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Theorem pj1val 16296
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 16295 . . 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 2453 . . . 4  |-  ( ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B
)  /\  X  e.  ( T  .(+)  U ) )  /\  z  =  X )  ->  (
z  =  ( x 
.+  y )  <->  X  =  ( x  .+  y ) ) )
98rexbidv 2844 . . 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 6153 . 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 6155 . . 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 5878 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 965    = wceq 1370    e. wcel 1758   E.wrex 2796   _Vcvv 3068    C_ wss 3426    |-> cmpt 4448   ` cfv 5516   iota_crio 6150  (class class class)co 6190   Basecbs 14276   +g cplusg 14340   LSSumclsm 16237   proj1cpj1 16238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-pj1 16240
This theorem is referenced by:  pj1id  16300
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