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Theorem mbfmco 28472
Description: The composition of two measurable functions is measurable. ( cf. cnmpt11 20330) (Contributed by Thierry Arnoux, 4-Jun-2017.)
Hypotheses
Ref Expression
mbfmco.1  |-  ( ph  ->  R  e.  U. ran sigAlgebra )
mbfmco.2  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
mbfmco.3  |-  ( ph  ->  T  e.  U. ran sigAlgebra )
mbfmco.4  |-  ( ph  ->  F  e.  ( RMblFnM
S ) )
mbfmco.5  |-  ( ph  ->  G  e.  ( SMblFnM
T ) )
Assertion
Ref Expression
mbfmco  |-  ( ph  ->  ( G  o.  F
)  e.  ( RMblFnM
T ) )

Proof of Theorem mbfmco
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 mbfmco.2 . . . . 5  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
2 mbfmco.3 . . . . 5  |-  ( ph  ->  T  e.  U. ran sigAlgebra )
3 mbfmco.5 . . . . 5  |-  ( ph  ->  G  e.  ( SMblFnM
T ) )
41, 2, 3mbfmf 28463 . . . 4  |-  ( ph  ->  G : U. S --> U. T )
5 mbfmco.1 . . . . 5  |-  ( ph  ->  R  e.  U. ran sigAlgebra )
6 mbfmco.4 . . . . 5  |-  ( ph  ->  F  e.  ( RMblFnM
S ) )
75, 1, 6mbfmf 28463 . . . 4  |-  ( ph  ->  F : U. R --> U. S )
8 fco 5723 . . . 4  |-  ( ( G : U. S --> U. T  /\  F : U. R --> U. S )  -> 
( G  o.  F
) : U. R --> U. T )
94, 7, 8syl2anc 659 . . 3  |-  ( ph  ->  ( G  o.  F
) : U. R --> U. T )
10 unielsiga 28358 . . . . 5  |-  ( T  e.  U. ran sigAlgebra  ->  U. T  e.  T )
112, 10syl 16 . . . 4  |-  ( ph  ->  U. T  e.  T
)
12 unielsiga 28358 . . . . 5  |-  ( R  e.  U. ran sigAlgebra  ->  U. R  e.  R )
135, 12syl 16 . . . 4  |-  ( ph  ->  U. R  e.  R
)
1411, 13elmapd 7426 . . 3  |-  ( ph  ->  ( ( G  o.  F )  e.  ( U. T  ^m  U. R )  <->  ( G  o.  F ) : U. R
--> U. T ) )
159, 14mpbird 232 . 2  |-  ( ph  ->  ( G  o.  F
)  e.  ( U. T  ^m  U. R ) )
16 cnvco 5177 . . . . . 6  |-  `' ( G  o.  F )  =  ( `' F  o.  `' G )
1716imaeq1i 5322 . . . . 5  |-  ( `' ( G  o.  F
) " a )  =  ( ( `' F  o.  `' G
) " a )
18 imaco 5495 . . . . 5  |-  ( ( `' F  o.  `' G ) " a
)  =  ( `' F " ( `' G " a ) )
1917, 18eqtri 2483 . . . 4  |-  ( `' ( G  o.  F
) " a )  =  ( `' F " ( `' G "
a ) )
205adantr 463 . . . . 5  |-  ( (
ph  /\  a  e.  T )  ->  R  e.  U. ran sigAlgebra )
211adantr 463 . . . . 5  |-  ( (
ph  /\  a  e.  T )  ->  S  e.  U. ran sigAlgebra )
226adantr 463 . . . . 5  |-  ( (
ph  /\  a  e.  T )  ->  F  e.  ( RMblFnM S ) )
232adantr 463 . . . . . 6  |-  ( (
ph  /\  a  e.  T )  ->  T  e.  U. ran sigAlgebra )
243adantr 463 . . . . . 6  |-  ( (
ph  /\  a  e.  T )  ->  G  e.  ( SMblFnM T ) )
25 simpr 459 . . . . . 6  |-  ( (
ph  /\  a  e.  T )  ->  a  e.  T )
2621, 23, 24, 25mbfmcnvima 28465 . . . . 5  |-  ( (
ph  /\  a  e.  T )  ->  ( `' G " a )  e.  S )
2720, 21, 22, 26mbfmcnvima 28465 . . . 4  |-  ( (
ph  /\  a  e.  T )  ->  ( `' F " ( `' G " a ) )  e.  R )
2819, 27syl5eqel 2546 . . 3  |-  ( (
ph  /\  a  e.  T )  ->  ( `' ( G  o.  F ) " a
)  e.  R )
2928ralrimiva 2868 . 2  |-  ( ph  ->  A. a  e.  T  ( `' ( G  o.  F ) " a
)  e.  R )
305, 2ismbfm 28460 . 2  |-  ( ph  ->  ( ( G  o.  F )  e.  ( RMblFnM T )  <->  ( ( G  o.  F )  e.  ( U. T  ^m  U. R )  /\  A. a  e.  T  ( `' ( G  o.  F ) " a
)  e.  R ) ) )
3115, 29, 30mpbir2and 920 1  |-  ( ph  ->  ( G  o.  F
)  e.  ( RMblFnM
T ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1823   A.wral 2804   U.cuni 4235   `'ccnv 4987   ran crn 4989   "cima 4991    o. ccom 4992   -->wf 5566  (class class class)co 6270    ^m cmap 7412  sigAlgebracsiga 28337  MblFnMcmbfm 28458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-map 7414  df-siga 28338  df-mbfm 28459
This theorem is referenced by:  rrvadd  28655  rrvmulc  28656
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