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Theorem restco 12125
Description: Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.)
Assertion
Ref Expression
restco  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( ( Jt  A )t  B )  =  ( Jt  ( A  i^i  B ) ) )

Proof of Theorem restco
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2644 . . . . 5  |-  y  e. 
_V
21inex1 4002 . . . 4  |-  ( y  i^i  A )  e. 
_V
3 ineq1 3217 . . . . 5  |-  ( x  =  ( y  i^i 
A )  ->  (
x  i^i  B )  =  ( ( y  i^i  A )  i^i 
B ) )
4 inass 3233 . . . . 5  |-  ( ( y  i^i  A )  i^i  B )  =  ( y  i^i  ( A  i^i  B ) )
53, 4syl6eq 2148 . . . 4  |-  ( x  =  ( y  i^i 
A )  ->  (
x  i^i  B )  =  ( y  i^i  ( A  i^i  B
) ) )
62, 5abrexco 5592 . . 3  |-  { z  |  E. x  e. 
{ w  |  E. y  e.  J  w  =  ( y  i^i 
A ) } z  =  ( x  i^i 
B ) }  =  { z  |  E. y  e.  J  z  =  ( y  i^i  ( A  i^i  B
) ) }
7 eqid 2100 . . . . . 6  |-  ( y  e.  J  |->  ( y  i^i  A ) )  =  ( y  e.  J  |->  ( y  i^i 
A ) )
87rnmpt 4725 . . . . 5  |-  ran  (
y  e.  J  |->  ( y  i^i  A ) )  =  { w  |  E. y  e.  J  w  =  ( y  i^i  A ) }
9 mpteq1 3952 . . . . 5  |-  ( ran  ( y  e.  J  |->  ( y  i^i  A
) )  =  {
w  |  E. y  e.  J  w  =  ( y  i^i  A
) }  ->  (
x  e.  ran  (
y  e.  J  |->  ( y  i^i  A ) )  |->  ( x  i^i 
B ) )  =  ( x  e.  {
w  |  E. y  e.  J  w  =  ( y  i^i  A
) }  |->  ( x  i^i  B ) ) )
108, 9ax-mp 7 . . . 4  |-  ( x  e.  ran  ( y  e.  J  |->  ( y  i^i  A ) ) 
|->  ( x  i^i  B
) )  =  ( x  e.  { w  |  E. y  e.  J  w  =  ( y  i^i  A ) }  |->  ( x  i^i  B ) )
1110rnmpt 4725 . . 3  |-  ran  (
x  e.  ran  (
y  e.  J  |->  ( y  i^i  A ) )  |->  ( x  i^i 
B ) )  =  { z  |  E. x  e.  { w  |  E. y  e.  J  w  =  ( y  i^i  A ) } z  =  ( x  i^i 
B ) }
12 eqid 2100 . . . 4  |-  ( y  e.  J  |->  ( y  i^i  ( A  i^i  B ) ) )  =  ( y  e.  J  |->  ( y  i^i  ( A  i^i  B ) ) )
1312rnmpt 4725 . . 3  |-  ran  (
y  e.  J  |->  ( y  i^i  ( A  i^i  B ) ) )  =  { z  |  E. y  e.  J  z  =  ( y  i^i  ( A  i^i  B ) ) }
146, 11, 133eqtr4i 2130 . 2  |-  ran  (
x  e.  ran  (
y  e.  J  |->  ( y  i^i  A ) )  |->  ( x  i^i 
B ) )  =  ran  ( y  e.  J  |->  ( y  i^i  ( A  i^i  B
) ) )
15 restval 11908 . . . . 5  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  A )  =  ran  ( y  e.  J  |->  ( y  i^i  A
) ) )
16153adant3 969 . . . 4  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( Jt  A )  =  ran  ( y  e.  J  |->  ( y  i^i  A
) ) )
1716oveq1d 5721 . . 3  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( ( Jt  A )t  B )  =  ( ran  ( y  e.  J  |->  ( y  i^i  A
) )t  B ) )
18 restfn 11906 . . . . . 6  |-t  Fn  ( _V  X.  _V )
19 simp1 949 . . . . . . 7  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  J  e.  V )
2019elexd 2654 . . . . . 6  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  J  e.  _V )
21 simp2 950 . . . . . . 7  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  A  e.  W )
2221elexd 2654 . . . . . 6  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  A  e.  _V )
23 fnovex 5736 . . . . . 6  |-  ( (t  Fn  ( _V  X.  _V )  /\  J  e.  _V  /\  A  e.  _V )  ->  ( Jt  A )  e.  _V )
2418, 20, 22, 23mp3an2i 1288 . . . . 5  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( Jt  A )  e.  _V )
2516, 24eqeltrrd 2177 . . . 4  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ran  ( y  e.  J  |->  ( y  i^i 
A ) )  e. 
_V )
26 simp3 951 . . . 4  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  B  e.  X )
27 restval 11908 . . . 4  |-  ( ( ran  ( y  e.  J  |->  ( y  i^i 
A ) )  e. 
_V  /\  B  e.  X )  ->  ( ran  ( y  e.  J  |->  ( y  i^i  A
) )t  B )  =  ran  ( x  e.  ran  ( y  e.  J  |->  ( y  i^i  A
) )  |->  ( x  i^i  B ) ) )
2825, 26, 27syl2anc 406 . . 3  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( ran  ( y  e.  J  |->  ( y  i^i  A ) )t  B )  =  ran  (
x  e.  ran  (
y  e.  J  |->  ( y  i^i  A ) )  |->  ( x  i^i 
B ) ) )
2917, 28eqtrd 2132 . 2  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( ( Jt  A )t  B )  =  ran  (
x  e.  ran  (
y  e.  J  |->  ( y  i^i  A ) )  |->  ( x  i^i 
B ) ) )
30 inex1g 4004 . . . 4  |-  ( A  e.  W  ->  ( A  i^i  B )  e. 
_V )
31303ad2ant2 971 . . 3  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( A  i^i  B
)  e.  _V )
32 restval 11908 . . 3  |-  ( ( J  e.  V  /\  ( A  i^i  B )  e.  _V )  -> 
( Jt  ( A  i^i  B ) )  =  ran  ( y  e.  J  |->  ( y  i^i  ( A  i^i  B ) ) ) )
3319, 31, 32syl2anc 406 . 2  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( Jt  ( A  i^i  B ) )  =  ran  ( y  e.  J  |->  ( y  i^i  ( A  i^i  B ) ) ) )
3414, 29, 333eqtr4a 2158 1  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( ( Jt  A )t  B )  =  ( Jt  ( A  i^i  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 930    = wceq 1299    e. wcel 1448   {cab 2086   E.wrex 2376   _Vcvv 2641    i^i cin 3020    |-> cmpt 3929    X. cxp 4475   ran crn 4478    Fn wfn 5054  (class class class)co 5706   ↾t crest 11902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-rest 11904
This theorem is referenced by:  restabs  12126  restin  12127
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