Metamath Proof Explorer


Theorem smfliminfmpt

Description: The inferior limit of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (e) of Fremlin1 p. 39 . A can contain m as a free variable, in other words it can be thought of as an indexed collection A ( m ) . B can be thought of as a collection with two indices B ( m , x ) . (Contributed by Glauco Siliprandi, 2-Jan-2022)

Ref Expression
Hypotheses smfliminfmpt.p m φ
smfliminfmpt.x x φ
smfliminfmpt.n n φ
smfliminfmpt.m φ M
smfliminfmpt.z Z = M
smfliminfmpt.s φ S SAlg
smfliminfmpt.b φ m Z x A B V
smfliminfmpt.f φ m Z x A B SMblFn S
smfliminfmpt.d D = x n Z m n A | lim inf m Z B
smfliminfmpt.g G = x D lim inf m Z B
Assertion smfliminfmpt φ G SMblFn S

Proof

Step Hyp Ref Expression
1 smfliminfmpt.p m φ
2 smfliminfmpt.x x φ
3 smfliminfmpt.n n φ
4 smfliminfmpt.m φ M
5 smfliminfmpt.z Z = M
6 smfliminfmpt.s φ S SAlg
7 smfliminfmpt.b φ m Z x A B V
8 smfliminfmpt.f φ m Z x A B SMblFn S
9 smfliminfmpt.d D = x n Z m n A | lim inf m Z B
10 smfliminfmpt.g G = x D lim inf m Z B
11 10 a1i φ G = x D lim inf m Z B
12 9 a1i φ D = x n Z m n A | lim inf m Z B
13 simpr φ x n Z m n A x n Z m n A
14 nfv m n Z
15 1 14 nfan m φ n Z
16 simpll φ n Z m n φ
17 5 uztrn2 n Z m n m Z
18 17 adantll φ n Z m n m Z
19 simpr φ m Z m Z
20 8 elexd φ m Z x A B V
21 eqid m Z x A B = m Z x A B
22 21 fvmpt2 m Z x A B V m Z x A B m = x A B
23 19 20 22 syl2anc φ m Z m Z x A B m = x A B
24 23 dmeqd φ m Z dom m Z x A B m = dom x A B
25 nfv x m Z
26 2 25 nfan x φ m Z
27 eqid x A B = x A B
28 7 3expa φ m Z x A B V
29 26 27 28 dmmptdf φ m Z dom x A B = A
30 24 29 eqtr2d φ m Z A = dom m Z x A B m
31 16 18 30 syl2anc φ n Z m n A = dom m Z x A B m
32 15 31 iineq2d φ n Z m n A = m n dom m Z x A B m
33 3 32 iuneq2df φ n Z m n A = n Z m n dom m Z x A B m
34 33 adantr φ x n Z m n A n Z m n A = n Z m n dom m Z x A B m
35 13 34 eleqtrd φ x n Z m n A x n Z m n dom m Z x A B m
36 35 adantrr φ x n Z m n A lim inf m Z B x n Z m n dom m Z x A B m
37 eliun x n Z m n A n Z x m n A
38 37 biimpi x n Z m n A n Z x m n A
39 38 adantl φ x n Z m n A n Z x m n A
40 nfv n lim inf m Z m Z x A B m x = lim inf m Z B
41 nfcv _ m x
42 nfii1 _ m m n A
43 41 42 nfel m x m n A
44 1 14 43 nf3an m φ n Z x m n A
45 23 fveq1d φ m Z m Z x A B m x = x A B x
46 16 18 45 syl2anc φ n Z m n m Z x A B m x = x A B x
47 46 3adantl3 φ n Z x m n A m n m Z x A B m x = x A B x
48 eliinid x m n A m n x A
49 48 3ad2antl3 φ n Z x m n A m n x A
50 simpl1 φ n Z x m n A m n φ
51 simp2 φ n Z x m n A n Z
52 51 17 sylan φ n Z x m n A m n m Z
53 50 52 49 7 syl3anc φ n Z x m n A m n B V
54 27 fvmpt2 x A B V x A B x = B
55 49 53 54 syl2anc φ n Z x m n A m n x A B x = B
56 47 55 eqtrd φ n Z x m n A m n m Z x A B m x = B
57 44 56 mpteq2da φ n Z x m n A m n m Z x A B m x = m n B
58 57 fveq2d φ n Z x m n A lim inf m n m Z x A B m x = lim inf m n B
59 nfcv _ m Z
60 nfcv _ m n
61 eqid n = n
62 5 eluzelz2 n Z n
63 62 uzidd n Z n n
64 63 3ad2ant2 φ n Z x m n A n n
65 fvexd φ n Z x m n A m n m Z x A B m x V
66 44 59 60 5 61 51 64 65 liminfequzmpt2 φ n Z x m n A lim inf m Z m Z x A B m x = lim inf m n m Z x A B m x
