Metamath Proof Explorer


Theorem cncombf

Description: The composition of a continuous function with a measurable function is measurable. (More generally, G can be a Borel-measurable function, but notably the condition that G be only measurable is too weak, the usual counterexample taking G to be the Cantor function and F the indicator function of the G -image of a nonmeasurable set, which is a subset of the Cantor set and hence null and measurable.) (Contributed by Mario Carneiro, 25-Aug-2014)

Ref Expression
Assertion cncombf ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) → ( 𝐺𝐹 ) ∈ MblFn )

Proof

Step Hyp Ref Expression
1 simp3 ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) → 𝐺 ∈ ( 𝐵cn→ ℂ ) )
2 cncff ( 𝐺 ∈ ( 𝐵cn→ ℂ ) → 𝐺 : 𝐵 ⟶ ℂ )
3 1 2 syl ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) → 𝐺 : 𝐵 ⟶ ℂ )
4 simp2 ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) → 𝐹 : 𝐴𝐵 )
5 fco ( ( 𝐺 : 𝐵 ⟶ ℂ ∧ 𝐹 : 𝐴𝐵 ) → ( 𝐺𝐹 ) : 𝐴 ⟶ ℂ )
6 3 4 5 syl2anc ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) → ( 𝐺𝐹 ) : 𝐴 ⟶ ℂ )
7 4 fdmd ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) → dom 𝐹 = 𝐴 )
8 mbfdm ( 𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol )
9 8 3ad2ant1 ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) → dom 𝐹 ∈ dom vol )
10 7 9 eqeltrrd ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) → 𝐴 ∈ dom vol )
11 mblss ( 𝐴 ∈ dom vol → 𝐴 ⊆ ℝ )
12 10 11 syl ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) → 𝐴 ⊆ ℝ )
13 cnex ℂ ∈ V
14 reex ℝ ∈ V
15 elpm2r ( ( ( ℂ ∈ V ∧ ℝ ∈ V ) ∧ ( ( 𝐺𝐹 ) : 𝐴 ⟶ ℂ ∧ 𝐴 ⊆ ℝ ) ) → ( 𝐺𝐹 ) ∈ ( ℂ ↑pm ℝ ) )
16 13 14 15 mpanl12 ( ( ( 𝐺𝐹 ) : 𝐴 ⟶ ℂ ∧ 𝐴 ⊆ ℝ ) → ( 𝐺𝐹 ) ∈ ( ℂ ↑pm ℝ ) )
17 6 12 16 syl2anc ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) → ( 𝐺𝐹 ) ∈ ( ℂ ↑pm ℝ ) )
18 coeq1 ( 𝑔 = ( ℜ ∘ 𝐺 ) → ( 𝑔𝐹 ) = ( ( ℜ ∘ 𝐺 ) ∘ 𝐹 ) )
19 coass ( ( ℜ ∘ 𝐺 ) ∘ 𝐹 ) = ( ℜ ∘ ( 𝐺𝐹 ) )
20 18 19 eqtrdi ( 𝑔 = ( ℜ ∘ 𝐺 ) → ( 𝑔𝐹 ) = ( ℜ ∘ ( 𝐺𝐹 ) ) )
21 20 cnveqd ( 𝑔 = ( ℜ ∘ 𝐺 ) → ( 𝑔𝐹 ) = ( ℜ ∘ ( 𝐺𝐹 ) ) )
22 21 imaeq1d ( 𝑔 = ( ℜ ∘ 𝐺 ) → ( ( 𝑔𝐹 ) “ 𝑥 ) = ( ( ℜ ∘ ( 𝐺𝐹 ) ) “ 𝑥 ) )
23 22 eleq1d ( 𝑔 = ( ℜ ∘ 𝐺 ) → ( ( ( 𝑔𝐹 ) “ 𝑥 ) ∈ dom vol ↔ ( ( ℜ ∘ ( 𝐺𝐹 ) ) “ 𝑥 ) ∈ dom vol ) )
24 cnvco ( 𝑔𝐹 ) = ( 𝐹 𝑔 )
25 24 imaeq1i ( ( 𝑔𝐹 ) “ 𝑥 ) = ( ( 𝐹 𝑔 ) “ 𝑥 )
