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Theorem restco 17220
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 2951 . . . . 5  |-  y  e. 
_V
21inex1 4336 . . . 4  |-  ( y  i^i  A )  e. 
_V
3 ineq1 3527 . . . . 5  |-  ( x  =  ( y  i^i 
A )  ->  (
x  i^i  B )  =  ( ( y  i^i  A )  i^i 
B ) )
4 inass 3543 . . . . 5  |-  ( ( y  i^i  A )  i^i  B )  =  ( y  i^i  ( A  i^i  B ) )
53, 4syl6eq 2483 . . . 4  |-  ( x  =  ( y  i^i 
A )  ->  (
x  i^i  B )  =  ( y  i^i  ( A  i^i  B
) ) )
62, 5abrexco 5978 . . 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 2435 . . . . . 6  |-  ( y  e.  J  |->  ( y  i^i  A ) )  =  ( y  e.  J  |->  ( y  i^i 
A ) )
87rnmpt 5108 . . . . 5  |-  ran  (
y  e.  J  |->  ( y  i^i  A ) )  =  { w  |  E. y  e.  J  w  =  ( y  i^i  A ) }
9 mpteq1 4281 . . . . 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 5108 . . 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 2435 . . . 4  |-  ( y  e.  J  |->  ( y  i^i  ( A  i^i  B ) ) )  =  ( y  e.  J  |->  ( y  i^i  ( A  i^i  B ) ) )
1312rnmpt 5108 . . 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 2465 . 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 13646 . . . . 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 6088 . . 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 6098 . . . . 5  |-  ( Jt  A )  e.  _V
1916, 18syl6eqelr 2524 . . . 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 13646 . . . 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 2467 . 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 4338 . . . 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 13646 . . 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 2493 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 1652    e. wcel 1725   {cab 2421   E.wrex 2698   _Vcvv 2948    i^i cin 3311    e. cmpt 4258   ran crn 4871  (class class class)co 6073   ↾t crest 13640
This theorem is referenced by:  restabs  17221  restin  17222  resstopn  17242  ressuss  18285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-rest 13642
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