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Theorem 0rrv 29280
Description: The constant function equal to zero is a random variable. (Contributed by Thierry Arnoux, 16-Jan-2017.) (Revised by Thierry Arnoux, 30-Jan-2017.)
Hypothesis
Ref Expression
0rrv.1  |-  ( ph  ->  P  e. Prob )
Assertion
Ref Expression
0rrv  |-  ( ph  ->  ( x  e.  U. dom  P  |->  0 )  e.  (rRndVar `  P )
)
Distinct variable group:    x, P
Allowed substitution hint:    ph( x)

Proof of Theorem 0rrv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 0re 9644 . . . . 5  |-  0  e.  RR
21rgenw 2786 . . . 4  |-  A. x  e.  U. dom  P 0  e.  RR
3 eqid 2422 . . . . 5  |-  ( x  e.  U. dom  P  |->  0 )  =  ( x  e.  U. dom  P 
|->  0 )
43fmpt 6055 . . . 4  |-  ( A. x  e.  U. dom  P
0  e.  RR  <->  ( x  e.  U. dom  P  |->  0 ) : U. dom  P --> RR )
52, 4mpbi 211 . . 3  |-  ( x  e.  U. dom  P  |->  0 ) : U. dom  P --> RR
65a1i 11 . 2  |-  ( ph  ->  ( x  e.  U. dom  P  |->  0 ) : U. dom  P --> RR )
7 fconstmpt 4894 . . . . . . . . . 10  |-  ( U. dom  P  X.  { 0 } )  =  ( x  e.  U. dom  P 
|->  0 )
87cnveqi 5025 . . . . . . . . 9  |-  `' ( U. dom  P  X.  { 0 } )  =  `' ( x  e.  U. dom  P  |->  0 )
9 cnvxp 5270 . . . . . . . . 9  |-  `' ( U. dom  P  X.  { 0 } )  =  ( { 0 }  X.  U. dom  P )
108, 9eqtr3i 2453 . . . . . . . 8  |-  `' ( x  e.  U. dom  P 
|->  0 )  =  ( { 0 }  X.  U.
dom  P )
1110imaeq1i 5181 . . . . . . 7  |-  ( `' ( x  e.  U. dom  P  |->  0 ) "
y )  =  ( ( { 0 }  X.  U. dom  P
) " y )
12 df-ima 4863 . . . . . . 7  |-  ( ( { 0 }  X.  U.
dom  P ) "
y )  =  ran  ( ( { 0 }  X.  U. dom  P )  |`  y )
13 df-rn 4861 . . . . . . 7  |-  ran  (
( { 0 }  X.  U. dom  P
)  |`  y )  =  dom  `' ( ( { 0 }  X.  U.
dom  P )  |`  y )
1411, 12, 133eqtri 2455 . . . . . 6  |-  ( `' ( x  e.  U. dom  P  |->  0 ) "
y )  =  dom  `' ( ( { 0 }  X.  U. dom  P )  |`  y )
15 df-res 4862 . . . . . . . . 9  |-  ( ( { 0 }  X.  U.
dom  P )  |`  y )  =  ( ( { 0 }  X.  U. dom  P
)  i^i  ( y  X.  _V ) )
16 inxp 4983 . . . . . . . . 9  |-  ( ( { 0 }  X.  U.
dom  P )  i^i  ( y  X.  _V ) )  =  ( ( { 0 }  i^i  y )  X.  ( U. dom  P  i^i  _V ) )
17 inv1 3789 . . . . . . . . . 10  |-  ( U. dom  P  i^i  _V )  =  U. dom  P
1817xpeq2i 4871 . . . . . . . . 9  |-  ( ( { 0 }  i^i  y )  X.  ( U. dom  P  i^i  _V ) )  =  ( ( { 0 }  i^i  y )  X. 
U. dom  P )
1915, 16, 183eqtri 2455 . . . . . . . 8  |-  ( ( { 0 }  X.  U.
dom  P )  |`  y )  =  ( ( { 0 }  i^i  y )  X. 
U. dom  P )
2019cnveqi 5025 . . . . . . 7  |-  `' ( ( { 0 }  X.  U. dom  P
)  |`  y )  =  `' ( ( { 0 }  i^i  y
)  X.  U. dom  P )
2120dmeqi 5052 . . . . . 6  |-  dom  `' ( ( { 0 }  X.  U. dom  P )  |`  y )  =  dom  `' ( ( { 0 }  i^i  y )  X.  U. dom  P )
22 cnvxp 5270 . . . . . . 7  |-  `' ( ( { 0 }  i^i  y )  X. 