26 imaco ( ( 𝐹 𝑔 ) “ 𝑥 ) = ( 𝐹 “ ( 𝑔𝑥 ) )
27 25 26 eqtri ( ( 𝑔𝐹 ) “ 𝑥 ) = ( 𝐹 “ ( 𝑔𝑥 ) )
28 simplll ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵cn→ ℝ ) ) → 𝐹 ∈ MblFn )
29 simpllr ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵cn→ ℝ ) ) → 𝐹 : 𝐴𝐵 )
30 cncfrss ( 𝑔 ∈ ( 𝐵cn→ ℝ ) → 𝐵 ⊆ ℂ )
31 30 adantl ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵cn→ ℝ ) ) → 𝐵 ⊆ ℂ )
32 simpr ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵cn→ ℝ ) ) → 𝑔 ∈ ( 𝐵cn→ ℝ ) )
33 ax-resscn ℝ ⊆ ℂ
34 eqid ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld )
35 eqid ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 ) = ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 )
36 34 tgioo2 ( topGen ‘ ran (,) ) = ( ( TopOpen ‘ ℂfld ) ↾t ℝ )
37 34 35 36 cncfcn ( ( 𝐵 ⊆ ℂ ∧ ℝ ⊆ ℂ ) → ( 𝐵cn→ ℝ ) = ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 ) Cn ( topGen ‘ ran (,) ) ) )
38 31 33 37 sylancl ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵cn→ ℝ ) ) → ( 𝐵cn→ ℝ ) = ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 ) Cn ( topGen ‘ ran (,) ) ) )
39 32 38 eleqtrd ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵cn→ ℝ ) ) → 𝑔 ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 ) Cn ( topGen ‘ ran (,) ) ) )
40 retopbas ran (,) ∈ TopBases
41 bastg ( ran (,) ∈ TopBases → ran (,) ⊆ ( topGen ‘ ran (,) ) )
42 40 41 ax-mp ran (,) ⊆ ( topGen ‘ ran (,) )
43 simplr ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵cn→ ℝ ) ) → 𝑥 ∈ ran (,) )
44 42 43 sselid ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵cn→ ℝ ) ) → 𝑥 ∈ ( topGen ‘ ran (,) ) )
45 cnima ( ( 𝑔 ∈ ( ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 ) Cn ( topGen ‘ ran (,) ) ) ∧ 𝑥 ∈ ( topGen ‘ ran (,) ) ) → ( 𝑔𝑥 ) ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 ) )
46 39 44 45 syl2anc ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵cn→ ℝ ) ) → ( 𝑔𝑥 ) ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 ) )
47 34 35 mbfimaopn2 ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐵 ⊆ ℂ ) ∧ ( 𝑔𝑥 ) ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝐵 ) ) → ( 𝐹 “ ( 𝑔𝑥 ) ) ∈ dom vol )
48 28 29 31 46 47 syl31anc ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵cn→ ℝ ) ) → ( 𝐹 “ ( 𝑔𝑥 ) ) ∈ dom vol )
