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Theorem mbfinf 25700

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 663	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ)
4	3	fmpttd 7085	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℝ)
5	4	frnd 6689	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) ⊆ ℝ)
6		mbfinf.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		uzid 12844	. . . . . . . . . . . 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 2867	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
11	10	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ 𝑍)
12		eqid 2756	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 ↦ 𝐵) = (𝑛 ∈ 𝑍 ↦ 𝐵)
13	12, 3	dmmptd 6655	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) = 𝑍)
14	11, 13	eleqtrrd 2859	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ dom (𝑛 ∈ 𝑍 ↦ 𝐵))
15	14	ne0d 4289	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → dom (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
16		dm0rn0 5893	. . . . . . . 8 ⊢ (dom (𝑛 ∈ 𝑍 ↦ 𝐵) = ∅ ↔ ran (𝑛 ∈ 𝑍 ↦ 𝐵) = ∅)
17	16	necon3bii 3003	. . . . . . 7 ⊢ (dom (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅ ↔ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
18	15, 17	sylib 220	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ 𝐵) ≠ ∅)
19		mbfinf.6	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)
20	4	ffnd 6681	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍)
21		breq2 5098	. . . . . . . . . . 11 ⊢ (𝑧 = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)))
22	21	ralrn 7058	. . . . . . . . . 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 2918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛𝑦
25		nfcv 2918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛 ≤
26		nffvmpt1 6867	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)
27	24, 25, 26	nfbr 5141	. . . . . . . . . . 11 ⊢ Ⅎ𝑛 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚)
28		nfv 1928	. . . . . . . . . . 11 ⊢ Ⅎ𝑚 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)
29		fveq2 6856	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))
30	29	breq2d 5106	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)))
31	27, 28, 30	cbvralw 3298	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))
32		simpr 487	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
33	12	fvmpt2 6976	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝐵 ∈ ℝ) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵)
34	32, 3, 33	syl2anc 592	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵)
35	34	breq2d 5106	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ 𝑦 ≤ 𝐵))
36	35	ralbidva 3177	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵))
37	31, 36	bitrid 285	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑚 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵))
38	23, 37	bitrd 281	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧 ↔ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵))
39	38	rexbidv 3180	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧 ↔ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵))
40	19, 39	mpbird 259	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑧 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)𝑦 ≤ 𝑧)
41		infrenegsup 12165	. . . . . 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 1386	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) = -sup({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}, ℝ, < ))
43		rabid 3429	. . . . . . . . . 10 ⊢ (𝑟 ∈ {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} ↔ (𝑟 ∈ ℝ ∧ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)))
44	3	recnd 11200	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℂ)
45	44	adantlr 723	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℂ)
46		simplr 776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝑟 ∈ ℝ)
47	46	recnd 11200	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝑟 ∈ ℂ)
48		negcon2 11474	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℂ ∧ 𝑟 ∈ ℂ) → (𝐵 = -𝑟 ↔ 𝑟 = -𝐵))
49	45, 47, 48	syl2anc 592	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝐵 = -𝑟 ↔ 𝑟 = -𝐵))
50		eqcom 2763	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 = -𝐵 ↔ -𝐵 = 𝑟)
51	49, 50	bitrdi 289	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝐵 = -𝑟 ↔ -𝐵 = 𝑟))
52	34	adantlr 723	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵)
53	52	eqeq1d 2758	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ 𝐵 = -𝑟))
54		negex 11418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -𝐵 ∈ V
55		eqid 2756	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ 𝑍 ↦ -𝐵) = (𝑛 ∈ 𝑍 ↦ -𝐵)
56	55	fvmpt2 6976	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ 𝑍 ∧ -𝐵 ∈ V) → ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = -𝐵)
57	54, 56	mpan2 699	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = -𝐵)
58	57	adantl 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = -𝐵)
59	58	eqeq1d 2758	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟 ↔ -𝐵 = 𝑟))
60	51, 53, 59	3bitr4d 313	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟))
61	60	ralrimiva 3148	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → ∀𝑛 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟))
62	26	nfeq1 2933	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟
63		nffvmpt1 6867	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚)
64	63	nfeq1 2933	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟
65	62, 64	nfbi 1917	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑛(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟)
66		nfv 1928	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚(((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟)
67		fveqeq2 6865	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = -𝑟))
68		fveqeq2 6865	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → (((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑛) = 𝑟))
