Metamath Proof Explorer


Theorem smfinfmpt

Description: The infimum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (c) of Fremlin1 p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypotheses smfinfmpt.n 𝑛 𝜑
smfinfmpt.x 𝑥 𝜑
smfinfmpt.y 𝑦 𝜑
smfinfmpt.m ( 𝜑𝑀 ∈ ℤ )
smfinfmpt.z 𝑍 = ( ℤ𝑀 )
smfinfmpt.s ( 𝜑𝑆 ∈ SAlg )
smfinfmpt.b ( ( 𝜑𝑛𝑍𝑥𝐴 ) → 𝐵𝑉 )
smfinfmpt.f ( ( 𝜑𝑛𝑍 ) → ( 𝑥𝐴𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) )
smfinfmpt.d 𝐷 = { 𝑥 𝑛𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 }
smfinfmpt.g 𝐺 = ( 𝑥𝐷 ↦ inf ( ran ( 𝑛𝑍𝐵 ) , ℝ , < ) )
Assertion smfinfmpt ( 𝜑𝐺 ∈ ( SMblFn ‘ 𝑆 ) )

Proof

Step Hyp Ref Expression
1 smfinfmpt.n 𝑛 𝜑
2 smfinfmpt.x 𝑥 𝜑
3 smfinfmpt.y 𝑦 𝜑
4 smfinfmpt.m ( 𝜑𝑀 ∈ ℤ )
5 smfinfmpt.z 𝑍 = ( ℤ𝑀 )
6 smfinfmpt.s ( 𝜑𝑆 ∈ SAlg )
7 smfinfmpt.b ( ( 𝜑𝑛𝑍𝑥𝐴 ) → 𝐵𝑉 )
8 smfinfmpt.f ( ( 𝜑𝑛𝑍 ) → ( 𝑥𝐴𝐵 ) ∈ ( SMblFn ‘ 𝑆 ) )
9 smfinfmpt.d 𝐷 = { 𝑥 𝑛𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 }
10 smfinfmpt.g 𝐺 = ( 𝑥𝐷 ↦ inf ( ran ( 𝑛𝑍𝐵 ) , ℝ , < ) )
11 eqidd ( 𝜑 → ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) = ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) )
12 11 8 fvmpt2d ( ( 𝜑𝑛𝑍 ) → ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) = ( 𝑥𝐴𝐵 ) )
13 12 dmeqd ( ( 𝜑𝑛𝑍 ) → dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) = dom ( 𝑥𝐴𝐵 ) )
14 nfcv 𝑥 𝑍
15 14 nfcri 𝑥 𝑛𝑍
16 2 15 nfan 𝑥 ( 𝜑𝑛𝑍 )
17 eqid ( 𝑥𝐴𝐵 ) = ( 𝑥𝐴𝐵 )
18 7 3expa ( ( ( 𝜑𝑛𝑍 ) ∧ 𝑥𝐴 ) → 𝐵𝑉 )
19 16 17 18 dmmptdf ( ( 𝜑𝑛𝑍 ) → dom ( 𝑥𝐴𝐵 ) = 𝐴 )
20 13 19 eqtr2d ( ( 𝜑𝑛𝑍 ) → 𝐴 = dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) )
21 1 20 iineq2d ( 𝜑 𝑛𝑍 𝐴 = 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) )
22 2 21 rabeqd ( 𝜑 → { 𝑥 𝑛𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 } = { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 } )
23 nfv 𝑦 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 )
24 3 23 nfan 𝑦 ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) )
25 nfii1 𝑛 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 )
26 25 nfcri 𝑛 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 )
27 1 26 nfan 𝑛 ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) )
28 simpll ( ( ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛𝑍 ) → 𝜑 )
29 simpr ( ( ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛𝑍 ) → 𝑛𝑍 )
30 eliinid ( ( 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∧ 𝑛𝑍 ) → 𝑥 ∈ dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) )
31 30 adantll ( ( ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛𝑍 ) → 𝑥 ∈ dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) )
32 13 19 eqtrd ( ( 𝜑𝑛𝑍 ) → dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) = 𝐴 )
33 32 adantlr ( ( ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛𝑍 ) → dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) = 𝐴 )
34 31 33 eleqtrd ( ( ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛𝑍 ) → 𝑥𝐴 )
35 12 fveq1d ( ( 𝜑𝑛𝑍 ) → ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑥𝐴𝐵 ) ‘ 𝑥 ) )
36 35 3adant3 ( ( 𝜑𝑛𝑍𝑥𝐴 ) → ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) = ( ( 𝑥𝐴𝐵 ) ‘ 𝑥 ) )
37 simp3 ( ( 𝜑𝑛𝑍𝑥𝐴 ) → 𝑥𝐴 )
