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Theorem pj1val 15004
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 15003 . . 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 451 . 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 447 . . . . 5  |-  ( ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B
)  /\  X  e.  ( T  .(+)  U ) )  /\  z  =  X )  ->  z  =  X )
87eqeq1d 2291 . . . 4  |-  ( ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B
)  /\  X  e.  ( T  .(+)  U ) )  /\  z  =  X )  ->  (
z  =  ( x 
.+  y )  <->  X  =  ( x  .+  y ) ) )
98rexbidv 2564 . . 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 6306 . 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 447 . 2  |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  X  e.  ( T  .(+) 
U ) )  ->  X  e.  ( T  .(+) 
U ) )
12 riotaex 6308 . . 3  |-  ( iota_ x  e.  T E. y  e.  U  X  =  ( x  .+  y ) )  e.  _V
1312a1i 10 . 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 5606 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    C_ wss 3152    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148   +g cplusg 13208   LSSumclsm 14945   proj
1cpj1 14946
This theorem is referenced by:  pj1id  15008
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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