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Theorem restco 12357
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 2689 . . . . 5  |-  y  e. 
_V
21inex1 4062 . . . 4  |-  ( y  i^i  A )  e. 
_V
3 ineq1 3270 . . . . 5  |-  ( x  =  ( y  i^i 
A )  ->  (
x  i^i  B )  =  ( ( y  i^i  A )  i^i 
B ) )
4 inass 3286 . . . . 5  |-  ( ( y  i^i  A )  i^i  B )  =  ( y  i^i  ( A  i^i  B ) )
53, 4syl6eq 2188 . . . 4  |-  ( x  =  ( y  i^i 
A )  ->  (
x  i^i  B )  =  ( y  i^i  ( A  i^i  B
) ) )
62, 5abrexco 5660 . . 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 2139 . . . . . 6  |-  ( y  e.  J  |->  ( y  i^i  A ) )  =  ( y  e.  J  |->  ( y  i^i 
A ) )
87rnmpt 4787 . . . . 5  |-  ran  (
y  e.  J  |->  ( y  i^i  A ) )  =  { w  |  E. y  e.  J  w  =  ( y  i^i  A ) }
9 mpteq1 4012 . . . . 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 4787 . . 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 2139 . . . 4  |-  ( y  e.  J  |->  ( y  i^i  ( A  i^i  B ) ) )  =  ( y  e.  J  |->  ( y  i^i  ( A  i^i  B ) ) )
1312rnmpt 4787 . . 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 2170 . 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 12140 . . . . 5  |-  ( ( J  e.  V  /\  A  e.  W )  ->  ( Jt  A )  =  ran  ( y  e.  J  |->  ( y  i^i  A
) ) )
16153adant3 1001 . . . 4  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( Jt  A )  =  ran  ( y  e.  J  |->  ( y  i^i  A
) ) )
1716oveq1d 5789 . . 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 12138 . . . . . 6  |-t  Fn  ( _V  X.  _V )
19 simp1 981 . . . . . . 7  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  J  e.  V )
2019elexd 2699 . . . . . 6  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  J  e.  _V )
21 simp2 982 . . . . . . 7  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  A  e.  W )
2221elexd 2699 . . . . . 6  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  A  e.  _V )
23 fnovex 5804 . . . . . 6  |-  ( (t  Fn  ( _V  X.  _V )  /\  J  e.  _V  /\  A  e.  _V )  ->  ( Jt  A )  e.  _V )
2418, 20, 22, 23mp3an2i 1320 . . . . 5  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( Jt  A )  e.  _V )
2516, 24eqeltrrd 2217 . . . 4  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ran  ( y  e.  J  |->  ( y  i^i 
A ) )  e. 
_V )
26 simp3 983 . . . 4  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  B  e.  X )
27 restval 12140 . . . 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 408 . . 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 2172 . 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 4064 . . . 4  |-  ( A  e.  W  ->  ( A  i^i  B )  e. 
_V )
31303ad2ant2 1003 . . 3  |-  ( ( J  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( A  i^i  B
)  e.  _V )
32 restval 12140 . . 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 408 . 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 2198 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 962    = wceq 1331    e. wcel 1480   {cab 2125   E.wrex 2417   _Vcvv 2686    i^i cin 3070    |-> cmpt 3989    X. cxp 4537   ran crn 4540    Fn wfn 5118  (class class class)co 5774   ↾t crest 12134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-rest 12136
This theorem is referenced by:  restabs  12358  restin  12359
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