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Theorem pj1val 16509
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
pj1val  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  X  e.  ( T  .(+) 
U ) )  -> 
( ( T P U ) `  X
)  =  ( iota_ x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) ) )
Distinct variable groups:    x, y, B    x, T, y    x, U, y    x,  .(+) , y    x, G, y    x, V, y   
x, X, y
Allowed substitution hints:    P( x, y)    .+ ( x, y)

Proof of Theorem pj1val
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 pj1fval.v . . . 4  |-  B  =  ( Base `  G
)
2 pj1fval.a . . . 4  |-  .+  =  ( +g  `  G )
3 pj1fval.s . . . 4  |-  .(+)  =  (
LSSum `  G )
4 pj1fval.p . . . 4  |-  P  =  ( proj1 `  G )
51, 2, 3, 4pj1fval 16508 . . 3  |-  ( ( 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 ) ) ) )
65adantr 465 . 2  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  X  e.  ( T  .(+) 
U ) )  -> 
( T P U )  =  ( z  e.  ( T  .(+)  U )  |->  ( iota_ x  e.  T  E. y  e.  U  z  =  ( x  .+  y ) ) ) )
7 simpr 461 . . . . 5  |-  ( ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B
)  /\  X  e.  ( T  .(+)  U ) )  /\  z  =  X )  ->  z  =  X )
87eqeq1d 2469 . . . 4  |-  ( ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B
)  /\  X  e.  ( T  .(+)  U ) )  /\  z  =  X )  ->  (
z  =  ( x 
.+  y )  <->  X  =  ( x  .+  y ) ) )
98rexbidv 2973 . . 3  |-  ( ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B
)  /\  X  e.  ( T  .(+)  U ) )  /\  z  =  X )  ->  ( E. y  e.  U  z  =  ( x  .+  y )  <->  E. y  e.  U  X  =  ( x  .+  y ) ) )
109riotabidv 6245 . 2  |-  ( ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B
)  /\  X  e.  ( T  .(+)  U ) )  /\  z  =  X )  ->  ( iota_ x  e.  T  E. y  e.  U  z  =  ( x  .+  y ) )  =  ( iota_ x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) ) )
11 simpr 461 . 2  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  X  e.  ( T  .(+) 
U ) )  ->  X  e.  ( T  .(+) 
U ) )
12 riotaex 6247 . . 3  |-  ( iota_ x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) )  e.  _V
1312a1i 11 . 2  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  X  e.  ( T  .(+) 
U ) )  -> 
( iota_ x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) )  e. 
_V )
146, 10, 11, 13fvmptd 5953 1  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  X  e.  ( T  .(+) 
U ) )  -> 
( ( T P U ) `  X
)  =  ( iota_ x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815   _Vcvv 3113    C_ wss 3476    |-> cmpt 4505   ` cfv 5586   iota_crio 6242  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   LSSumclsm 16450   proj1cpj1 16451
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-pj1 16453
This theorem is referenced by:  pj1id  16513
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