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Theorem pj1fval 16562
Description: The left projection function (for a direct product of group subspaces). (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
pj1fval.v  |-  B  =  ( Base `  G
)
pj1fval.a  |-  .+  =  ( +g  `  G )
pj1fval.s  |-  .(+)  =  (
LSSum `  G )
pj1fval.p  |-  P  =  ( proj1 `  G )
Assertion
Ref Expression
pj1fval  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  ( T P U )  =  ( z  e.  ( T  .(+)  U )  |->  ( iota_ x  e.  T  E. y  e.  U  z  =  ( x  .+  y ) ) ) )
Distinct variable groups:    z,  .+    x, y, z, B    x, T, y, z    x, U, y, z    x,  .(+) , y, z    x, G, y, z    x, V, y, z
Allowed substitution hints:    P( x, y, z)    .+ ( x, y)

Proof of Theorem pj1fval
Dummy variables  t 
g  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pj1fval.p . . 3  |-  P  =  ( proj1 `  G )
2 elex 3127 . . . . 5  |-  ( G  e.  V  ->  G  e.  _V )
323ad2ant1 1017 . . . 4  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  G  e.  _V )
4 fveq2 5871 . . . . . . . 8  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
5 pj1fval.v . . . . . . . 8  |-  B  =  ( Base `  G
)
64, 5syl6eqr 2526 . . . . . . 7  |-  ( g  =  G  ->  ( Base `  g )  =  B )
76pweqd 4020 . . . . . 6  |-  ( g  =  G  ->  ~P ( Base `  g )  =  ~P B )
8 fveq2 5871 . . . . . . . . 9  |-  ( g  =  G  ->  ( LSSum `  g )  =  ( LSSum `  G )
)
9 pj1fval.s . . . . . . . . 9  |-  .(+)  =  (
LSSum `  G )
108, 9syl6eqr 2526 . . . . . . . 8  |-  ( g  =  G  ->  ( LSSum `  g )  = 
.(+)  )
1110oveqd 6311 . . . . . . 7  |-  ( g  =  G  ->  (
t ( LSSum `  g
) u )  =  ( t  .(+)  u ) )
12 fveq2 5871 . . . . . . . . . . . 12  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
13 pj1fval.a . . . . . . . . . . . 12  |-  .+  =  ( +g  `  G )
1412, 13syl6eqr 2526 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
1514oveqd 6311 . . . . . . . . . 10  |-  ( g  =  G  ->  (
x ( +g  `  g
) y )  =  ( x  .+  y
) )
1615eqeq2d 2481 . . . . . . . . 9  |-  ( g  =  G  ->  (
z  =  ( x ( +g  `  g
) y )  <->  z  =  ( x  .+  y ) ) )
1716rexbidv 2978 . . . . . . . 8  |-  ( g  =  G  ->  ( E. y  e.  u  z  =  ( x
( +g  `  g ) y )  <->  E. y  e.  u  z  =  ( x  .+  y ) ) )
1817riotabidv 6257 . . . . . . 7  |-  ( g  =  G  ->  ( iota_ x  e.  t  E. y  e.  u  z  =  ( x ( +g  `  g ) y ) )  =  ( iota_ x  e.  t  E. y  e.  u  z  =  ( x  .+  y ) ) )
1911, 18mpteq12dv 4530 . . . . . 6  |-  ( g  =  G  ->  (
z  e.  ( t ( LSSum `  g )
u )  |->  ( iota_ x  e.  t  E. y  e.  u  z  =  ( x ( +g  `  g ) y ) ) )  =  ( z  e.  ( t 
.(+)  u )  |->  ( iota_ x  e.  t  E. y  e.  u  z  =  ( x  .+  y ) ) ) )
207, 7, 19mpt2eq123dv 6353 . . . . 5  |-  ( g  =  G  ->  (
t  e.  ~P ( Base `  g ) ,  u  e.  ~P ( Base `  g )  |->  ( z  e.  ( t ( LSSum `  g )
u )  |->  ( iota_ x  e.  t  E. y  e.  u  z  =  ( x ( +g  `  g ) y ) ) ) )  =  ( t  e.  ~P B ,  u  e.  ~P B  |->  ( z  e.  ( t  .(+)  u )  |->  ( iota_ x  e.  t  E. y  e.  u  z  =  ( x  .+  y ) ) ) ) )
21 df-pj1 16507 . . . . 5  |-  proj1 
=  ( g  e. 
_V  |->  ( t  e. 
~P ( Base `  g
) ,  u  e. 
~P ( Base `  g
)  |->  ( z  e.  ( t ( LSSum `  g ) u ) 
|->  ( iota_ x  e.  t  E. y  e.  u  z  =  ( x
( +g  `  g ) y ) ) ) ) )
22 fvex 5881 . . . . . . . 8  |-  ( Base `  G )  e.  _V
235, 22eqeltri 2551 . . . . . . 7  |-  B  e. 