U. dom  P )  =  ( U. dom  P  X.  ( { 0 }  i^i  y ) )
2322dmeqi 5052 . . . . . 6  |-  dom  `' ( ( { 0 }  i^i  y )  X.  U. dom  P
)  =  dom  ( U. dom  P  X.  ( { 0 }  i^i  y ) )
2414, 21, 233eqtri 2455 . . . . 5  |-  ( `' ( x  e.  U. dom  P  |->  0 ) "
y )  =  dom  ( U. dom  P  X.  ( { 0 }  i^i  y ) )
25 xpeq2 4865 . . . . . . . . . 10  |-  ( ( { 0 }  i^i  y )  =  (/)  ->  ( U. dom  P  X.  ( { 0 }  i^i  y ) )  =  ( U. dom  P  X.  (/) ) )
26 xp0 5271 . . . . . . . . . 10  |-  ( U. dom  P  X.  (/) )  =  (/)
2725, 26syl6eq 2479 . . . . . . . . 9  |-  ( ( { 0 }  i^i  y )  =  (/)  ->  ( U. dom  P  X.  ( { 0 }  i^i  y ) )  =  (/) )
2827dmeqd 5053 . . . . . . . 8  |-  ( ( { 0 }  i^i  y )  =  (/)  ->  dom  ( U. dom  P  X.  ( { 0 }  i^i  y ) )  =  dom  (/) )
29 dm0 5064 . . . . . . . 8  |-  dom  (/)  =  (/)
3028, 29syl6eq 2479 . . . . . . 7  |-  ( ( { 0 }  i^i  y )  =  (/)  ->  dom  ( U. dom  P  X.  ( { 0 }  i^i  y ) )  =  (/) )
3130adantl 467 . . . . . 6  |-  ( (
ph  /\  ( {
0 }  i^i  y
)  =  (/) )  ->  dom  ( U. dom  P  X.  ( { 0 }  i^i  y ) )  =  (/) )
32 0rrv.1 . . . . . . . 8  |-  ( ph  ->  P  e. Prob )
33 domprobsiga 29240 . . . . . . . 8  |-  ( P  e. Prob  ->  dom  P  e.  U.
ran sigAlgebra )
34 0elsiga 28932 . . . . . . . 8  |-  ( dom 
P  e.  U. ran sigAlgebra  ->  (/)  e.  dom  P )
3532, 33, 343syl 18 . . . . . . 7  |-  ( ph  -> 
(/)  e.  dom  P )
3635adantr 466 . . . . . 6  |-  ( (
ph  /\  ( {
0 }  i^i  y
)  =  (/) )  ->  (/) 
e.  dom  P )
3731, 36eqeltrd 2510 . . . . 5  |-  ( (
ph  /\  ( {
0 }  i^i  y
)  =  (/) )  ->  dom  ( U. dom  P  X.  ( { 0 }  i^i  y ) )  e.  dom  P )
3824, 37syl5eqel 2514 . . . 4  |-  ( (
ph  /\  ( {
0 }  i^i  y
)  =  (/) )  -> 
( `' ( x  e.  U. dom  P  |->  0 ) " y
)  e.  dom  P
)
39 dmxp 5069 . . . . . . 7  |-  ( ( { 0 }  i^i  y )  =/=  (/)  ->  dom  ( U. dom  P  X.  ( { 0 }  i^i  y ) )  = 
U. dom  P )
4039adantl 467 . . . . . 6  |-  ( (
ph  /\  ( {
0 }  i^i  y
)  =/=  (/) )  ->  dom  ( U. dom  P  X.  ( { 0 }  i^i  y ) )  =  U. dom  P
)
4132unveldomd 29244 . . . . . . 7  |-  ( ph  ->  U. dom  P  e. 
dom  P )
4241adantr 466 . . . . . 6  |-  ( (
ph  /\  ( {
0 }  i^i  y
)  =/=  (/) )  ->  U. dom  P  e.  dom  P )
4340, 42eqeltrd 2510 . . . . 5  |-  ( (
ph  /\  ( {
0 }  i^i  y
)  =/=  (/) )  ->  dom  ( U. dom  P  X.  ( { 0 }  i^i  y ) )  e.  dom  P )
4424, 43syl5eqel 2514 . . . 4  |-  ( (
ph  /\  ( {
0 }  i^i  y
)  =/=  (/) )  -> 
( `' ( x  e.  U. dom  P  |->  0 ) " y
)  e.  dom  P
)
4538, 44pm2.61dane 2742 . . 3  |-  ( ph  ->  ( `' ( x  e.  U. dom  P  |->  0 ) " y
)  e.  dom  P
)
4645ralrimivw 2840 . 2  |-  ( ph  ->  A. y  e. 𝔅  ( `' ( x  e.  U. dom  P  |->  0 ) " y
)  e.  dom  P
)
4732isrrvv 29272 . 2  |-  ( ph  ->  ( ( x  e. 
U. dom  P  |->  0 )  e.  (rRndVar `  P
)  <->  ( ( x  e.  U. dom  P  |->  0 ) : U. dom  P --> RR  /\  A. y  e. 𝔅  ( `' ( x  e.  U. dom  P  |->  0 ) " y
)  e.  dom  P
) ) )
486, 46, 47mpbir2and 930 1  |-  ( ph  ->  ( x  e.  U. dom  P  |->  0 )  e.  (rRndVar `  P )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   _Vcvv 3081    i^i cin 3435   (/)c0 3761   {csn 3996   U.cuni 4216    |-> cmpt 4479    X. cxp 4848   `'ccnv 4849   dom cdm 4850   ran crn 4851    |` cres 4852   "cima 4853   -->wf 5594   ` cfv 5598   RRcr 9539   0cc0 9540  sigAlgebracsiga 28925  𝔅cbrsiga 28999  Probcprb 29236  rRndVarcrrv 29269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-i2m1 9608  ax-1ne0 9609  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-po 4771  df-so 4772  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-1st 6804  df-2nd 6805  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-ioo 11640  df-topgen 15330  df-top 19908  df-bases 19909  df-esum 28845  df-siga 28926  df-sigagen 28957  df-brsiga 29000  df-meas 29014  df-mbfm 29069  df-prob 29237  df-rrv 29270
This theorem is referenced by: (None)
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