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Theorem pj1fval 15003
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  =  ( proj 1 `  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  =  ( proj 1 `  G )
2 elex 2796 . . . . 5  |-  ( G  e.  V  ->  G  e.  _V )
323ad2ant1 976 . . . 4  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  G  e.  _V )
4 fveq2 5525 . . . . . . . 8  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
5 pj1fval.v . . . . . . . 8  |-  B  =  ( Base `  G
)
64, 5syl6eqr 2333 . . . . . . 7  |-  ( g  =  G  ->  ( Base `  g )  =  B )
76pweqd 3630 . . . . . 6  |-  ( g  =  G  ->  ~P ( Base `  g )  =  ~P B )
8 fveq2 5525 . . . . . . . . 9  |-  ( g  =  G  ->  ( LSSum `  g )  =  ( LSSum `  G )
)
9 pj1fval.s . . . . . . . . 9  |-  .(+)  =  (
LSSum `  G )
108, 9syl6eqr 2333 . . . . . . . 8  |-  ( g  =  G  ->  ( LSSum `  g )  = 
.(+)  )
1110oveqd 5875 . . . . . . 7  |-  ( g  =  G  ->  (
t ( LSSum `  g
) u )  =  ( t  .(+)  u ) )
12 fveq2 5525 . . . . . . . . . . . 12  |-  ( g  =  G  ->  ( +g  `  g )  =  ( +g  `  G
) )
13 pj1fval.a . . . . . . . . . . . 12  |-  .+  =  ( +g  `  G )
1412, 13syl6eqr 2333 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( +g  `  g )  = 
.+  )
1514oveqd 5875 . . . . . . . . . 10  |-  ( g  =  G  ->  (
x ( +g  `  g
) y )  =  ( x  .+  y
) )
1615eqeq2d 2294 . . . . . . . . 9  |-  ( g  =  G  ->  (
z  =  ( x ( +g  `  g
) y )  <->  z  =  ( x  .+  y ) ) )
1716rexbidv 2564 . . . . . . . 8  |-  ( g  =  G  ->  ( E. y  e.  u  z  =  ( x
( +g  `  g ) y )  <->  E. y  e.  u  z  =  ( x  .+  y ) ) )
1817riotabidv 6306 . . . . . . 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 4098 . . . . . 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 5910 . . . . 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 14948 . . . . 5  |-  proj 1  =  ( 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 5539 . . . . . . . 8  |-  ( Base `  G )  e.  _V
235, 22eqeltri 2353 . . . . . . 7  |-  B  e. 
_V
2423pwex 4193 . . . . . 6  |-  ~P B  e.  _V
2524, 24mpt2ex 6198 . . . . 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 5602 . . . 4  |-  ( G  e.  _V  ->  ( proj 1 `  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 15 . . 3  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  ( proj 1 `  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 2327 . 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 5867 . . . 4  |-  ( ( t  =  T  /\  u  =  U )  ->  ( t  .(+)  u )  =  ( T  .(+)  U ) )
3029adantl 452 . . 3  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  ( t  =  T  /\  u  =  U ) )  ->  (
t  .(+)  u )  =  ( T  .(+)  U ) )
31 simprl 732 . . . 4  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  ( t  =  T  /\  u  =  U ) )  ->  t  =  T )
32 simprr 733 . . . . 5  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  ( t  =  T  /\  u  =  U ) )  ->  u  =  U )
3332rexeqdv 2743 . . . 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 6307 . . 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 4098 . 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 956 . . 3  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  T  C_  B )
3723elpw2 4175 . . 3  |-  ( T  e.  ~P B  <->  T  C_  B
)
3836, 37sylibr 203 . 2  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  T  e.  ~P B )
39 simp3 957 . . 3  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  U  C_  B )
4023elpw2 4175 . . 3  |-  ( U  e.  ~P B  <->  U  C_  B
)
4139, 40sylibr 203 . 2  |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  U  e.  ~P B )
42 ovex 5883 . . . 4  |-  ( T 
.(+)  U )  e.  _V
4342mptex 5746 . . 3  |-  ( z  e.  ( T  .(+)  U )  |->  ( iota_ x  e.  T E. y  e.  U  z  =  ( x  .+  y ) ) )  e.  _V
4443a1i 10 . 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 5975 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   iota_crio 6297   Basecbs 13148   +g cplusg 13208   LSSumclsm 14945   proj
1cpj1 14946
This theorem is referenced by:  pj1val  15004  pj1f  15006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-pj1 14948
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