69	67, 68	bibi12d 347	. . . . . . . . . . . . . . . . 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 3298	. . . . . . . . . . . . . . . 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 236	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → ∀𝑚 ∈ 𝑍 (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟))
72	71	r19.21bi 3248	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) ∧ 𝑚 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟))
73	72	rexbidva 3178	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (∃𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑚) = -𝑟 ↔ ∃𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ -𝐵)‘𝑚) = 𝑟))
74	20	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (𝑛 ∈ 𝑍 ↦ 𝐵) Fn 𝑍)
75		fvelrnb 6916	. . . . . . . . . . . . . 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 11604	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → -𝐵 ∈ ℝ)
78	77	fmpttd 7085	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ -𝐵):𝑍⟶ℝ)
79	78	adantr 483	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (𝑛 ∈ 𝑍 ↦ -𝐵):𝑍⟶ℝ)
80	79	ffnd 6681	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (𝑛 ∈ 𝑍 ↦ -𝐵) Fn 𝑍)
81		fvelrnb 6916	. . . . . . . . . . . . . 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 313	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑟 ∈ ℝ) → (-𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵) ↔ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵)))
84	83	pm5.32da 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑟 ∈ ℝ ∧ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)) ↔ (𝑟 ∈ ℝ ∧ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵))))
85	78	frnd 6689	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ran (𝑛 ∈ 𝑍 ↦ -𝐵) ⊆ ℝ)
86	85	sseld 3930	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵) → 𝑟 ∈ ℝ))
87	86	pm4.71rd 569	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵) ↔ (𝑟 ∈ ℝ ∧ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵))))
88	84, 87	bitr4d 284	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑟 ∈ ℝ ∧ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)) ↔ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵)))
89	43, 88	bitrid 285	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑟 ∈ {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} ↔ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵)))
90	89	alrimiv 1941	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑟(𝑟 ∈ {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} ↔ 𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ -𝐵)))
91		nfrab1 3428	. . . . . . . . 9 ⊢ Ⅎ𝑟{𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}
92		nfcv 2918	. . . . . . . . 9 ⊢ Ⅎ𝑟ran (𝑛 ∈ 𝑍 ↦ -𝐵)
93	91, 92	cleqf 2946	. . . . . . . 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 236	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)} = ran (𝑛 ∈ 𝑍 ↦ -𝐵))
95	94	supeq1d 9382	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → sup({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}, ℝ, < ) = sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ))
96	95	negeqd 11414	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -sup({𝑟 ∈ ℝ ∣ -𝑟 ∈ ran (𝑛 ∈ 𝑍 ↦ 𝐵)}, ℝ, < ) = -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ))
97	42, 96	eqtrd 2791	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < ) = -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ))
98	97	mpteq2dva 5187	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ 𝐵), ℝ, < )) = (𝑥 ∈ 𝐴 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )))
99	1, 98	eqtrid 2803	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )))
100		ltso 11253	. . . . 5 ⊢ < Or ℝ
101	100	supex 9400	. . . 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 2756	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) = (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < ))
104	2	anassrs 470	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
105		mbfinf.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
106	104, 105	mbfneg 25685	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ -𝐵) ∈ MblFn)
107	2	renegcld 11604	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → -𝐵 ∈ ℝ)
108		renegcl 11484	. . . . . . 7 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
109	108	ad2antrl 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)) → -𝑦 ∈ ℝ)
110		simplr 776	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝑦 ∈ ℝ)
111	3	adantlr 723	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℝ)
112	110, 111	lenegd 11756	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) ∧ 𝑛 ∈ 𝑍) → (𝑦 ≤ 𝐵 ↔ -𝐵 ≤ -𝑦))
113	112	ralbidva 3177	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) → (∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 ↔ ∀𝑛 ∈ 𝑍 -𝐵 ≤ -𝑦))
114	113	biimpd 231	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) → (∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵 → ∀𝑛 ∈ 𝑍 -𝐵 ≤ -𝑦))
115	114	impr 457	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)) → ∀𝑛 ∈ 𝑍 -𝐵 ≤ -𝑦)
116		brralrspcev 5154	. . . . . 6 ⊢ ((-𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 -𝐵 ≤ -𝑦) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -𝐵 ≤ 𝑧)
117	109, 115, 116	syl2anc 592	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝑦 ∈ ℝ ∧ ∀𝑛 ∈ 𝑍 𝑦 ≤ 𝐵)) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -𝐵 ≤ 𝑧)
118	19, 117	rexlimddv 3163	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑧 ∈ ℝ ∀𝑛 ∈ 𝑍 -𝐵 ≤ 𝑧)
119	9, 103, 6, 106, 107, 118	mbfsup 25699	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) ∈ MblFn)
120	102, 119	mbfneg 25685	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -sup(ran (𝑛 ∈ 𝑍 ↦ -𝐵), ℝ, < )) ∈ MblFn)
121	99, 120	eqeltrd 2856	1 ⊢ (𝜑 → 𝐺 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1552 = wceq 1554 ∈ wcel 2136 ≠ wne 2951 ∀wral 3070 ∃wrex 3080 {crab 3408 Vcvv 3448 ⊆ wss 3899 ∅c0 4280 class class class wbr 5094 ↦ cmpt 5175 dom cdm 5640 ran crn 5641 Fn wfn 6505 ⟶wf 6506 ‘cfv 6510 supcsup 9376 infcinf 9377 ℂcc 11061 ℝcr 11062 < clt 11206 ≤ cle 11207 -cneg 11405 ℤcz 12558 ℤ_≥cuz 12829 MblFncmbf 25649

