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Theorem restco 17182
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 2919 . . . . 5  |-  y  e. 
_V
21inex1 4304 . . . 4  |-  ( y  i^i  A )  e. 
_V
3 ineq1 3495 . . . . 5  |-  ( x  =  ( y  i^i 
A )  ->  (
x  i^i  B )  =  ( ( y  i^i  A )  i^i 
B ) )
4 inass 3511 . . . . 5  |-  ( ( y  i^i  A )  i^i  B )  =  ( y  i^i  ( A  i^i  B ) )
53, 4syl6eq 2452 . . . 4  |-  ( x  =  ( y  i^i 
A )  ->  (
x  i^i  B )  =  ( y  i^i  ( A  i^i  B
) ) )
62, 5abrexco 5945 . . 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 2404 . . . . . 6  |-  ( y  e.  J  |->  ( y  i^i  A ) )  =  ( y  e.  J  |->  ( y  i^i 
A ) )
87rnmpt 5075 . . . . 5  |-  ran  (
y  e.  J  |->  ( y  i^i  A ) )  =  { w  |  E. y  e.  J  w  =  ( y  i^i  A ) }
9 mpteq1 4249 . . . . 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 8 . . . 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 5075 . . 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 2404 . . . 4  |-  ( y  e.  J  |->  ( y  i^i  ( A  i^i  B ) ) )  =  ( y  e.  J  |->  ( y  i^i  ( A  i^i  B ) ) )
1312rnmpt 5075 . . 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 2434 . 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 13609 . . . . 5  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  A )  =  ran  ( y  e.  J  |->  ( y  i^i  A
) ) )
16153adant3 977 . . . 4  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( Jt  A )  =  ran  ( y  e.  J  |->  ( y  i^i  A
) ) )
1716oveq1d 6055 . . 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 ovex 6065 . . . . 5  |-  ( Jt  A )  e.  _V
1916, 18syl6eqelr 2493 . . . 4  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ran  ( y  e.  J  |->  ( y  i^i 
A ) )  e. 
_V )
20 simp3 959 . . . 4  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  B  e.  X )
21 restval 13609 . . . 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 ) ) )
2219, 20, 21syl2anc 643 . . 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 ) ) )
2317, 22eqtrd 2436 . 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 ) ) )
24 simp1 957 . . 3  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  J  e.  V )
25 inex1g 4306 . . . 4  |-  ( A  e.  W  ->  ( A  i^i  B )  e. 
_V )
26253ad2ant2 979 . . 3  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( A  i^i  B
)  e.  _V )
27 restval 13609 . . 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 ) ) ) )
2824, 26, 27syl2anc 643 . 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 ) ) ) )
2914, 23, 283eqtr4a 2462 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 936    = wceq 1649    e. wcel 1721   {cab 2390   E.wrex 2667   _Vcvv 2916    i^i cin 3279    e. cmpt 4226   ran crn 4838  (class class class)co 6040   ↾t crest 13603
This theorem is referenced by:  restabs  17183  restin  17184  resstopn  17204  ressuss  18246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-rest 13605
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