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Theorem restco 19533
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 3121 . . . . 5  |-  y  e. 
_V
21inex1 4594 . . . 4  |-  ( y  i^i  A )  e. 
_V
3 ineq1 3698 . . . . 5  |-  ( x  =  ( y  i^i 
A )  ->  (
x  i^i  B )  =  ( ( y  i^i  A )  i^i 
B ) )
4 inass 3713 . . . . 5  |-  ( ( y  i^i  A )  i^i  B )  =  ( y  i^i  ( A  i^i  B ) )
53, 4syl6eq 2524 . . . 4  |-  ( x  =  ( y  i^i 
A )  ->  (
x  i^i  B )  =  ( y  i^i  ( A  i^i  B
) ) )
62, 5abrexco 6155 . . 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 2467 . . . . . 6  |-  ( y  e.  J  |->  ( y  i^i  A ) )  =  ( y  e.  J  |->  ( y  i^i 
A ) )
87rnmpt 5254 . . . . 5  |-  ran  (
y  e.  J  |->  ( y  i^i  A ) )  =  { w  |  E. y  e.  J  w  =  ( y  i^i  A ) }
9 mpteq1 4533 . . . . 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 5 . . . 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 5254 . . 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 2467 . . . 4  |-  ( y  e.  J  |->  ( y  i^i  ( A  i^i  B ) ) )  =  ( y  e.  J  |->  ( y  i^i  ( A  i^i  B ) ) )
1312rnmpt 5254 . . 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 2506 . 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 14699 . . . . 5  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  A )  =  ran  ( y  e.  J  |->  ( y  i^i  A
) ) )
16153adant3 1016 . . . 4  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( Jt  A )  =  ran  ( y  e.  J  |->  ( y  i^i  A
) ) )
1716oveq1d 6310 . . 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 6320 . . . . 5  |-  ( Jt  A )  e.  _V
1916, 18syl6eqelr 2564 . . . 4  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ran  ( y  e.  J  |->  ( y  i^i 
A ) )  e. 
_V )
20 simp3 998 . . . 4  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  B  e.  X )
21 restval 14699 . . . 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 661 . . 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 2508 . 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 996 . . 3  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  J  e.  V )
25 inex1g 4596 . . . 4  |-  ( A  e.  W  ->  ( A  i^i  B )  e. 
_V )
26253ad2ant2 1018 . . 3  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( A  i^i  B
)  e.  _V )
27 restval 14699 . . 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 661 . 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 2534 1  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( ( Jt  A )t  B )  =  ( Jt  ( A  i^i  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2818   _Vcvv 3118    i^i cin 3480    |-> cmpt 4511   ran crn 5006  (class class class)co 6295   ↾t crest 14693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-rest 14695
This theorem is referenced by:  restabs  19534  restin  19535  resstopn  19555  ressuss  20634
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