 111	lenegd 11728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑦 ≤ 𝐵 ↔ -𝐵 ≤ -𝑦))
113	112	ralbidva 3159	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) → (∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ↔ ∀𝑛 ∈ 𝑍 -𝐵 ≤ -𝑦))
114	113	biimpd 229	. . . . . . 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 5160	. . . . . 6 ⊢ ((-𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 -𝐵 ≤ -𝑦) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -𝐵 ≤ 𝑧)
117	109, 115, 116	syl2anc 585	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -𝐵 ≤ 𝑧)
118	19, 117	rexlimddv 3145	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -𝐵 ≤ 𝑧)
119	9, 103, 6, 106, 107, 118	mbfsup 25633	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) ∈ MblFn)
120	102, 119	mbfneg 25619	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) ∈ MblFn)
121	99, 120	eqeltrd 2837	1 ⊢ (𝜑 → 𝐺 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1540 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 {crab 3401 Vcvv 3442 ⊆ wss 3903 ∅c0 4287 class class class wbr 5100 ↦ cmpt 5181 dom cdm 5632 ran crn 5633 Fn wfn 6495 ⟶wf 6496 ‘cfv 6500 supcsup 9355 infcinf 9356 ℂcc 11036 ℝcr 11037 < clt 11178 ≤ cle 11179 -cneg 11377 ℤcz 12500 ℤ_≥cuz 12763 MblFncmbf 25583

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

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-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-disj 5068 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-omul 8412 df-er 8645 df-map 8777 df-pm 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-oi 9427 df-dju 9825 df-card 9863 df-acn 9866 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-z 12501 df-uz 12764 df-q 12874 df-rp 12918 df-xadd 13039 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-rlim 15424 df-sum 15622 df-xmet 21314 df-met 21315 df-ovol 25433 df-vol 25434 df-mbf 25588

This theorem is referenced by: mbflimsup 25635

W3C validator