38 fvmpt4 ( ( 𝑥𝐴𝐵𝑉 ) → ( ( 𝑥𝐴𝐵 ) ‘ 𝑥 ) = 𝐵 )
39 37 7 38 syl2anc ( ( 𝜑𝑛𝑍𝑥𝐴 ) → ( ( 𝑥𝐴𝐵 ) ‘ 𝑥 ) = 𝐵 )
40 36 39 eqtr2d ( ( 𝜑𝑛𝑍𝑥𝐴 ) → 𝐵 = ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) )
41 40 breq2d ( ( 𝜑𝑛𝑍𝑥𝐴 ) → ( 𝑦𝐵𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
42 28 29 34 41 syl3anc ( ( ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ) ∧ 𝑛𝑍 ) → ( 𝑦𝐵𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
43 27 42 ralbida ( ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ) → ( ∀ 𝑛𝑍 𝑦𝐵 ↔ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
44 24 43 rexbid ( ( 𝜑𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ) → ( ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 ↔ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
45 2 44 rabbida ( 𝜑 → { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 } = { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } )
46 22 45 eqtrd ( 𝜑 → { 𝑥 𝑛𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 } = { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } )
47 9 46 eqtrid ( 𝜑𝐷 = { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } )
48 nfcv 𝑛
49 nfra1 𝑛𝑛𝑍 𝑦𝐵
50 48 49 nfrexw 𝑛𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵
51 nfii1 𝑛 𝑛𝑍 𝐴
52 50 51 nfrabw 𝑛 { 𝑥 𝑛𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 }
53 9 52 nfcxfr 𝑛 𝐷
54 53 nfcri 𝑛 𝑥𝐷
55 1 54 nfan 𝑛 ( 𝜑𝑥𝐷 )
56 simpll ( ( ( 𝜑𝑥𝐷 ) ∧ 𝑛𝑍 ) → 𝜑 )
57 simpr ( ( ( 𝜑𝑥𝐷 ) ∧ 𝑛𝑍 ) → 𝑛𝑍 )
58 rabidim1 ( 𝑥 ∈ { 𝑥 𝑛𝑍 𝐴 ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦𝐵 } → 𝑥 𝑛𝑍 𝐴 )
59 58 9 eleq2s ( 𝑥𝐷𝑥 𝑛𝑍 𝐴 )
60 eliinid ( ( 𝑥 𝑛𝑍 𝐴𝑛𝑍 ) → 𝑥𝐴 )
61 59 60 sylan ( ( 𝑥𝐷𝑛𝑍 ) → 𝑥𝐴 )
62 61 adantll ( ( ( 𝜑𝑥𝐷 ) ∧ 𝑛𝑍 ) → 𝑥𝐴 )
63 56 57 62 40 syl3anc ( ( ( 𝜑𝑥𝐷 ) ∧ 𝑛𝑍 ) → 𝐵 = ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) )
64 55 63 mpteq2da ( ( 𝜑𝑥𝐷 ) → ( 𝑛𝑍𝐵 ) = ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
65 64 rneqd ( ( 𝜑𝑥𝐷 ) → ran ( 𝑛𝑍𝐵 ) = ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) )
66 65 infeq1d ( ( 𝜑𝑥𝐷 ) → inf ( ran ( 𝑛𝑍𝐵 ) , ℝ , < ) = inf ( ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) )
67 2 47 66 mpteq12da ( 𝜑 → ( 𝑥𝐷 ↦ inf ( ran ( 𝑛𝑍𝐵 ) , ℝ , < ) ) = ( 𝑥 ∈ { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) )
68 10 67 eqtrid ( 𝜑𝐺 = ( 𝑥 ∈ { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) )
69 nfmpt1 𝑛 ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) )
70 nfmpt1 𝑥 ( 𝑥𝐴𝐵 )
71 14 70 nfmpt 𝑥 ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) )
72 1 8 fmptd2f ( 𝜑 → ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) : 𝑍 ⟶ ( SMblFn ‘ 𝑆 ) )
73 eqid { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } = { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) }
74 eqid ( 𝑥 ∈ { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) = ( 𝑥 ∈ { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) )
75 69 71 4 5 6 72 73 74 smfinf ( 𝜑 → ( 𝑥 ∈ { 𝑥 𝑛𝑍 dom ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ∣ ∃ 𝑦 ∈ ℝ ∀ 𝑛𝑍 𝑦 ≤ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) } ↦ inf ( ran ( 𝑛𝑍 ↦ ( ( ( 𝑛𝑍 ↦ ( 𝑥𝐴𝐵 ) ) ‘ 𝑛 ) ‘ 𝑥 ) ) , ℝ , < ) ) ∈ ( SMblFn ‘ 𝑆 ) )
76 68 75 eqeltrd ( 𝜑𝐺 ∈ ( SMblFn ‘ 𝑆 ) )