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Theorem pj1fval 16196
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 2986 . . . . 5  |-  ( G  e.  V  ->  G  e.  _V )
323ad2ant1 1009 . . . 4  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  G  e.  _V )
4 fveq2 5696 . . . . . . . 8  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
5 pj1fval.v . . . . . . . 8  |-  B  =  ( Base `  G
)
64, 5syl6eqr 2493 . . . . . . 7  |-  ( g  =  G  ->  ( Base `  g )  =  B )
76pweqd 3870 . . . . . 6  |-  ( g  =  G  ->  ~P ( Base `  g )  =  ~P B )
8 fveq2 5696 . . . . . . . . 9  |-  ( g  =  G  ->  ( LSSum `  g )  =  ( LSSum `  G )
)
9 pj1fval.s . . . . . . . . 9  |-  .(+)  =  (
LSSum `  G )
108, 9syl6eqr 2493 . . . . . . . 8  |-  ( g  =  G  ->  ( LSSum `  g )  = 
.(+)  )
1110oveqd 6113 . . . . . . 7  |-  ( g  =  G  ->  (
t ( LSSum `  g
) u )  =  ( t  .(+)  u ) )
12 fveq2 5696 . . . . . . . . . . . 12  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
13 pj1fval.a . . . . . . . . . . . 12  |-  .+  =  ( +g  `  G )
1412, 13syl6eqr 2493 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
1514oveqd 6113 . . . . . . . . . 10  |-  ( g  =  G  ->  (
x ( +g  `  g
) y )  =  ( x  .+  y
) )
1615eqeq2d 2454 . . . . . . . . 9  |-  ( g  =  G  ->  (
z  =  ( x ( +g  `  g
) y )  <->  z  =  ( x  .+  y ) ) )
1716rexbidv 2741 . . . . . . . 8  |-  ( g  =  G  ->  ( E. y  e.  u  z  =  ( x
( +g  `  g ) y )  <->  E. y  e.  u  z  =  ( x  .+  y ) ) )
1817riotabidv 6059 . . . . . . 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 4375 . . . . . 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 6153 . . . . 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 16141 . . . . 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 5706 . . . . . . . 8  |-  ( Base `  G )  e.  _V
235, 22eqeltri 2513 . . . . . . 7  |-  B  e. 
_V
2423pwex 4480 . . . . . 6  |-  ~P B  e.  _V
2524, 24mpt2ex 6655 . . . . 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 5779 . . . 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 2487 . 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 6105 . . . 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 2929 . . . 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 6060 . . 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 4375 . 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 989 . . 3  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  T  C_  B )
3723elpw2 4461 . . 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 990 . . 3  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  U  C_  B )
4023elpw2 4461 . . 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 6121 . . . 4  |-  ( T 
.(+)  U )  e.  _V
4342mptex 5953 . . 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 6223 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 965    = wceq 1369    e. wcel 1756   E.wrex 2721   _Vcvv 2977    C_ wss 3333   ~Pcpw 3865    e. cmpt 4355   ` cfv 5423   iota_crio 6056  (class class class)co 6096    e. cmpt2 6098   Basecbs 14179   +g cplusg 14243   LSSumclsm 16138   proj1cpj1 16139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-pj1 16141
This theorem is referenced by:  pj1val  16197  pj1f  16199
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