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Theorem pj1val 15286
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 15285 . . 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 452 . 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 448 . . . . 5  |-  ( ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B
)  /\  X  e.  ( T  .(+)  U ) )  /\  z  =  X )  ->  z  =  X )
87eqeq1d 2416 . . . 4  |-  ( ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B
)  /\  X  e.  ( T  .(+)  U ) )  /\  z  =  X )  ->  (
z  =  ( x 
.+  y )  <->  X  =  ( x  .+  y ) ) )
98rexbidv 2691 . . 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 6514 . 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 448 . 2  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  X  e.  ( T  .(+) 
U ) )  ->  X  e.  ( T  .(+) 
U ) )
12 riotaex 6516 . . 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 5773 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2671   _Vcvv 2920    C_ wss 3284    e. cmpt 4230   ` cfv 5417  (class class class)co 6044   iota_crio 6505   Basecbs 13428   +g cplusg 13488   LSSumclsm 15227   proj
1cpj1 15228
This theorem is referenced by:  pj1id  15290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-pj1 15230
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