67 44 59 60 5 61 51 64 53 liminfequzmpt2 φ n Z x m n A lim inf m Z B = lim inf m n B
68 58 66 67 3eqtr4d φ n Z x m n A lim inf m Z m Z x A B m x = lim inf m Z B
69 68 3exp φ n Z x m n A lim inf m Z m Z x A B m x = lim inf m Z B
70 3 40 69 rexlimd φ n Z x m n A lim inf m Z m Z x A B m x = lim inf m Z B
71 70 adantr φ x n Z m n A n Z x m n A lim inf m Z m Z x A B m x = lim inf m Z B
72 39 71 mpd φ x n Z m n A lim inf m Z m Z x A B m x = lim inf m Z B
73 72 adantrr φ x n Z m n A lim inf m Z B lim inf m Z m Z x A B m x = lim inf m Z B
74 simprr φ x n Z m n A lim inf m Z B lim inf m Z B
75 73 74 eqeltrd φ x n Z m n A lim inf m Z B lim inf m Z m Z x A B m x
76 36 75 jca φ x n Z m n A lim inf m Z B x n Z m n dom m Z x A B m lim inf m Z m Z x A B m x
77 simpl φ x n Z m n dom m Z x A B m lim inf m Z m Z x A B m x φ
78 simpr φ x n Z m n dom m Z x A B m x n Z m n dom m Z x A B m
79 33 eqcomd φ n Z m n dom m Z x A B m = n Z m n A
80 79 adantr φ x n Z m n dom m Z x A B m n Z m n dom m Z x A B m = n Z m n A
81 78 80 eleqtrd φ x n Z m n dom m Z x A B m x n Z m n A
82 81 adantrr φ x n Z m n dom m Z x A B m lim inf m Z m Z x A B m x x n Z m n A
83 simprr φ x n Z m n dom m Z x A B m lim inf m Z m Z x A B m x lim inf m Z m Z x A B m x
84 simp2 φ x n Z m n A lim inf m Z m Z x A B m x x n Z m n A
85 72 eqcomd φ x n Z m n A lim inf m Z B = lim inf m Z m Z x A B m x
86 85 3adant3 φ x n Z m n A lim inf m Z m Z x A B m x lim inf m Z B = lim inf m Z m Z x A B m x
87 simp3 φ x n Z m n A lim inf m Z m Z x A B m x lim inf m Z m Z x A B m x
88 86 87 eqeltrd φ x n Z m n A lim inf m Z m Z x A B m x lim inf m Z B
89 84 88 jca φ x n Z m n A lim inf m Z m Z x A B m x x n Z m n A lim inf m Z B
90 77 82 83 89 syl3anc φ x n Z m n dom m Z x A B m lim inf m Z m Z x A B m x x n Z m n A lim inf m Z B
91 76 90 impbida φ x n Z m n A lim inf m Z B x n Z m n dom m Z x A B m lim inf m Z m Z x A B m x
92 2 91 rabbida3 φ x n Z m n A | lim inf m Z B = x n Z m n dom m Z x A B m | lim inf m Z m Z x A B m x
93 12 92 eqtrd φ D = x n Z m n dom m Z x A B m | lim inf m Z m Z x A B m x
94 9 eleq2i x D x x n Z m n A | lim inf m Z B
95 94 biimpi x D x x n Z m n A | lim inf m Z B
96 rabidim1 x x n Z m n A | lim inf m Z B x n Z m n A
97 95 96 syl x D x n Z m n A
98 97 85 sylan2 φ x D lim inf m Z B = lim inf m Z m Z x A B m x
99 2 93 98 mpteq12da φ x D lim inf m Z B = x x n Z m n dom m Z x A B m | lim inf m Z m Z x A B m x lim inf m Z m Z x A B m x
100 11 99 eqtrd φ G = x x n Z m n dom m Z x A B m | lim inf m Z m Z x A B m x lim inf m Z m Z x A B m x
101 nfmpt1 _ m m Z x A B
102 nfcv _ x Z
103 nfmpt1 _ x x A B
104 102 103 nfmpt _ x m Z x A B
105 1 8 fmptd2f φ m Z x A B : Z SMblFn S
106 eqid x n Z m n dom m Z x A B m | lim inf m Z m Z x A B m x = x n Z m n dom m Z x A B m | lim inf m Z m Z x A B m x
107 eqid x x n Z m n dom m Z x A B m | lim inf m Z m Z x A B m x lim inf m Z m Z x A B m x = x x n Z m n dom m Z x A B m | lim inf m Z m Z x A B m x lim inf m Z m Z x A B m x
108 101 104 4 5 6 105 106 107 smfliminf φ x x n Z m n dom m Z x A B m | lim inf m Z m Z x A B m x lim inf m Z m Z x A B m x SMblFn S
109 100 108 eqeltrd φ G SMblFn S