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-rep 5221 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707 ax-inf2 9586 ax-cc 10382 ax-cnex 11119 ax-resscn 11120 ax-1cn 11121 ax-icn 11122 ax-addcl 11123 ax-addrcl 11124 ax-mulcl 11125 ax-mulrcl 11126 ax-mulcom 11127 ax-addass 11128 ax-mulass 11129 ax-distr 11130 ax-i2m1 11131 ax-1ne0 11132 ax-1rid 11133 ax-rnegex 11134 ax-rrecex 11135 ax-cnre 11136 ax-pre-lttri 11137 ax-pre-lttrn 11138 ax-pre-ltadd 11139 ax-pre-mulgt0 11140 ax-pre-sup 11141

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-nel 3056 df-ral 3071 df-rex 3081 df-rmo 3361 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-int 4900 df-iun 4945 df-disj 5062 df-br 5095 df-opab 5157 df-mpt 5176 df-tr 5202 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-isom 6519 df-riota 7342 df-ov 7388 df-oprab 7389 df-mpo 7390 df-of 7649 df-om 7836 df-1st 7959 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8330 df-rdg 8369 df-1o 8425 df-2o 8426 df-oadd 8429 df-omul 8430 df-er 8666 df-map 8798 df-pm 8799 df-en 8917 df-dom 8918 df-sdom 8919 df-fin 8920 df-sup 9378 df-inf 9379 df-oi 9448 df-dju 9849 df-card 9887 df-acn 9890 df-pnf 11208 df-mnf 11209 df-xr 11210 df-ltxr 11211 df-le 11212 df-sub 11406 df-neg 11407 df-div 11835 df-nn 12201 df-2 12270 df-3 12271 df-n0 12472 df-z 12559 df-uz 12830 df-q 12940 df-rp 12984 df-xadd 13105 df-ioo 13343 df-ioc 13344 df-ico 13345 df-icc 13346 df-fz 13503 df-fzo 13650 df-fl 13792 df-seq 14005 df-exp 14065 df-hash 14334 df-cj 15102 df-re 15103 df-im 15104 df-sqrt 15238 df-abs 15239 df-clim 15491 df-rlim 15492 df-sum 15690 df-xmet 21390 df-met 21391 df-ovol 25499 df-vol 25500 df-mbf 25654

This theorem is referenced by: mbflimsup 25701

W3C validator