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Theorem pj1val 15332
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
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  =  ( proj 1 `  G )
51, 2, 3, 4pj1fval 15331 . . 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 453 . 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 449 . . . . 5  |-  ( ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B
)  /\  X  e.  ( T  .(+)  U ) )  /\  z  =  X )  ->  z  =  X )
87eqeq1d 2446 . . . 4  |-  ( ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B
)  /\  X  e.  ( T  .(+)  U ) )  /\  z  =  X )  ->  (
z  =  ( x 
.+  y )  <->  X  =  ( x  .+  y ) ) )
98rexbidv 2728 . . 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 6554 . 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 449 . 2  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  X  e.  ( T  .(+) 
U ) )  ->  X  e.  ( T  .(+) 
U ) )
12 riotaex 6556 . . 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 5813 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 set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708   _Vcvv 2958    C_ wss 3322    e. cmpt 4269   ` cfv 5457  (class class class)co 6084   iota_crio 6545   Basecbs 13474   +g cplusg 13534   LSSumclsm 15273   proj
1cpj1 15274
This theorem is referenced by:  pj1id  15336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-pj1 15276
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