Metamath Proof Explorer


Theorem issmfgt

Description: The predicate " F is a real-valued measurable function w.r.t. to the sigma-algebra S ". A function is measurable iff the preimages of all left-open intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of F is required to be b subset of the underlying set of S . Definition 121C of Fremlin1 p. 36, and Proposition 121B (iii) of Fremlin1 p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021)

Ref Expression
Hypotheses issmfgt.s ( 𝜑𝑆 ∈ SAlg )
issmfgt.d 𝐷 = dom 𝐹
Assertion issmfgt ( 𝜑 → ( 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ↔ ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑎 ∈ ℝ { 𝑥𝐷𝑎 < ( 𝐹𝑥 ) } ∈ ( 𝑆t 𝐷 ) ) ) )

Proof

Step Hyp Ref Expression
1 issmfgt.s ( 𝜑𝑆 ∈ SAlg )
2 issmfgt.d 𝐷 = dom 𝐹
3 1 adantr ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → 𝑆 ∈ SAlg )
4 simpr ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → 𝐹 ∈ ( SMblFn ‘ 𝑆 ) )
5 3 4 2 smfdmss ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → 𝐷 𝑆 )
6 3 4 2 smff ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → 𝐹 : 𝐷 ⟶ ℝ )
7 nfv 𝑏 𝜑
8 nfv 𝑏 𝐹 ∈ ( SMblFn ‘ 𝑆 )
9 7 8 nfan 𝑏 ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) )
10 3 5 restuni4 ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ( 𝑆t 𝐷 ) = 𝐷 )
11 10 eqcomd ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → 𝐷 = ( 𝑆t 𝐷 ) )
12 11 rabeqdv ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } = { 𝑦 ( 𝑆t 𝐷 ) ∣ 𝑏 < ( 𝐹𝑦 ) } )
13 12 adantr ( ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ ) → { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } = { 𝑦 ( 𝑆t 𝐷 ) ∣ 𝑏 < ( 𝐹𝑦 ) } )
14 nfv 𝑦 𝜑
15 nfv 𝑦 𝐹 ∈ ( SMblFn ‘ 𝑆 )
16 14 15 nfan 𝑦 ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) )
17 nfv 𝑦 𝑏 ∈ ℝ
18 16 17 nfan 𝑦 ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ )
19 nfv 𝑐 ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ )
20 1 uniexd ( 𝜑 𝑆 ∈ V )
21 20 adantr ( ( 𝜑𝐷 𝑆 ) → 𝑆 ∈ V )
22 simpr ( ( 𝜑𝐷 𝑆 ) → 𝐷 𝑆 )
23 21 22 ssexd ( ( 𝜑𝐷 𝑆 ) → 𝐷 ∈ V )
24 5 23 syldan ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → 𝐷 ∈ V )
25 eqid ( 𝑆t 𝐷 ) = ( 𝑆t 𝐷 )
26 3 24 25 subsalsal ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ( 𝑆t 𝐷 ) ∈ SAlg )
27 26 adantr ( ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ ) → ( 𝑆t 𝐷 ) ∈ SAlg )
28 eqid ( 𝑆t 𝐷 ) = ( 𝑆t 𝐷 )
29 6 adantr ( ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑦 ( 𝑆t 𝐷 ) ) → 𝐹 : 𝐷 ⟶ ℝ )
30 simpr ( ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑦 ( 𝑆t 𝐷 ) ) → 𝑦 ( 𝑆t 𝐷 ) )
31 10 adantr ( ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑦 ( 𝑆t 𝐷 ) ) → ( 𝑆t 𝐷 ) = 𝐷 )
32 30 31 eleqtrd ( ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑦 ( 𝑆t 𝐷 ) ) → 𝑦𝐷 )
