Description: A constant function is measurable. Cf. mbfconst . (Contributed by Thierry Arnoux, 26-Jan-2017)
Ref | Expression | ||
---|---|---|---|
Hypotheses | mbfmcst.1 | |
|
mbfmcst.2 | |
||
mbfmcst.3 | |
||
mbfmcst.4 | |
||
Assertion | mbfmcst | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbfmcst.1 | |
|
2 | mbfmcst.2 | |
|
3 | mbfmcst.3 | |
|
4 | mbfmcst.4 | |
|
5 | 4 | adantr | |
6 | 3 5 | fmpt3d | |
7 | unielsiga | |
|
8 | 2 7 | syl | |
9 | unielsiga | |
|
10 | 1 9 | syl | |
11 | 8 10 | elmapd | |
12 | 6 11 | mpbird | |
13 | fconstmpt | |
|
14 | 13 | cnveqi | |
15 | cnvxp | |
|
16 | 14 15 | eqtr3i | |
17 | 16 | imaeq1i | |
18 | df-ima | |
|
19 | df-rn | |
|
20 | 17 18 19 | 3eqtri | |
21 | df-res | |
|
22 | inxp | |
|
23 | inv1 | |
|
24 | 23 | xpeq2i | |
25 | 21 22 24 | 3eqtri | |
26 | 25 | cnveqi | |
27 | 26 | dmeqi | |
28 | cnvxp | |
|
29 | 28 | dmeqi | |
30 | 20 27 29 | 3eqtri | |
31 | xpeq2 | |
|
32 | xp0 | |
|
33 | 31 32 | eqtrdi | |
34 | 33 | dmeqd | |
35 | dm0 | |
|
36 | 34 35 | eqtrdi | |
37 | 36 | adantl | |
38 | 0elsiga | |
|
39 | 1 38 | syl | |
40 | 39 | adantr | |
41 | 37 40 | eqeltrd | |
42 | 30 41 | eqeltrid | |
43 | dmxp | |
|
44 | 43 | adantl | |
45 | 10 | adantr | |
46 | 44 45 | eqeltrd | |
47 | 30 46 | eqeltrid | |
48 | 42 47 | pm2.61dane | |
49 | 48 | ralrimivw | |
50 | 3 | cnveqd | |
51 | 50 | imaeq1d | |
52 | 51 | eleq1d | |
53 | 52 | ralbidv | |
54 | 49 53 | mpbird | |
55 | 1 2 | ismbfm | |
56 | 12 54 55 | mpbir2and | |