_V
2423pwex 4635 . . . . . 6  |-  ~P B  e.  _V
2524, 24mpt2ex 6870 . . . . 5  |-  ( t  e.  ~P B ,  u  e.  ~P B  |->  ( z  e.  ( t  .(+)  u )  |->  ( iota_ x  e.  t  E. y  e.  u  z  =  ( x  .+  y ) ) ) )  e.  _V
2620, 21, 25fvmpt 5956 . . . 4  |-  ( G  e.  _V  ->  ( proj1 `  G )  =  ( t  e. 
~P B ,  u  e.  ~P B  |->  ( z  e.  ( t  .(+)  u )  |->  ( iota_ x  e.  t  E. y  e.  u  z  =  ( x  .+  y ) ) ) ) )
273, 26syl 16 . . 3  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  ( proj1 `  G )  =  ( t  e. 
~P B ,  u  e.  ~P B  |->  ( z  e.  ( t  .(+)  u )  |->  ( iota_ x  e.  t  E. y  e.  u  z  =  ( x  .+  y ) ) ) ) )
281, 27syl5eq 2520 . 2  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  P  =  ( t  e. 
~P B ,  u  e.  ~P B  |->  ( z  e.  ( t  .(+)  u )  |->  ( iota_ x  e.  t  E. y  e.  u  z  =  ( x  .+  y ) ) ) ) )
29 oveq12 6303 . . . 4  |-  ( ( t  =  T  /\  u  =  U )  ->  ( t  .(+)  u )  =  ( T  .(+)  U ) )
3029adantl 466 . . 3  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  ( t  =  T  /\  u  =  U ) )  ->  (
t  .(+)  u )  =  ( T  .(+)  U ) )
31 simprl 755 . . . 4  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  ( t  =  T  /\  u  =  U ) )  ->  t  =  T )
32 simprr 756 . . . . 5  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  ( t  =  T  /\  u  =  U ) )  ->  u  =  U )
3332rexeqdv 3070 . . . 4  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  ( t  =  T  /\  u  =  U ) )  ->  ( E. y  e.  u  z  =  ( x  .+  y )  <->  E. y  e.  U  z  =  ( x  .+  y ) ) )
3431, 33riotaeqbidv 6258 . . 3  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  ( t  =  T  /\  u  =  U ) )  ->  ( iota_ x  e.  t  E. y  e.  u  z  =  ( x  .+  y ) )  =  ( iota_ x  e.  T  E. y  e.  U  z  =  ( x  .+  y ) ) )
3530, 34mpteq12dv 4530 . 2  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  ( t  =  T  /\  u  =  U ) )  ->  (
z  e.  ( t 
.(+)  u )  |->  ( iota_ x  e.  t  E. y  e.  u  z  =  ( x  .+  y ) ) )  =  ( z  e.  ( T 
.(+)  U )  |->  ( iota_ x  e.  T  E. y  e.  U  z  =  ( x  .+  y ) ) ) )
36 simp2 997 . . 3  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  T  C_  B )
3723elpw2 4616 . . 3  |-  ( T  e.  ~P B  <->  T  C_  B
)
3836, 37sylibr 212 . 2  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  T  e.  ~P B )
39 simp3 998 . . 3  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  U  C_  B )
4023elpw2 4616 . . 3  |-  ( U  e.  ~P B  <->  U  C_  B
)
4139, 40sylibr 212 . 2  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  U  e.  ~P B )
42 ovex 6319 . . . 4  |-  ( T 
.(+)  U )  e.  _V
4342mptex 6141 . . 3  |-  ( z  e.  ( T  .(+)  U )  |->  ( iota_ x  e.  T  E. y  e.  U  z  =  ( x  .+  y ) ) )  e.  _V
4443a1i 11 . 2  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  (
z  e.  ( T 
.(+)  U )  |->  ( iota_ x  e.  T  E. y  e.  U  z  =  ( x  .+  y ) ) )  e.  _V )
4528, 35, 38, 41, 44ovmpt2d 6424 1  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  ( T P U )  =  ( z  e.  ( T  .(+)  U )  |->  ( iota_ x  e.  T  E. y  e.  U  z  =  ( x  .+  y ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2818   _Vcvv 3118    C_ wss 3481   ~Pcpw 4015    |-> cmpt 4510   ` cfv 5593   iota_crio 6254  (class class class)co 6294    |-> cmpt2 6296   Basecbs 14502   +g cplusg 14567   LSSumclsm 16504   proj1cpj1 16505
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 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
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 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-1st 6794  df-2nd 6795  df-pj1 16507
This theorem is referenced by:  pj1val  16563  pj1f  16565
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