33 29 32 ffvelrnd ( ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑦 ( 𝑆t 𝐷 ) ) → ( 𝐹𝑦 ) ∈ ℝ )
34 33 rexrd ( ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑦 ( 𝑆t 𝐷 ) ) → ( 𝐹𝑦 ) ∈ ℝ* )
35 34 adantlr ( ( ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ ) ∧ 𝑦 ( 𝑆t 𝐷 ) ) → ( 𝐹𝑦 ) ∈ ℝ* )
36 3 2 issmfle ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ( 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ↔ ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑐 ∈ ℝ { 𝑦𝐷 ∣ ( 𝐹𝑦 ) ≤ 𝑐 } ∈ ( 𝑆t 𝐷 ) ) ) )
37 4 36 mpbid ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑐 ∈ ℝ { 𝑦𝐷 ∣ ( 𝐹𝑦 ) ≤ 𝑐 } ∈ ( 𝑆t 𝐷 ) ) )
38 37 simp3d ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ∀ 𝑐 ∈ ℝ { 𝑦𝐷 ∣ ( 𝐹𝑦 ) ≤ 𝑐 } ∈ ( 𝑆t 𝐷 ) )
39 10 rabeqdv ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → { 𝑦 ( 𝑆t 𝐷 ) ∣ ( 𝐹𝑦 ) ≤ 𝑐 } = { 𝑦𝐷 ∣ ( 𝐹𝑦 ) ≤ 𝑐 } )
40 39 eleq1d ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ( { 𝑦 ( 𝑆t 𝐷 ) ∣ ( 𝐹𝑦 ) ≤ 𝑐 } ∈ ( 𝑆t 𝐷 ) ↔ { 𝑦𝐷 ∣ ( 𝐹𝑦 ) ≤ 𝑐 } ∈ ( 𝑆t 𝐷 ) ) )
41 40 ralbidv ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ( ∀ 𝑐 ∈ ℝ { 𝑦 ( 𝑆t 𝐷 ) ∣ ( 𝐹𝑦 ) ≤ 𝑐 } ∈ ( 𝑆t 𝐷 ) ↔ ∀ 𝑐 ∈ ℝ { 𝑦𝐷 ∣ ( 𝐹𝑦 ) ≤ 𝑐 } ∈ ( 𝑆t 𝐷 ) ) )
42 38 41 mpbird ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ∀ 𝑐 ∈ ℝ { 𝑦 ( 𝑆t 𝐷 ) ∣ ( 𝐹𝑦 ) ≤ 𝑐 } ∈ ( 𝑆t 𝐷 ) )
43 42 adantr ( ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑐 ∈ ℝ ) → ∀ 𝑐 ∈ ℝ { 𝑦 ( 𝑆t 𝐷 ) ∣ ( 𝐹𝑦 ) ≤ 𝑐 } ∈ ( 𝑆t 𝐷 ) )
44 simpr ( ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑐 ∈ ℝ ) → 𝑐 ∈ ℝ )
45 rspa ( ( ∀ 𝑐 ∈ ℝ { 𝑦 ( 𝑆t 𝐷 ) ∣ ( 𝐹𝑦 ) ≤ 𝑐 } ∈ ( 𝑆t 𝐷 ) ∧ 𝑐 ∈ ℝ ) → { 𝑦 ( 𝑆t 𝐷 ) ∣ ( 𝐹𝑦 ) ≤ 𝑐 } ∈ ( 𝑆t 𝐷 ) )
46 43 44 45 syl2anc ( ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑐 ∈ ℝ ) → { 𝑦 ( 𝑆t 𝐷 ) ∣ ( 𝐹𝑦 ) ≤ 𝑐 } ∈ ( 𝑆t 𝐷 ) )
47 46 adantlr ( ( ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ ) ∧ 𝑐 ∈ ℝ ) → { 𝑦 ( 𝑆t 𝐷 ) ∣ ( 𝐹𝑦 ) ≤ 𝑐 } ∈ ( 𝑆t 𝐷 ) )
48 simpr ( ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ ) → 𝑏 ∈ ℝ )
49 18 19 27 28 35 47 48 salpreimalegt ( ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ ) → { 𝑦 ( 𝑆t 𝐷 ) ∣ 𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) )
50 13 49 eqeltrd ( ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) ∧ 𝑏 ∈ ℝ ) → { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) )
51 50 ex ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ( 𝑏 ∈ ℝ → { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) ) )
52 9 51 ralrimi ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) )
53 5 6 52 3jca ( ( 𝜑𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) → ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) ) )
54 53 ex ( 𝜑 → ( 𝐹 ∈ ( SMblFn ‘ 𝑆 ) → ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) ) ) )