49 27 48 eqeltrid ( ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵 ) ∧ 𝑥 ∈ ran (,) ) ∧ 𝑔 ∈ ( 𝐵cn→ ℝ ) ) → ( ( 𝑔𝐹 ) “ 𝑥 ) ∈ dom vol )
50 49 ralrimiva ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵 ) ∧ 𝑥 ∈ ran (,) ) → ∀ 𝑔 ∈ ( 𝐵cn→ ℝ ) ( ( 𝑔𝐹 ) “ 𝑥 ) ∈ dom vol )
51 50 3adantl3 ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) ∧ 𝑥 ∈ ran (,) ) → ∀ 𝑔 ∈ ( 𝐵cn→ ℝ ) ( ( 𝑔𝐹 ) “ 𝑥 ) ∈ dom vol )
52 recncf ℜ ∈ ( ℂ –cn→ ℝ )
53 52 a1i ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) → ℜ ∈ ( ℂ –cn→ ℝ ) )
54 1 53 cncfco ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) → ( ℜ ∘ 𝐺 ) ∈ ( 𝐵cn→ ℝ ) )
55 54 adantr ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) ∧ 𝑥 ∈ ran (,) ) → ( ℜ ∘ 𝐺 ) ∈ ( 𝐵cn→ ℝ ) )
56 23 51 55 rspcdva ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) ∧ 𝑥 ∈ ran (,) ) → ( ( ℜ ∘ ( 𝐺𝐹 ) ) “ 𝑥 ) ∈ dom vol )
57 coeq1 ( 𝑔 = ( ℑ ∘ 𝐺 ) → ( 𝑔𝐹 ) = ( ( ℑ ∘ 𝐺 ) ∘ 𝐹 ) )
58 coass ( ( ℑ ∘ 𝐺 ) ∘ 𝐹 ) = ( ℑ ∘ ( 𝐺𝐹 ) )
59 57 58 eqtrdi ( 𝑔 = ( ℑ ∘ 𝐺 ) → ( 𝑔𝐹 ) = ( ℑ ∘ ( 𝐺𝐹 ) ) )
60 59 cnveqd ( 𝑔 = ( ℑ ∘ 𝐺 ) → ( 𝑔𝐹 ) = ( ℑ ∘ ( 𝐺𝐹 ) ) )
61 60 imaeq1d ( 𝑔 = ( ℑ ∘ 𝐺 ) → ( ( 𝑔𝐹 ) “ 𝑥 ) = ( ( ℑ ∘ ( 𝐺𝐹 ) ) “ 𝑥 ) )
62 61 eleq1d ( 𝑔 = ( ℑ ∘ 𝐺 ) → ( ( ( 𝑔𝐹 ) “ 𝑥 ) ∈ dom vol ↔ ( ( ℑ ∘ ( 𝐺𝐹 ) ) “ 𝑥 ) ∈ dom vol ) )
63 imcncf ℑ ∈ ( ℂ –cn→ ℝ )
64 63 a1i ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) → ℑ ∈ ( ℂ –cn→ ℝ ) )
65 1 64 cncfco ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) → ( ℑ ∘ 𝐺 ) ∈ ( 𝐵cn→ ℝ ) )
66 65 adantr ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) ∧ 𝑥 ∈ ran (,) ) → ( ℑ ∘ 𝐺 ) ∈ ( 𝐵cn→ ℝ ) )
67 62 51 66 rspcdva ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) ∧ 𝑥 ∈ ran (,) ) → ( ( ℑ ∘ ( 𝐺𝐹 ) ) “ 𝑥 ) ∈ dom vol )
68 56 67 jca ( ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) ∧ 𝑥 ∈ ran (,) ) → ( ( ( ℜ ∘ ( 𝐺𝐹 ) ) “ 𝑥 ) ∈ dom vol ∧ ( ( ℑ ∘ ( 𝐺𝐹 ) ) “ 𝑥 ) ∈ dom vol ) )
69 68 ralrimiva ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) → ∀ 𝑥 ∈ ran (,) ( ( ( ℜ ∘ ( 𝐺𝐹 ) ) “ 𝑥 ) ∈ dom vol ∧ ( ( ℑ ∘ ( 𝐺𝐹 ) ) “ 𝑥 ) ∈ dom vol ) )
70 ismbf1 ( ( 𝐺𝐹 ) ∈ MblFn ↔ ( ( 𝐺𝐹 ) ∈ ( ℂ ↑pm ℝ ) ∧ ∀ 𝑥 ∈ ran (,) ( ( ( ℜ ∘ ( 𝐺𝐹 ) ) “ 𝑥 ) ∈ dom vol ∧ ( ( ℑ ∘ ( 𝐺𝐹 ) ) “ 𝑥 ) ∈ dom vol ) ) )
71 17 69 70 sylanbrc ( ( 𝐹 ∈ MblFn ∧ 𝐹 : 𝐴𝐵𝐺 ∈ ( 𝐵cn→ ℂ ) ) → ( 𝐺𝐹 ) ∈ MblFn )