55 nfv 𝑦 𝐷 𝑆
56 nfv 𝑦 𝐹 : 𝐷 ⟶ ℝ
57 nfcv 𝑦
58 nfrab1 𝑦 { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) }
59 nfcv 𝑦 ( 𝑆t 𝐷 )
60 58 59 nfel 𝑦 { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 )
61 57 60 nfralw 𝑦𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 )
62 55 56 61 nf3an 𝑦 ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) )
63 14 62 nfan 𝑦 ( 𝜑 ∧ ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) ) )
64 nfv 𝑏 𝐷 𝑆
65 nfv 𝑏 𝐹 : 𝐷 ⟶ ℝ
66 nfra1 𝑏𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 )
67 64 65 66 nf3an 𝑏 ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) )
68 7 67 nfan 𝑏 ( 𝜑 ∧ ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) ) )
69 1 adantr ( ( 𝜑 ∧ ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) ) ) → 𝑆 ∈ SAlg )
70 simpr1 ( ( 𝜑 ∧ ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) ) ) → 𝐷 𝑆 )
71 simpr2 ( ( 𝜑 ∧ ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) ) ) → 𝐹 : 𝐷 ⟶ ℝ )
72 simpr3 ( ( 𝜑 ∧ ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) ) ) → ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) )
73 63 68 69 2 70 71 72 issmfgtlem ( ( 𝜑 ∧ ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) ) ) → 𝐹 ∈ ( SMblFn ‘ 𝑆 ) )
74 73 ex ( 𝜑 → ( ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) ) → 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ) )
75 54 74 impbid ( 𝜑 → ( 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ↔ ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) ) ) )
76 breq1 ( 𝑏 = 𝑎 → ( 𝑏 < ( 𝐹𝑦 ) ↔ 𝑎 < ( 𝐹𝑦 ) ) )
77 76 rabbidv ( 𝑏 = 𝑎 → { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } = { 𝑦𝐷𝑎 < ( 𝐹𝑦 ) } )
78 fveq2 ( 𝑦 = 𝑥 → ( 𝐹𝑦 ) = ( 𝐹𝑥 ) )
79 78 breq2d ( 𝑦 = 𝑥 → ( 𝑎 < ( 𝐹𝑦 ) ↔ 𝑎 < ( 𝐹𝑥 ) ) )
80 79 cbvrabv { 𝑦𝐷𝑎 < ( 𝐹𝑦 ) } = { 𝑥𝐷𝑎 < ( 𝐹𝑥 ) }
81 80 a1i ( 𝑏 = 𝑎 → { 𝑦𝐷𝑎 < ( 𝐹𝑦 ) } = { 𝑥𝐷𝑎 < ( 𝐹𝑥 ) } )
82 77 81 eqtrd ( 𝑏 = 𝑎 → { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } = { 𝑥𝐷𝑎 < ( 𝐹𝑥 ) } )
83 82 eleq1d ( 𝑏 = 𝑎 → ( { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) ↔ { 𝑥𝐷𝑎 < ( 𝐹𝑥 ) } ∈ ( 𝑆t 𝐷 ) ) )
84 83 cbvralvw ( ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) ↔ ∀ 𝑎 ∈ ℝ { 𝑥𝐷𝑎 < ( 𝐹𝑥 ) } ∈ ( 𝑆t 𝐷 ) )
85 84 3anbi3i ( ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) ) ↔ ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑎 ∈ ℝ { 𝑥𝐷𝑎 < ( 𝐹𝑥 ) } ∈ ( 𝑆t 𝐷 ) ) )
86 85 a1i ( 𝜑 → ( ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑏 ∈ ℝ { 𝑦𝐷𝑏 < ( 𝐹𝑦 ) } ∈ ( 𝑆t 𝐷 ) ) ↔ ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑎 ∈ ℝ { 𝑥𝐷𝑎 < ( 𝐹𝑥 ) } ∈ ( 𝑆t 𝐷 ) ) ) )
87 75 86 bitrd ( 𝜑 → ( 𝐹 ∈ ( SMblFn ‘ 𝑆 ) ↔ ( 𝐷 𝑆𝐹 : 𝐷 ⟶ ℝ ∧ ∀ 𝑎 ∈ ℝ { 𝑥𝐷𝑎 < ( 𝐹𝑥 ) } ∈ ( 𝑆t 𝐷 ) ) ) )